Research Article A Modified Hybrid Genetic Algorithm for Solving Nonlinear...

Research ArticleA Modified Hybrid Genetic Algorithm forSolving Nonlinear Optimal Control Problems

Saeed Nezhadhosein1 Aghileh Heydari1 and Reza Ghanbari2

1Department of Applied Mathematics Payame Noor University Tehran 193953697 Iran2Department of Applied Mathematics Faculty of Mathematical Science Ferdowsi University of MashhadMashhad 9177948953 Iran

Correspondence should be addressed to Saeed Nezhadhosein s nejhadhoseinpnuacir

Received 4 September 2014 Accepted 30 January 2015

Academic Editor Alain Vande Wouwer

Copyright copy 2015 Saeed Nezhadhosein et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Here a two-phase algorithm is proposed for solving bounded continuous-time nonlinear optimal control problems (NOCP) Ineach phase of the algorithm a modified hybrid genetic algorithm (MHGA) is applied which performs a local search on offspringsIn first phase a random initial population of control input values in time nodes is constructed Next MHGA starts with thispopulation After phase 1 to achieve more accurate solutions the number of time nodes is increased The values of the associatednew control inputs are estimated by Linear interpolation (LI) or Spline interpolation (SI) using the curves obtained from the phase1 In addition to maintain the diversity in the population some additional individuals are added randomly Next in the secondphase MHGA restarts with the new population constructed by above procedure and tries to improve the obtained solutions at theend of phase 1 We implement our proposed algorithm on 20 well-known benchmark and real world problems then the results arecompared with some recently proposed algorithms Moreover two statistical approaches are considered for the comparison of theLI and SI methods and investigation of sensitivity analysis for the MHGA parameters

1 Introduction

NOCPs are dynamic optimization problemswithmany appli-cations in industrial processes such as airplane robotic armbio-process system biomedicine electric power systems andplasma physics [1]

High-quality solutions and the less required computa-tional time aremain issues for solvingNOCPsThenumericalmethods direct [2] or indirect [3] usually have two maindeficiencies less accuracy and convergence to a local solu-tion In direct methods the quality of solution depends ondiscretization resolution Since these methods using controlparametrization convert the continuous problem to discreteproblem they have less accuracy However the adaptivestrategies [4 5] can overcome these defects but they maybe trapped by a local optimal yet In indirect approachesthe problem through the use of the Pontryagins minimumprinciple (PMP) is converted into a two-boundary valueproblem (TBVP) that can be solved by numerical methods

such as shooting method [6] These methods need the goodinitial guesses that lie within the domain of convergenceTherefore the numerical methods usually are not suitablefor solving NOCPs especially for large-scale andmultimodalmodels

Metaheuristics as the global optimization methods canovercome these problems but they usually need morecomputational time though they do not really need goodinitial guesses and deterministic rules Several researchersused metaheuristics to solve optimal control problems ForinstanceMichalewicz et al [7] applied floating-point Geneticalgorithms (GA) to solve discrete time optimal control prob-lems Yamashita and Shima [8] used the classical GAs to solvethe free final time optimal control problems with terminalconstraints Abo-Hammour et al [9] used continuous GA forsolvingNOCPsMoreover the other usages ofGA for optimalcontrol problems can be found in [6 10] Lopez Cruz et al[11] applied differential evolution (DE) algorithms for solvingthe multimodal optimal control problems Recently Ghosh

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 139036 21 pageshttpdxdoiorg1011552015139036

2 Mathematical Problems in Engineering

et al [12] developed an ecologically inspired optimizationtechnique called invasive weed optimization (IWO) forsolving optimal control problems The other well-knownmetaheuristic algorithms used for solving NOCPs are geneticprogramming (GP) [13] particle swarm optimization (PSO)[14 15] ant colony optimization (ACO) [16] and DE [17 18]

To increase the quality of solutions and decrease therunning time hybrid methods were introduced which useda local search in the implementation of a population-basedmetaheuristics [19] Modares and Naghibi-Sistani [20] pro-posed a hybrid algorithm by integrating an improved PSOwith successive quadratic programming (SQP) for solvingNOCPs Recently Sun et al [21] proposed a hybrid improvedGA which used simplex method (SM) to perform a localsearch for solving NOCPs and applied it for chemicalprocesses

Based on the success of the hybrid methods for solvingNOCPs mentioned above we here use a modified hybridgenetic algorithm (MHGA) which combines GA with SQPsee [22] as a local search SQP is an iterative algorithm forsolving nonlinear programming (NLP) problems which usesgradient information It can moreover be used for solvingNOCPs see [23ndash25] For decreasing the running time in theearly generations (iterations) of MHGA a less number ofiterations for SQP was used and then when the promisingregion of search space was found we increase the number ofiterations of SQP gradually

To perform MHGA for solving an NOCP the timeinterval is uniformly divided by using a constant number oftime nodes Next in each of these time nodes the controlvariable is approximated by a scaler vector of control inputvalues Thus an infinite dimensional NOCP is changed toa finite dimensional NLP Now we encounter two conflictsituations the quality of the global solution and the neededcomputational time In other words when the number oftime nodes is increased then we expect that the quality ofthe global solution is also increased but we know that in thissituation the computational time is increased dramatically Inother situation if we consider less number of time nodesthen the computational time is decreased but we may finda poor local solution To conquer these problems MHGAperforms in two phases In the first phase (explorationphase) to decrease the computational time and to find apromising region of search space MHGA uses a less numberof time nodes After phase 1 to increase the quality ofsolutions obtained from phase 1 the number of time nodesis increased Using the population obtained in phase 1 thevalues of the new control inputs are estimated by Linear orSpline interpolations Next in the second phase (exploitationphase) MHGA uses the solutions constructed by the aboveprocedure as an initial population

The paper is organized as follows in Section 2 theformulation of the problem is introduced In Section 3 theproposed MHGA is presented In Section 4 we introduceour algorithm for solving NOCP In Section 5 we provide20 numerical benchmark examples to compare the proposedalgorithm with the other recently proposed algorithmsIn Section 6 we consider two statistical approaches for

the comparison of the LI and SI methods and investigation ofsensitivity analysis of the algorithm parameters The impactof SQP as local search in the proposed algorithm is surveyedin Section 7 We conclude in Section 8

2 Formulation of Problem

The bounded continuous-time NOCP is considered as find-ing the control input 119906(119905) isin R119898 over the planning horizon[1199050 119905119891] which minimizes the cost functional

119869 = 120601 (119909 (119905119891) 119905119891) + int

119905119891

1199050

119892 (119909 (119905) 119906 (119905) 119905) 119889119905 (1)

subject to

(119905) = 119891 (119909 (119905) 119906 (119905) 119905) (2)

119888 (119909 119906 119905) = 0 (3)

119889 (119909 119906 119905) le 0 (4)

120595 (119909 (119905119891) 119905119891) = 0 (5)

119909 (1199050) = 1199090 (6)

where 119909(119905) isin R119899 denotes the state vector for the system and1199090isin R119899 is the initial state The functions 119891 R119899 timesR119898 timesR rarr

R119899 119892 R119899 timesR119898 timesR rarr R 119888 R119899 timesR119898 timesR rarr R119899119888 119889 R119899 times

R119898timesR rarr R119899119889 120595 R119899timesR rarr R119899120595 and 120601 R119899timesR rarr R areassumed to be sufficiently smooth on appropriate open setsCost function (1) must be minimized subject to dynamic (2)control and state equality constraints (3) and control and stateinequality constraints (4) the final state constraints (5) andthe initial condition (6) A special case of the NOCPs is thelinear quadratic regulator (LQR) problemwhere the dynamicequations are linear and the objective function is a quadraticfunction of 119909 and 119906 The minimum time problems trackingproblem terminal control problem andminimum energy areanother special case of NOCPs

3 Modified Hybrid Genetic Algorithm

In this section first MHGA as a subprocedure for the mainalgorithm is introduced To perform MHGA the controlvariables are discretized Next NOCP is changed into a finitedimensional NLP see [21 26] Now we can imply a GA tofind the global solution of the corresponding NLP In thefollowing we introduce GA operators

31 Underlying GA GAs introduced by Holland in 1975 areheuristics and probabilistic methods [27] These algorithmsstart with an initial population of solutions This populationis evaluated by using genetic operators that include selectioncrossover andmutation Here inMHGA the underlying GAhas the following steps

InitializationThe time interval is divided into119873119905minus1 subinter-

vals using time nodes 1199050 119905

119873119905minus1and then the control input

Mathematical Problems in Engineering 3

Initialization Input the the size of tournament119873tour and select119873tour

individuals from population randomly Let 119896119894 119894 = 1 2 119873tour be the indexes of them

repeatif 119873tour = 2 thenif 119868(1198801198961 ) lt 119868(119880

1198962) then

Let 119896 = 1198961

elseLet 119896 = 119896

2

end ifend if

Perform Tournament algorithm with size119873tour2 with output 1198961


until 1198961

= 1198962

Return the 119896th individual of population

Algorithm 1 Tournament algorithm

values are computed (or selected randomly)This can be doneby the following stages

(1) Let 119905119895= 1199050+ 119895ℎ where ℎ = (119905

119891minus 1199050)(119873119905minus 1) 119895 =

0 1 119873119905minus1 be time nodes where 119905

0and 119905119891are the

initial and final times respectively

(2) The corresponding control input value at each timenode 119905

119895is an119898times1 vector 119906

119895 which can be calculated

randomly with the following components

119906119894119895= 119906119895(119894) = 119906left (119894) + (119906right (119894) minus 119906left (119894)) sdot 119903119894119895

119894 = 1 2 119898 119895 = 0 1 119873119905minus 1

(7)

where 119903119894119895is a random number in [0 1]with a uniform

distribution and 119906left 119906right isin R119898 are the lower and theupper bound vectors of control input values whichcan be given by the problemrsquos definition or the user(eg see the NOCPs numbers (6) and (5) in theAppendix resp) So each individual of the popula-tion is an 119898 times 119873

119905matrix as 119880 = (119906

119894119895)119898times119873119905

= [119906119895]119873119905minus1

119895=0

Next we let 119880(119896) = (119906119894119895)119898times119873119905

119896 = 1 2 119873119901as 119896th

individual of the population which 119873119901is the size of

the population

Evaluation For each control input matrix 119880(119896) 119896 =

1 2 119873119901 the corresponding state variable is an 119899 times 119873

119905

matrix119883(119896) and it can be computed by the forthRunge-Kuttamethod on dynamic system (2) with the initial condition(6) approximately Then the performance index 119869(119880(119896))is approximated by a numerical method (denoted by 119869) IfNOCP includes equality or inequality constraints (3) or (4)then we add some penalty terms to the corresponding fitness

value of the solution Finally we assign 119868(119880(119896)

) to 119880(119896) as the

fitness value as follows

119868 (119880(119896)

) = 119869 +

119899119889

sum

119897=1

119873119905minus1

sum

119895=0

1198721119897max 0 119889

119897(119909119895 119906119895 119905119895)

+

119899119888

sum

ℎ=1

119873119905minus1

sum

119895=0

1198722ℎ1198882

ℎ(119909119895 119906119895 119905119895)

+

119899120595

sum

119894=119901

11987231199011205952

119901(119909119873119905minus1

119905119873119905minus1

)

(8)

where 1198721= [119872

11 119872

1119899119889]119879 1198722= [119872

21 119872

2119899119888]119879 and

1198723

= [11987231 119872

3119899120595]119879 are big numbers as the penalty

parameters 119888ℎ(sdot sdot) ℎ = 1 2 119899

119888 119889119897(sdot sdot) 119897 = 1 2 119899

119889

and 120595119901(sdot sdot) 119901 = 1 2 119899

120595are defined in (3) (4) and (5)

respectively

Selection To select two parents we use a tournament selec-tion [27] It can be applied for parallel GA The tournamentoperator applies competition among the same individualsand the best of them is selected for next generation At firstwe select a specified number of individuals from populationrandomly This number is tournament selection parameterwhich is denoted by119873tourThe tournament algorithm is givenin Algorithm 1

Crossover When two parents 119880(1) and 119880(2) are selected we

use the following stages to construct an offspring(1) Select the following numbers

1205821isin [0 1] 120582

2isin [minus120582max 0] 120582

3isin [1 1 + 120582max]

(9)randomly where 120582max is a random number in [0 1]

(2) Let

of119896 = 120582119896119880(1)

+ (1 minus 120582119896) 119880(2)

119896 = 1 2 3 (10)


where 120582119896 119896 = 1 2 3 is defined in (9) For 119894 = 1 119898

and 119895 = 1 119873119905 if (of119896)

119894119895gt 119906right(119894) then let (of

119896

)119894119895=

119906right(119894) and if (of119896

)119894119895lt 119906left(119894) then let (of

119896

)119894119895= 119906left(119894)

(3) Let of = oflowast where oflowast is the best of119896 119896 = 1 2 3

constructed by (10)

MutationWe apply a perturbation on each component of theoffspring as follows

(of)119894119895= (of)

119894119895+ 119903119894119895sdot 120572 119894 = 1 2 119898 119895 = 1 2 119873

119905

(11)

where 119903119894119895is selected randomly in minus1 1 and 120572 is a random

number with a uniform distribution in [0 1] If (of)119894119895

gt

119906right(119894) then let (of)119894119895= 119906right(119894) and if (of)

119894119895lt 119906left(119894) then

let (of)119894119895= 119906left(119894)

ReplacementHere in the underling GA we use a traditionalreplacement strategy The replacement is done if the newoffspring has two properties first it is better than the worstperson in the population 119868(of) lt max

1le119894le119873119901119868(119880(119894)

) secondit is not very similar to a person in the population that is foreach 119894 = 1 119873

119901 at least one of the following conditions is

satisfied10038161003816100381610038161003816119868 (of) minus 119868 (119880

(119894)

)

10038161003816100381610038161003816gt 120576

10038171003817100381710038171003817of minus 119880

(119894)10038171003817100381710038171003817gt 120576

(12)

where 120576 is the machine epsilon

Termination Conditions Underlying GA is terminated whenat least one of the following conditions is occurred

(1) Themaximumnumber of generations119873119892 is reached

(2) Over a specified number of generations 119873119894 we do

not have any improvement (the best individual is notchanged) or the two-norm or error of final stateconstraints will reach a small number as the desiredprecision 120576 that is

120593119891=100381710038171003817100381712059510038171003817100381710038172

lt 120576 (13)

where120595 = [1205951 1205952 120595

119899120595]119879 is the vector of final state

constraints defined in (5)

32 SQP The fitness value in (8) can be viewed as anonlinear objective function with the decision variable as119906 = [119906

0 1199061 119906

119873119905minus1] This cost function with upper and

lower bounds of input signals construct a finite dimensionalNLP problem as following

min 119868 (119906) = 119868 (1199060 1199061 119906

119873119905minus1) (14)

st 119906left le 119906119895le 119906right 119895 = 0 1 119873

119905minus 1 (15)

SQP algorithm [22 28] is performed on the NLP (14)-(15)using 1199060 = of and constructed in (11) as the initial solutionwhen the maximum number of iteration is 119904119902119901119898119886119909119894119905119890119903

SQP is an effective and iterative algorithm for thenumerical solution of the constrained NLP problem Thistechnique is based on finding a solution to the system ofnonlinear equations that arise from the first-order necessaryconditions for an extremum of the NLP problem Usingan initial solution of NLP 119906119896 119896 = 0 1 a sequence ofsolutions as 119906119896+1 = 119906

119896

+ 119889119896 is constructed which 119889

119896 is theoptimal solution of the constructed quadratic programming(QP) that approximates NLP in the iteration 119896 based on119906119896 as the search direction in the line search procedure

For the NLP (15) the principal idea is the formulation of aQP subproblem based on a quadratic approximation of theLagrangian function as 119871(119906 120582) = 119868(119906) + 120582

119879

ℎ(119906) where thevector 120582 is Lagrangian multiplier and ℎ(119906) return the vectorof inequality constraints evaluated at 119906 The QP is obtainedby linearizing the nonlinear functions as follows

min 1

2

119889119879

119867(119906119896

) 119889 + nabla119868 (119906119896

)

119879

119889 (16)

nablaℎ (119906119896

)

119879

119889 + ℎ (119906119896

) le 0 (17)

Similar to [26] here a finite difference approximation isapplied to compute the gradient of the cost function and theconstraints with the following components

120597119868

120597119906119895

=

119868 (sdot sdot sdot 119906119895+ 120575 sdot sdot sdot ) minus 119868 (119906)

120575

119895 = 0 1 119873119905minus 1 (18)

where 120575 is the double precision of machine So the gradientvector is nabla119868 = [120597119868120597119906

0 120597119868120597119906

119873119905minus1]119879 Also at each major

iteration a positive definite quasi-Newton approximation oftheHessian of the Lagrangian function119867 is calculated usingthe BFGS method [28] where 120582

119894 119894 = 1 119898 is an estimate

of the Lagrange multipliers The general procedure for SQPfor NLP (14)-(15) is as follows

(1) Given an initial solution 1199060 Let 119896 = 0

(2) Construct the QP subproblem (16)-(17) based on 1199060

using the approximations of the gradient and theHessian of the the Lagrangian function

(3) Compute the new point as 119906119896+1 = 119906119896

+ 119889119896 where 119889119896

is the optimal solution of the current QP(4) Let 119896 = 119896 + 1 and go to step (2)

Here in MHGA SQP is used as the local search and weuse the maximum number of iterations as the main criterionfor stopping SQP In other words we terminate SQP when itconverges either to local solution or themaximumnumber ofSQPrsquos iterations is reached

33 MHGA In MHGA GA uses a local search method toimprove solutions Here we use SQP as a local search UsingSQP as a local search in the hybrid metaheuristic is commonfor example see [20]

MHGA can be seen as a multi start local search whereinitial solutions are constructed by GA From another per-spective MHGA can be seen as a GA that the quality of


Initialization

sqpmaxiter

an initial population

EvaluationEvaluate the fitnessof each individual

using (8)

Local search(Section 32)

ReturnThe best individual inthe final populationas an approximate

of the global solution of NOCP

Stoppingconditions

Selection(Algorithm 1)

Crossover(using Equation (10))

Replacement(using Equations

(12))

Mutation(using Equation(11))


Input NtNpNiNg Pm

120576Mi i = 1 2 3 and

=sqpmaxitersqpmaxiter + 1

Figure 1 Flowchart of the MHGA algorithm

its population is intensified by SQP In the beginning ofMHGA a less number of iterations for SQP was used Thenwhen the promising regions of search space were found byGA operators we increase the number of iterations of SQPgradually Using this approach we may decrease the neededrunning time (in [19] the philosophy of this approach isdiscussed)

Finally we give our modified MHGA to find the globalsolution by the flowchart in Figure 1

4 Proposed Algorithm

Here we give a new algorithm which is a direct approachbased on MHGA for solving NOCPs The proposed algo-rithm has two main phases In the first phase we performMHGA with a completely random initial population con-structed by (7) In the first phase to find the promisingregions of the search space in a less running time we usea few numbers of time nodes In addition to have a fasterMHGA the size of the population in the first phase is usuallyless than the size of the population in the second phase

After phase 1 to maintain the property of individuals inthe last population of phase 1 and to increase the accuratelyof solutions we add some additional time nodes Whenthe number of time nodes is increased it is estimated thatthe quality of solution obtained by numerical methods (eg

Runge-Kutta and Simpson) is increased Thus we increasetime nodes from 119873

1199051in phase 1 to 119873

1199052in phase 2 The

corresponding control input values of the new time nodesare added to individuals To use the information of theobtained solutions from phase 1 in the construction of theinitial population of phase 2 we use either Linear or Splineinterpolation to estimate the value of the control inputs inthe new time nodes in each individual of the last populationof phase 1 Moreover to maintain the diversity in the initialpopulation of phase 2 we add new random individuals tothe population using (7) In the second phase MHGA startswith this population and new value of parameters Finally theproposed algorithm is given in Algorithm 2

5 Numerical Experiments

In this section to investigate the efficiency of the proposedalgorithm 20 well-known and real world NOCPs as bench-mark problems are considered which are presented in termsof (1)ndash(6) in the Appendix These NOCPs are selected withsingle control signal and multi control signals which will bedemonstrated in a general manner

The numerical behaviour of algorithms can be studiedfrom two view of points the relative error of the performanceindex and the status of the final state constraints Let 119869 be theobtained performance index by an algorithm 120593

119891 defined in


Initialization Input the desired precision 120576 in (13) the penalty parameters119872119894 119894 = 1 2 3 in (8) the bounds of control input values in (7) 119906left and 119906right

Phase 1 Perform MHGA with a random population and1198731199051 1198731199011 1198731198941 1198731198921 1198751198981

and 1199041199021199011198981198861199091198941199051198901199031

Construction of the initial population of the phase 2 Increase time nodes uniformly to1198731199052

and estimate the corresponding control input values of the new time nodes in eachindividual obtained from phase 1 using either Linear or Spline interpolationCreate119873

1199012minus 1198731199011new different individuals with119873

1199052time nodes randomly

Phase 2 Perform MHGA with the constructed population and1198731199052 1198731199012 1198731198942 1198731198922 1198751198982

and 1199041199021199011198981198861199091198941199051198901199032

Algorithm 2 The proposed algorithm

(13) be the error of final state constraints and 119869lowast be the best

obtained solution among all implementations or the exactsolution (when exists) Now the relative error of 119869 119864

119869 of the

algorithm can be defined as

119864119869=

10038161003816100381610038161003816100381610038161003816

119869 minus 119869lowast

119869lowast

10038161003816100381610038161003816100381610038161003816

(19)

Tomore accurate study we now define a new criterion calledfactor to compare the algorithms as follows

119870120595= 119864119869+ 120593119891 (20)

Note that 119870120595shows the summation of two important errors

Thus based on 119870120595

we can study the behaviour of thealgorithms on the quality and feasibility of given solutionssimultaneously

To solve any NOCP described in the Appendix wemust know the algorithmrsquos parameters including MHGArsquosparameters including 119873

119905 119873119901 119873119894 119873119892 119875119898and 119904119902119901119898119886119909119894119905119890119903

in both phases in Algorithm 2 and the problemrsquos parametersincluding 120576 in (13) 119872

119894 119894 = 1 2 3 in (8) 119906left and 119906right in

(7) To estimate the best value of the algorithmrsquos parameterswe ran the proposed algorithm with different values ofparameters and then select the best However the sensitivityof MHGA parameters are studied in the next section In bothof phases of Algorithm 2 in MHGA we let 119904119902119901119898119886119909119894119905119890119903

119894= 4

and 119875119898119894

= 08 for 119894 = 1 2 Also we consider 1198731198921

= 1198731198922

1198731198941

= 1198731198942 1198731199011

= 9 and 1198731199012

= 12 The other MHGAparameters are given in the associated subsection and theproblemrsquos parameters in Table 2 For each NOCP 12 differentruns were done and the best results are reported in Table 1which the best value of each column is seen in the bold

The reported numerical results of the proposed algo-rithm for each NOCP include the value of performanceindex 119869 the relative error of 119869119864

119869 defined in (19) the required

computational time Time the norm of final state constraints120593119891 defined in (13) and the factor 119870

120595 defined in (20)

The algorithm was implemented in Matlab R2011a envi-ronment on a Notebook with Windows 7 Ultimate CPU253GHz and 400GB RAM Also to implement SQP in ourproposed algorithm we used ldquofminconrdquo in Matlab when theldquoAlgorithmrdquo was set to ldquoSQPrdquo Moreover we use compositeSimpsonrsquos method [29] to approximate integrations

Remark 1 We use the following abbreviations to show theused interpolation method in our proposed algorithm

(1) LI linear interpolation(2) SI spline interpolation

For comparing the numerical results of the proposedalgorithm two subsections are considered comparison withsome metaheuristic algorithms in Section 51 and com-parison with some numerical methods in Section 52 Wegive more details of these comparisons in the followingsubsections

51 Comparison with Metaheuristic Algorithms The numeri-cal results for the NOCPs numbers (1)ndash(3) in the Appendixare compared with a continuous GA CGA as a meta-heuristic proposed in [9] which gave better solutions thanshooting method and gradient algorithm from the indirectmethods category [2 30] and SUMT from the directmethodscategory [26] For NOCPs numbers (4) and (5) the resultsare compared with another metaheuristic which is a hybridimproved PSO called IPSO proposed in [20]

511 VDP Problem [9] The first NOCP in the Appendix isVanDer Pol Problem VDP which has two state variables andone control variable VDPproblemhas a final state constraintwhich is 120595 = 119909

1(119905119891) minus 1199092(119905119891) + 1 = 0 The results of the

proposed algorithm with the MHGArsquos parameters as 1198731199051

=

311198731199052= 71119873

119892= 300 and119873

119894= 200 are reported in Table 1

FromTable 1 it is obvious that the numerical results of LI andSI methods are more accurate than CGA with less amount of119870120595

512 CRP Problem [9] The second NOCP in the Appendixis Chemical Reactor Problem CRP which has two statevariables and one control variableThe results of the proposedalgorithm with the MHGArsquos parameters as 119873

1199051= 31 119873

1199052=

71 119873119892

= 300 and 119873119894= 200 are shown in the second

row of Table 1 CRP problem has two final state constraints120595 = [119909

1 1199092]119879 Although from Table 1 the norm of final state

constraints 120593119891 for the CGA equals 120593lowast

119891= 757 times 10

minus10 isless than 120593

119891rsquos of LI and SI methods which equals 115 times 10

minus9

and 599 times 10minus9 respectively but the performance index


Table 1 The best numerical results in 12 different runs of the SI and LI methods for NOCPs in the Appendix

Problem Algorithm 119869 119864119869

120593119891

119870120595

Time

VDPCGA 17404 00912 267 times 10

minus11

00912 50128LI 15949 0 108 times 10

minus13 108 times 10minus13 28282SI 15950 627 times 10

minus5 999 times 10minus14 627 times 10minus5 23914

CRPCGA 163 times 10

minus2 02835 757 times 10minus10 02835 50128LI 127 times 10minus2 0 115 times 10

minus9 115 times 10minus9 17194

SI 127 times 10minus2 0 599 times 10minus9

599 times 10minus9

12405

FFRPCGA 8363 13256 465 times 10

minus3

13302 1413LI 3596 0 122 times 10


SI 3599 834 times 10minus4 901 times 10minus6 843 times 10

minus4 134340

MSNICIPSO 01727 00165 mdash mdash 12626LI 01699 0 mdash mdash 13882

SI 01700 589 times 10minus4 mdash mdash 13868

CSTCRIPSO 01355 02098 mdash mdash 3134

LI 01120 0 mdash mdash 10358

SI 01120 0 mdash mdash 10673

No 6Bezier minus53898 00251 mdash mdash NRa

LI minus54309 00177 mdash mdash 13838

SI minus54309 00177 mdash mdash 10734

Number 7HPM 02353 01677 420 times 10

minus6 01677 NRLI 02015 0 235 times 10

minus9

235 times 10minus9 21392

SI 02015 0 282 times 10minus10 282 times 10minus10 20696

Number 8

SQP 636 times 10minus6

220 times 109 mdash mdash NR

SUMT 515 times 10minus6


LI 289 times 10minus15 0 mdash mdash 7282SI 364 times 10

minus15

02595 mdash mdash 6868

Number 9

SQP 17950 00873 mdash mdash NRSUMT 17980 00891 mdash mdash NRLI 16509 0 mdash mdash 63910SI 16509 0 mdash mdash 5950

Number 10

SQP 02163 03964 mdash mdash NRSUMT 01703 00994 mdash mdash NRLI 01549 0 mdash mdash 62570


Number 11

SQP 325 01648 0 01648 NRSUMT 325 01648 0 01648 NRLI 27901 0 162 times 10

minus9



Number 12

SQP minus02490 40 times 10minus3 0 40 times 10

minus3 NRSUMT minus02490 40 times 10

minus3 0 40 times 10minus3 NR

LI minus02500 0 285 times 10minus8

285 times 10minus8

54771

SI minus02500 0 390 times 10minus10 390 times 10minus10 54548

Number 13

SQP 168 times 10minus2 01748 0 01748 NR

SUMT 167 times 10minus2 01678 0 01678 NR

LI 143 times 10minus2 0 118 times 10minus9 118 times 10minus9 42599SI 144 times 10

minus2

70 times 10minus3

632 times 10minus9

70 times 10minus3

49027

Number 14

SQP 37220 00961 0 00961 NRSUMT 37700 01103 0 01103 NRLI 33956 0 786 times 10minus7 786 times 10minus7 104124SI 33965 265 times 10

minus4

276 times 10minus6

267 times 10minus4

115144


Table 1 Continued


120593119891

119870120595

Time

Number 15




minus4

520 times 10minus3

164 times 10minus9

520 times 10minus3

34014

Number 16


minus12

394 times 10minus12

46982


Number 17




LI minus88692 0 145 times 10minus10 145 times 10minus10 32183SI minus88692 0 279 times 10

minus10

279 times 10minus10

30959

Number 18



Number 19


minus3

01919 164509


Number 20

SQP 7752 12554 0 12554 NRSUMT 7683 12353 0 12353 NRLI 343716 232 times 10

minus5

225 times 10minus4



aNot reported

the relative error of 119869 and the factor of the proposed algorithmis better and so the proposed algorithm is more robust thanCGA

513 FFRP Problem [9] The third NOCP in the Appendixis Free Floating Robot Problem FFRP which has six statevariables and four control variables FFRP has been solvedby CGA and the proposed algorithm with the MHGArsquosparameters as 119873

1199051= 31 119873

1199052= 71 119873

119892= 300 and

119873119894

= 200 This problem has six final state constraints120595 = [119909

1minus 4 119909

2 1199093minus 4 119909

4 1199095 1199096]119879 The numerical results

are shown in Table 1 The values of 119869 119864119869 120593119891and 119870

120595for LI

and SI methods separately are less than CGA Thereforethe proposed algorithm can achieve much better qualitysolutions than the CGA with reasonable computational time

514 MSNIC Problem [20] For the forth NOCP in theAppendix which is a Mathematical System with NonlinearInequality Constraint NSNIC the numerical results are com-pared with IPSO MSNIC contains an inequality constraint119889(119909 119905) = 119909

2(119905) + 05 minus 8(119905 minus 05)

2

le 0 The problem solvedby several numerical methods as [24 31] From [20] IPSOmethod could achieved more accurate results than men-tioned numericalmethods AlsoMSNIC can be solved by theproposed algorithm with the MHGArsquos parameters as 119873

1199051=

31 1198731199052

= 91 119873119892= 100 and 119873

119894= 60 From the forth row

of Table 1 the absolute error of 119869 119864119869 for LI and SI methods

equal 0 and 589 times 10minus4 respectively which are less than

IPSOrsquos 00165Subplots (a) and (b) in Figure 2 show the graphs of

the convergence rate for the performance index and theinequality constraint respect to the number of iterationrespectively

515 CSTCR Problem [20] The fifth NOCP in the Appendixis a model of a nonlinear Continuous Stirred-tank ChemicalReactor CSTCR It has two state variables 119909

1(119905) and 119909

2(119905)

as the deviation from the steady-state temperature and con-centration and one control variable 119906(119905) which represent theeffect of the flow rate of cooling fluid on chemical reactorTheobjective is to maintain the temperature and concentrationclose to steady-state values without expending large amountof control effort Also this is a benchmark problem in thehandbook of test problems in local and global optimization[32] which is a multimodal optimal control problem [33]It involves two different local minima The values of theperformance indexes for these solutions equal 0244 and0133 Similarly to the MSNIC the numerical results of theproposed algorithm with the MHGArsquos parameters as 119873

1199051=

31 1198731199052

= 51 119873119892= 100 and 119873

119894= 50 are compared with

IPSO From Table 1 the performance index 119869 for LI and SImethods is equal to 119869

lowast

= 01120 which is less than IPSOs01355


Table 2 The problem parameters for NOCPs in the Appendix

Parameters Problem number1 2 3 4 5 6 7 8 9 10

119906left minus05 minus15 minus15 minus20 0 minus2 0 minus2 minus1 minus20

119906right 2 2 10 20 5 2 1 2 1 20119872119894

103

102 70 1 mdash 1 mdash 10

2 mdash 102

120576 10minus12

10minus10

10minus3 mdash mdash mdash mdash mdash mdash 10

minus9

Parameters Problem no11 12 13 14 15 16 17 18 19 20

119906left minus5 minus1 minus2 minus120587 minus1 minus3 minus30 minus1 minus15 minus15

119906right 5 1 2 120587 1 3 30 1 10 10119872119894

10 102 mdash 10

2

102

102 mdash 10 70 10

minus3

120576 10minus9

10minus11

10minus11 mdash 10

minus10

10minus10

10minus10 mdash 10

minus3

10minus4

0 5 10 15 20 25 30 35 40 45 50017

0175

018

0185

019

0195

02

0205

Iter

Perfo

rman

ce in

dex

(a)

0 01 02 03 04 05 06 07 08 09 1minus25

minus2

minus15

minus1

minus05

0

Time

Ineq

ualit

y co

nstr

aint

(b)

Figure 2 Graphical results of MSNIC problem using SI method (a) convergence rate of the performance index and (b) the inequalityconstraint 119889(119909 119905) le 0 in respect to the number of iterations

52 Comparison with Numerical Methods For NOCPs num-bers (6)ndash(20) the comparison are done with some numericalmethods Unfortunately for these methods usually the finalstate constraints and the required computational time arenot reported which are shown with NR in Table 1 but thesevalues are reported for both LI and SI methods in Table 1For all NOCPs in this section the MHGArsquos parameters areconsidered as 119873

1199051= 31 119873

1199052= 51 119873

119892= 100 119873

119894= 50 with

the problem parameters in Table 2

521 Compared with Bezier [34] The NOCP number (6) inthe Appendix has exact solution which has an inequalityconstraint as 119889(119909

1 119905) = minus6 minus 119909

1(119905) le 0 The exact value of

performance index equals 119869lowast = minus55285 [35] This problemhas been solved by a numerical method proposed in [34]called Bezier From sixth row of Table 1 the absolute error

of the LI and SI methods equal 00177 which is less thanBezierrsquos 00251

Figure 3 shows the graphs of the convergence rate of theperformance index subplot (a) and inequality constraintsubplot (b) respect to the number of iteration using SImethod

522 Compared with HPM [36] For NOCP number (7) inthe Appendix which is a constraint nonlinear model thenumerical results of the proposed algorithm are comparedwith HPM proposed in [36] This NOCP has a final stateconstraint as 120595 = 119909 minus 05 = 0 From [36] the norm offinal state constraint for HPM equals 42times10minus6 however thiscriterion for the LI and SI methods equals 235 times 10

minus9 and120593lowast

119891= 282 times 10

minus10 respectively From Table 1 it is obviousthat the obtained values of the performance index the norm


2 4 6 8 10 12 14minus5353

minus5352

minus5351

minus535

minus5349

minus5348

minus5347

minus5346

minus5345

minus5344

minus5343

Iter

Perfo

rman

ce in

dex

(a)

0 05 1 15 2 25 3minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Time

Ineq

ualit

y co

nstr

aint

(b)

Figure 3 Graphical results for NOCP number (6) using SI method (a) convergence rate for the performance index and (b) the inequalityconstraint 119889(119909

1 119905) le 0 in respect to the number of iterations

of final state constraint and 119870120595for the SI method are more

accurate than LI than HPMmethods

523 Compared with SQP and SUMT [26] For the NOCPsnumbers (8)ndash(20) in the Appendix the numerical resultsof LI and SI methods are compared with two numericalmethods contain SQP and SUMT proposed in [26] Amongthese NOCPs only two problems numbers (8) and (18) areunconstrained and the others have at least one constraintfinal state constraint or inequality constraint All of theseNOCPs solved by the proposed algorithm with the problemparameters in Table 2 and their results are summarized inTable 1 Because the final state constraints in these methodsare not reported we let 120593

119891= 0 to calculate the factor119870

120595

Figures 4ndash16 show the graphical results of NOCPsnumbers (8)ndash(20) in the Appendix using SI method Forunconstrained NOCPs numbers (8) and (18) only the graphof convergent rate of performance index is shown see Figures4 and 14 For constraint NOCPs and NOCPs numbers(11)ndash(17) and numbers (19)-(20) with final state constraintthe graphs of convergent rates of performance index and theerror of final state constraint are shown see Figures 7ndash13 and15-16 For the constraint NOCPs with inequality constraintsNOCPs numbers (9) and (10) the graphs of convergent rateof performance index and inequality constraint are shownsee Figures 5 and 6

Table 1 shows that the proposed algorithm LI and SImethods was 100 percent successful in point of views theperformance index 119869 and the factor119870

120595 numerically So the

proposed algorithm provides robust solutions with respect tothe other mentioned numerical or metaheuristic methodsTo compare 119869 in LI and SI methods in 35 percent of NOCPsLI is more accurate than SI in 10 percent SI is more accurate

1 15 2 25 3 35 40

002

004

006

008

01

012

Iter

Perfo

rman

ce in

dex

Figure 4 Convergence rate for the performance index of SI methodfor NOCP number (8)

than LI and in 55 percent are same In point of view the finalconditions 65 percent of NOCPs in the Appendix have finalstate constraints In all of them 120593

119891in the proposed algorithm

is improved except CRP problem however the factor of theproposed algorithm is the best yet In 54 percent of theseNOCPs LI is better performance and in 46 percent SI isbetter Also in point of the running time Time LI is same asSI that is in 50 percent of NOCPs LI and in 50 percent SI isbetter than another So the proposed algorithmcould providevery suitable solutions in a reasonable computational timeAlso for more accurate comparison of LI and SI methods astatistical approach will done in next section

To compare with CGA the mean of relative error of 119869 119864119869

for CGA LI and SImethods inNOCPs (1)ndash(3) equal 056680 and 298 times 10

minus4 respectively Also the mean for the error


5 10 15 20 25165

17

175

18

185

Iter

Perfo

rman

ce in

dex

(a)

0 05 1 15 2 25 3 35 4 45 50

005

01

015

02

025

TimeIn

equa

lity

cons

trai

nt(b)

Figure 5 (a) Convergence rate for the performance index and (b) the inequality constraint of SI method for NOCP number (9)

5 10 15 20 25 30 35 40 45 500155

016

0165

017

0175

018

0185

019

0195

02

0205

Iter

Perfo

rman

ce in

dex

(a)

0 01 02 03 04 05 06 07 08 09 1minus25

minus2

minus15

minus1

minus05

0

Time

Ineq

ualit

y co

nstr

aint

(b)

Figure 6 (a) Convergence rate of the performance index and (b) the inequality constraint for SI method for NOCP number (10)

of the final state constraints 120593119891 are 16 times 10

minus3 407 times 10minus6

and 301 times 10minus6 respectively and for 119870

120595 these values equal

05883 407 times 10minus6 and 302 times 10

minus4 Thus we can say that thefeasibility of the solutions given by the proposed algorithmand CGA are competitive

From Table 1 to compare with numerical methods SQPand SUMT in NOCPs (8)ndash(20) the mean of 119864

119869for LI SI

SQP and SUMT equals 00145 00209 169 times 108 and 136 times

108 respectively Also the mean for the error of final state

constraints for these NOCPs equal 334 times 10minus4 441 times 10

minus50 and 0 respectively For 119870

120595 these values are 00213 00014

12263 and 11644 Therefore the performance index 119869 andthe factor 119870

120595 for the LI and SI methods are more accurate

than SQP and SUMT So the proposed algorithm gave morebetter solution in comparison with the numerical methods


5 10 15 20 25minus025

minus0245

minus024

minus0235

minus023

minus0225

Iter

Perfo

rman

ce in

dex

(a)

1 15 2 25 3 35 4 45 5 55 60

02

04

06

08

1

12

14

16

IterFi

nal c

ondi

tion

times10minus5

(b)

Figure 7 (a) Convergence rates of the performance index and (b) the error of final state constraint for SI method for NOCP number (11)

2 4 6 8 10 12 14minus025

minus0245

minus024

minus0235

minus023

minus0225

minus022

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

14

16

18

2

Iter

Fina

l con

ditio

n

times10minus5

(b)


Therefore based on this numerical studywe can concludethat the proposed algorithm outperform well-known numer-ical method Since the algorithms were not implementedon the same PC the computational times of them are notcompetitive Therefore we did not give the computationaltimes in bold in Table 1

6 Sensitivity Analysis andComparing LI and SI

In this section two statistical analysis based on the one-way analysis of variance (ANOVA) used for investigatingthe sensitivity of MHGA parameters and Mann-Whitney


1 2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

07

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

IterFi

nal c

ondi

tion

times10minus3

(b)


5 10 15 20 25 30

35

4

45

5

55

6

65

7

75

8

85

9

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

Iter

Fina

l con

ditio

n

(b)


applied comparing LI and SI methods are done by thestatistical software IBM SPSS version 21

61 Sensitivity Analysis In order to survey the sensitivityof the MHGA parameters from the proposed algorithmthe VDP problem is selected for example The influenceof these parameters are investigated for this NOCP on

the dependent outputs consist of the performance index119869 the relative error of 119869 119864

119869 the required computational

time Time the error of final state constraints 120593119891and the

factor 119870120595 The independent parameters are consist of the

number of time nodes in both two phases 1198731199051and 119873

1199052 the

size of population in both two phases 1198731199011

and 1198731199012 the

maximum number of generations without improvement119873119894


5 10 15 20 25 30 35 40 45 50

00955

0096

00965

0097

00975

0098

00985

0099

00995

Iter

Perfo

rman

ce in

dex

(a)

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

35

4

45

IterFi

nal c

ondi

tion

times10minus3

(b)


5 10 15 20 25

21

22

23

24

25

26

27

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 80

1

2

3

4

5

6

Iter

Fina

l con

ditio

n

times10minus3

(b)


and the maximum number of generations 119873119892 Because the

mutation implementation probability119875119898 has a less influence

in numerical results it is not considered Also 119904119902119901119898119886119909119894119905119890119903 ischanged in each iteration of the proposed algorithm so it isnot considered too

At first we selected at least four constant value foreach of parameters and then in each pose 35 differentruns were made independently The statistical analysis isdone based on ANOVA The descriptive statistics whichcontains the number of different runs (119873) the mean of each


1 2 3 4 5 6 7minus887

minus886

minus885

minus884

minus883

minus882

minus881

minus88

minus879

minus878

minus877

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 70

01

02

03

04

05

06

07

08

09

1

IterFi

nal c

ondi

tion

times10minus3

(b)


1 15 2 25 3 35 4 45 5 55 600331003320033200332003320033200333003330033300333

Iter

Perfo

rman

ce in

dex

Figure 14 Convergence rate for the performance index of SImethod for NOCP numbers (18)

output in119873 different runs (Mean) standard deviation (Sd)maximum (Max) and minimum (Min) of each outputcould be achieved Among of the MHGA parameters wepresent the descriptive statistics of the ANOVA only for the1198731199051parameter which is reported in Table 3Table 4 summarized the statistical data contain the test

statistics (119865) and 119875-values of ANOVA tests Sensitivityanalysis for each parameter separately are done based on thistable as follows

1198731199051 From Table 4 the significant level or 119875-value for119870120595

is equal to 0279 which is greater than 005So from ANOVA 119873

1199051parameter has no significant

effect on the factor 119870120595 With similar analysis this

parameter has no effect on the other outputs except

required computational time Time because its 119875-value is equal to 0 which is less than 005 that is thisoutput sensitive to the parameter119873

1199051

1198731199052 From the second row of Table 4 the outputs 119869 119864

119869119870120595

and Time are sensitive to the parameter 1198731199052 and the

only output 120593119891is independent

1198731199011 From the third row of Table 4 only the computationalrequired time Time is sensitive because the its 119875-value is equal to 0006 which is less than 005 and theother outputs contain 119869 119864

119869 120593119891119870120595 are independent

1198731199012 From the forth row of Table 4 the output 120593

119891is

independent respect to the parameter 1198731199012 but other

outputs contain 119869 119864119869 Time 119870

120601are sensitive to this

parameter119873119892 From the fifth row of Table 4 the outputs 119869 119864

119869and

119870120595are independent to the parameter 119873

119892 and other

outputs contain 120593119891and Time are sensitive respect to

this parameter119873119894 The sensitivity analysis is similar to119873

1199011

From above cases it is obvious that all parameters canbe effect on the required computational time except 119873

119892

Moreover intuitions shows the normof final state constraints120593119891is independent output with respect to all parameters that

is any of the parameters could not effect on this output and itis not sensitive with respect to any of the MHGA parameters

62 Comparison of LI and SI To compare the efficiency ofthe LI and SI methods for NOCPs in the Appendix weused the Mann-Whitney nonparametric statistical test [37]


1 15 2 25 3 35 40

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 80

200

400

600

800

1000

1200

1400

1600

IterFi

nal c

ondi

tion

(b)


2 4 6 8 10 12 14 16 18 20

50

100

150

200

250

300

350

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 10 11 120

05

1

15

2

25

3

Iter

Fina

l con

ditio

n

(b)


Using the nonparametric statistical tests for comparing theperformance of several algorithms presented in [38] Forthis purpose At first 15 different runs for each of NOCPwere made Table 5 shows the mean of numerical resultsseparately Here we apply the fixMHGAparameters as119873

1199011=

91198731199012

= 121198731199051= 31119873

1199052= 91119873

119892= 100 and119873

119894= 50 with

the previous problem parameters in Table 2 Comparison

criterions contain 119869 120593119891 119870120595and Time The results of Mann-

Whitney test are shown in Tables 6ndash9 From Table 6 since119885 = 0 gt 119885

001= minus234 then the average relative error of

the LI and SI methods are same with the significant level 001In other words with probability 99 we can say that LI andSI methods have same effectiveness form the perspective oferror for 119869 Similarly the results of Tables 7ndash9 indicate that LI


Table 3 Descriptive statistics for the sensitivity analysis of1198731199051 for VDP problem

Output 1198731199051

119873 Mean Sd Min Max

119869

11 35 15333 00069 15286 1559821 35 15328 00097 15285 1584931 35 15329 00076 15285 1569941 35 15455 00607 15285 18449

Total 140 15357 00294 15285 18449

120593119891

11 35 281 times 10minus8

952 times 10minus8

1 times 10minus12

452 times 10minus7


235 times 10minus7

1 times 10minus12

141 times 10minus6


491 times 10minus8

1 times 10minus12

221 times 10minus7


285 times 10minus7

1 times 10minus12

126 times 10minus6

Total 140 426 times 10minus8 191 times 10minus7 1 times 10minus12 141 times 10minus6

Time

11 35 897838 272081 456614 155439421 35 1098147 460264 604192 284031031 35 1163591 300227 772673 179510341 35 1448290 393091 810425 2849514

Total 140 1142131 410891 456615 2849514

119864119869

11 35 00031 00045 0 0020421 35 00028 00064 0 0036931 35 00029 00050 0 0027141 35 00111 00397 0 02070

Total 140 00047 00192 0 02070

119870120595

11 35 00031 00045 475 times 10minus9

00204

21 35 00027 00064 550 times 10minus10

00369

31 35 00029 00050 154 times 10minus13

00271

41 35 00111 00397 193 times 10minus10

02070

Total 140 00047 00192 154 times 10minus13 02070

Table 4 Summary statistical data of ANOVA test for the parameters1198731199051 1198731199052 1198731199011 1198731199012 119873119892 119873119894

Parameters 119869 119864119869

120593119891

Time 119870120595

Test statistic (119865)

1198731199051

1294 1296 0624 1079 12961198731199052

110572 393 0868 5939 3931198731199011

1945 1835 1271 4317 18351198731199012

2740 2478 1011 2302 2478119873119892

3541 3591 0491 0890 3591119873119894

0495 0309 0301 5849 1046

119875 value

1198731199051

0280 0279 0601 0 02791198731199052

0 0005 0489 0 00051198731199011

0125 0143 0286 0006 01431198731199012

0021 0034 0413 0 0034119873119892

0009 0008 0743 0472 0008119873119894

0739 0819 0877 0 0386

and SI methods from the perspective of errors for 120593119891 Time

and119870120595 have same behaviour

7 The Impact of SQP

In this section we investigation the impact of the SQP on theproposed algorithm For this purpose we remove the SQPfrom Algorithm 2 as without SQP algorithm For comparingthe numerical results with the previous results the required

running time in each NOCP is considered fixed which is themaximum of running time in 12 different runs were done inSection 5 Also the all of the parameters for each problemwasset as same as Section 5 The numerical results of the withoutSQP algorithm is summarized in Table 10 By comparing theresults of Tables 10 and 1 it is obvious that for all NOCPsin the Appendix the obtained values of the performanceindex and the norm of final state constraint for the proposedalgorithm (Algorithm 2) are more accurate than the withoutSQP algorithm


Table 5 The mean of numerical results of 15 different runs for NOCPs in the Appendix using LI and SI methods

Problem LI SI119869 120593

119891119870120595

Time 119869 120593119891

119870120595

TimeVDP 15977 157 times 10

minus9

170 times 10minus3 17720 16002 667 times 10

minus8


CRP 00126 307 times 10minus8


minus8


FFPP 4774 805 times 10minus5 03277 133280 5351 592 times 10

minus5 04868 137989MSNIC 01702 mdash mdash 13678 01703 mdash mdash 14092CSTCR 01120 mdash mdash 17385 01120 mdash mdash 17606Number 6 minus54307 mdash mdash 10541 minus54308 mdash mdash 10372Number 7 02015 326 times 10

minus9


minus9


Number 8 630 times 10minus15 mdash mdash 6376 124 times 10

minus14 mdash mdash 5837Number 9 16512 mdash mdash 59239 16512 mdash mdash 58125Number 10 01550 mdash mdash 63153 01550 mdash mdash 65597Number 11 29701 270 times 10

minus9


minus9


Number 12 minus02499 201 times 10minus8

464 times 10minus5 54718 minus02499 115 times 10

minus9


Number 13 00147 115 times 10minus8

00252 57797 00151 139 times 10minus8

00472 66098Number 14 34761 672 times 10

minus6

00237 104585 34290 757 times 10minus6


Number 15 195 times 10minus4

111 times 10minus8

569 times 10minus3 363 194 times 10

minus4

704 times 10minus9




minus10




minus8


Number 18 00326 mdash mdash 65957 00326 mdash mdash 70897Number 19 011588 655 times 10

minus4 06828 161813 01091 884 times 10minus4 08808 159990

Number 20 5242 00176 05429 84165 4464 453 times 10minus4 03765 86891

Table 6 Results of Mann-Whitney test on relative errors of the pair (LI SI) for 119869

Method Mean rank Sum of ranks Test statistics ValueLI 2050 4100 Mann-Whitney 119880 200SI 2050 4100 Wilcoxon119882 410mdash mdash mdash 119885 0


Method Mean rank Sum of ranks Test statistics ValueLI 1362 177 Mann-Whitney 119880 83SI 1338 174 Wilcoxon119882 174mdash mdash mdash 119885 minus0077

Table 8 Results of Mann-Whitney test on relative errors of the pair (LI SI) for Time


Table 9 Results of Mann-Whitney test on relative errors of the pair (LI SI) for119870120595



Table 10 The numerical results of the without SQP algorithm for NOCPs in the Appendix

Problem no1 2 3 4 5 6 7 8 9 10

119869 17296 00174 21380 01702 01130 minus52489 02390 169 times 10minus1 18606 02048

120601119891

330 times 10minus9

135 times 10minus5 11445 mdash mdash mdash 758 times 10

minus9 mdash mdash mdashProblem no

11 12 13 14 15 16 17 18 19 20119869 48816 minus01273 00153 45727 979 times 10

minus4 23182 minus90882 00336 2513 20274120601119891

167 times 10minus5


minus4

594 times 10minus3

156 times 10minus5 164 mdash 1706 25283

8 Conclusions

Here a two-phase algorithm was proposed for solvingbounded continuous-time NOCPs In each phase of thealgorithm a MHGA was applied which performed a localsearch on offsprings In first phase a random initial popu-lation of control input values in time nodes was constructedNext MHGA started with this population After phase 1 toachieve more accurate solutions the number of time nodeswas increased The values of the associated new controlinputs were estimated by Linear interpolation (LI) or Splineinterpolation (SI) using the curves obtained from phase 1 Inaddition to maintain the diversity in the population someadditional individuals were added randomly Next in thesecond phase MHGA restarted with the new populationconstructed by above procedure and tried to improve theobtained solutions at the end of phase 1 We implementedour proposed algorithm on 20 well-known benchmark andreal world problems then the results were compared withsome recently proposed algorithms Moreover two statisticalapproaches were considered for the comparison of the LI andSI methods and investigation of sensitivity analysis for theMHGA parameters

Appendix

Test Problems

The following problems are presented using notation given in(1)ndash(6)

(1) (VDP) [9] 119892 = (12)(1199092

1+ 1199092

2+ 1199062

) 1199050= 0 119905119891= 5 119891 =

[1199092 minus1199092+(1minus119909

2

1)1199092+119906]119879 1199090= [1 0]

119879120595 = 1199091minus1199092+1

(2) (CRP) [9] 119892 = (12)(1199092

1+1199092

2+01119906

2

) 1199050= 0 119905119891= 078

119891 = [1199091minus2(1199091+025)+(119909

2+05) exp(25119909

1(1199091+2))minus

(1199091+025)119906 05minus119909

2minus(1199092+05) exp(25119909

1(1199091+2))]119879

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(3) (FFRP) [9] 119892 = (12)(1199062

1+ 1199062

2+ 1199062

3+ 1199062

4) 1199050

=

0 119905119891

= 5 119891 = [1199092 (110)((119906

1+ 1199062) cos119909

5minus

(1199062+ 1199064) sin119909

5) 1199094 (110)((119906

1+ 1199063) sin119909

5+ (1199062+

1199064) cos119909

5) 1199096 (512)((119906

1+ 1199063) minus (119906

2+ 1199064))]119879 1199090=

[0 0 0 0 0 0]119879 120595 = [119909

1minus 4 1199092 1199093minus 4 1199094 1199095 1199096]119879

(4) (MSNIC) [20] 120601 = 1199093 1199050= 0 119905119891= 1 119891 = [119909

2 minus1199092+

119906 1199092

1+ 1199092

2+ 0005119906

2

]119879 119889 = [minus(119906 minus 20)(119906 + 20) 119909

2+

05 minus 8(119905 minus 05)2

]119879 1199090= [0 minus1 0]

119879

(5) (CSTCR) [20] 119892 = 1199092

1+ 1199092

2+ 01119906

2 1199050= 0 119905119891= 078

119891 = [minus(2 + 119906)(1199091+ 025) + (119909

2+ 05) exp(25119909

1(1199091+

2)) 05 minus 1199092minus (1199092+ 05) exp(25119909

1(1199091+ 2))]

119879 1199090=

[009 009]119879

(6) [34] 119892 = 21199091 1199050= 0 119905119891= 3 119891 = [119909

2 119906]119879 119889 = [minus(2 +

119906)(2 minus 119906) minus(6 + 1199091)]119879 1199090= [2 0]

119879(7) [36] 119892 = 119906

2 1199050= 0 119905

119891= 1 119891 = (12)119909

2 sin119909 + 1199061199090= 0 120595 = 119909 minus 05

(8) [26] 119892 = 1199092cos2119906 119905

0= 0 119905119891= 120587 119891 = sin(1199062) 119909

0=

1205872(9) [26] 119892 = (12)(119909

2

1+ 1199092

2+ 1199062

) 1199050= 0 119905

119891= 5 119891 =

[1199092 minus1199091+ (1 minus 119909

2

1)1199092+ 119906]119879 119889 = minus(119909

2+ 025) 119909

0=

[1 0]119879

(10) [26] 119892 = 1199092

1+ 1199092

2+ 0005119906

2

1199050= 0 119905

119891= 1 119891 =

[1199092 minus1199092+119906]119879

119889 = [minus(20 + 119906)(20 minus 119906) minus(8(119905 minus 05)(119905 minus

05) minus 05 minus 1199092)]119879

1199090= [0 minus1]

119879(11) [26] 119892 = (12)119906

2 1199050= 0 119905

119891= 2 119891 = [119909

2 119906]119879 1199090=

[1 1]119879 120595 = [119909

1 1199092]119879

(12) [26] 119892 = minus1199092 1199050= 0 119905

119891= 1 119891 = [119909

2 119906]119879 119889 =

minus(1 minus 119906)(1 + 119906) 1199090= [0 0]

119879 120595 = 1199092

(13) [26] 119892 = (12)(1199092

1+ 1199092

2+ 01119906

2

) 1199050= 0 119905

119891= 078

119891 = [minus2(1199091+ 025) + (119909

2+ 05) exp(25119909

1(1199091+ 2)) minus

(1199091+025)119906 05minus119909

2minus(1199092+05) exp(25119909

1(1199091+2))]119879

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(14) [26] 119892 = (12)1199062 1199050= 0 119905

119891= 10 119891 = [cos 119906 minus

1199092 sin 119906]119879 119889 = minus(120587 + 119906)(120587 minus 119906) 119909

0= [366 minus186]

119879120595 = [119909

1 1199092]119879

(15) [26]119892 = (12)(1199092

1+1199092

2) 1199050= 0 119905119891= 078119891 = [minus2(119909

1+

025)+(1199092+05) exp(25119909

1(1199091+2))minus(119909

1+025)119906 05minus

1199092minus(1199092+05) exp(25119909

1(1199091+2))]119879 119889 = minus(1minus119906)(1+119906)

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(16) [26] 120601 = 1199093 1199050= 0 119905

119891= 1 119891 = [119909

2 119906 (12)119906

2

]119879

119889 = 1199091minus 19 119909

0= [0 0 0]

119879 120595 = [1199091 1199092+ 1]119879

(17) [26] 120601 = minus1199093 1199050= 0 119905

119891= 5 119891 = [119909

2 minus2 + 119906119909

3

minus001119906]119879 119889 = minus(30 minus 119906)(30 + 119906) 119909

0= [10 minus2 10]

119879120595 = [119909

1 1199092]119879

(18) [26] 120601 = (1199091minus1)2

+1199092

2+1199092

3 119892 = (12)119906

2 1199050= 0 119905119891= 5

119891 = [1199093cos 119906 119909

3sin 119906 sin 119906]119879 119909

0= [0 0 0]

119879


(19) [26] 119892 = 45(1199092

3+ 1199092

6) + 05(119906

2

1+ 1199062

2) 1199050= 0 119905

119891= 1

119891 = [91199094 91199095 91199096 9(1199061+ 1725119909

3) 91199062 minus(9119909

2)(1199061minus

270751199093+ 211990951199096)]119879 1199090= [0 22 0 0 minus1 0]

119879 120595 =

[1199091minus 10 119909

2minus 14 119909

3 1199094minus 25 119909

5 1199096]119879

(20) [26] similar to problem (3) with 120595 = [1199091minus 4 119909

2 1199093minus

4 1199094 1199095minus 1205874 119909

6]119879

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] J T Betts Practical Methods for Optimal Control and Estima-tion Using Nonlinear Programming Society for Industrial andApplied Mathematics 2010

[2] D E Kirk Optimal Control Theory An Introduction DoverPublications 2004

[3] B Srinivasan S Palanki and D Bonvin ldquoDynamic optimiza-tion of batch processes I Characterization of the nominalsolutionrdquo Computers and Chemical Engineering vol 27 no 1pp 1ndash26 2003

[4] T Binder L Blank W Dahmen and W Marquardt ldquoIterativealgorithms for multiscale state estimation I Conceptsrdquo Journalof OptimizationTheory and Applications vol 111 no 3 pp 501ndash527 2001

[5] M Schlegel K Stockmann T Binder and W MarquardtldquoDynamic optimization using adaptive control vector param-eterizationrdquo Computers and Chemical Engineering vol 29 no8 pp 1731ndash1751 2005

[6] Y C Sim S B Leng andV Subramaniam ldquoA combined geneticalgorithms-shooting method approach to solving optimal con-trol problemsrdquo International Journal of Systems Science vol 31no 1 pp 83ndash89 2000

[7] Z Michalewicz C Z Janikow and J B Krawczyk ldquoA modifiedgenetic algorithm for optimal control problemsrdquoComputers andMathematics with Applications vol 23 no 12 pp 83ndash94 1992

[8] Y Yamashita andM Shima ldquoNumerical computationalmethodusing genetic algorithm for the optimal control problem withterminal constraints and free parametersrdquo Nonlinear AnalysisTheory Methods amp Applications vol 30 no 4 pp 2285ndash22901997

[9] Z S Abo-Hammour A G Asasfeh A M Al-Smadi and O MAlsmadi ldquoA novel continuous genetic algorithm for the solutionof optimal control problemsrdquo Optimal Control Applications ampMethods vol 32 no 4 pp 414ndash432 2011

[10] X H Shi L M Wan H P Lee X W Yang L M Wangand Y C Liang ldquoAn improved genetic algorithm with vari-able population-size and a PSO-GA based hybrid evolution-ary algorithmrdquo in Proceedings of the International Conferenceon Machine Learning and Cybernetics vol 3 pp 1735ndash1740November 2003

[11] I L Lopez Cruz L G van Willigenburg and G van StratenldquoEfficient differential evolution algorithms formultimodal opti-mal control problemsrdquo Applied Soft Computing Journal vol 3no 2 pp 97ndash122 2003

[12] A Ghosh S Das A Chowdhury and R Giri ldquoAn ecologicallyinspired direct searchmethod for solving optimal control prob-lems with Bezier parameterizationrdquo Engineering Applications ofArtificial Intelligence vol 24 no 7 pp 1195ndash1203 2011

[13] A V A Kumar and P Balasubramaniam ldquoOptimal controlfor linear system using genetic programmingrdquoOptimal ControlApplications amp Methods vol 30 no 1 pp 47ndash60 2009

[14] M S Arumugam and M V C Rao ldquoOn the improvedperformances of the particle swarm optimization algorithmswith adaptive parameters cross-over operators and root meansquare (RMS) variants for computing optimal control of a classof hybrid systemsrdquo Applied Soft Computing Journal vol 8 no 1pp 324ndash336 2008

[15] M Senthil Arumugam G RamanaMurthy and C K Loo ldquoOnthe optimal control of the steel annealing processes as a two-stage hybrid systems via PSO algorithmsrdquo International Journalof Bio-Inspired Computation vol 1 no 3 pp 198ndash209 2009

[16] J M van Ast R Babuska and B de Schutter ldquoNovel antcolony optimization approach to optimal controlrdquo InternationalJournal of Intelligent Computing andCybernetics vol 2 no 3 pp414ndash434 2009

[17] M H Lee C Han and K S Chang ldquoDynamic optimizationof a continuous polymer reactor using a modified differentialevolution algorithmrdquo Industrial and Engineering ChemistryResearch vol 38 no 12 pp 4825ndash4831 1999

[18] F-S Wang and J-P Chiou ldquoOptimal control and optimaltime location problems of differential-algebraic systems bydifferential evolutionrdquo Industrial and Engineering ChemistryResearch vol 36 no 12 pp 5348ndash5357 1997

[19] S Babaie-Kafaki R Ghanbari and N Mahdavi-Amiri ldquoTwoeffective hybrid metaheuristic algorithms for minimizationof multimodal functionsrdquo International Journal of ComputerMathematics vol 88 no 11 pp 2415ndash2428 2011

[20] H Modares and M-B Naghibi-Sistani ldquoSolving nonlinearoptimal control problems using a hybrid IPSO-SQP algorithmrdquoEngineering Applications of Artificial Intelligence vol 24 no 3pp 476ndash484 2011

[21] F SunWDu R Qi F Qian andW Zhong ldquoA hybrid improvedgenetic algorithm and its application in dynamic optimizationproblems of chemical processesrdquo Chinese Journal of ChemicalEngineering vol 21 no 2 pp 144ndash154 2013

[22] J J F Bonnans J C Gilbert C Lemarechal and C ASagastizabal Numerical Optimization Theoretical and PracticalAspects Springer London UK 2006

[23] K L Teo C J Goh and K H Wong A Unified ComputationalApproach to Optimal Control Problems Pitman Monographsand Surveys in Pure and Applied Mathematics LongmanScientific and Technical 1991

[24] C J Goh and K L Teo ldquoControl parametrization a unifiedapproach to optimal control problemswith general constraintsrdquoAutomatica vol 24 no 1 pp 3ndash18 1988

[25] B C Fabien ldquoNumerical solution of constrained optimalcontrol problems with parametersrdquo Applied Mathematics andComputation vol 80 no 1 pp 43ndash62 1996

[26] B C Fabien ldquoSome tools for the direct solution of optimalcontrol problemsrdquo Advances Engineering Software vol 29 no1 pp 45ndash61 1998

[27] A P Engelbrecht Computational Intelligence An IntroductionJohn Wiley amp Sons New York NY USA 2007

[28] J Nocedal and S J Wright Numerical Optimization SpringerSeries in Operations Research Springer Berlin Germany 1999


[29] K Atkinson and W Han Theoretical Numerical Analysis AFunctional Analysis Framework vol 39 ofTexts inAppliedMath-ematics Springer Dordrecht The Netherlands 3rd edition2009

[30] A E Bryson Applied Optimal Control Optimization Estima-tion and Control Halsted Press Taylor amp Francis 1975

[31] W Mekarapiruk and R Luus ldquoOptimal control of inequalitystate constrained systemsrdquo Industrial and Engineering Chem-istry Research vol 36 no 5 pp 1686ndash1694 1997

[32] C A Floudas and P M PardalosHandbook of Test Problems inLocal and Global Optimization Nonconvex Optimization andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 1999

[33] M M Ali C Storey and A Torn ldquoApplication of stochasticglobal optimization algorithms to practical problemsrdquo Journalof OptimizationTheory and Applications vol 95 no 3 pp 545ndash563 1997

[34] F Ghomanjani M H Farahi and M Gachpazan ldquoBeziercontrol points method to solve constrained quadratic optimalcontrol of time varying linear systemsrdquo Computational ampApplied Mathematics vol 31 no 3 pp 433ndash456 2012

[35] J Vlassenbroeck ldquoAChebyshev polynomialmethod for optimalcontrol with state constraintsrdquo Automatica vol 24 no 4 pp499ndash506 1988

[36] S Effati and H Saberi Nik ldquoSolving a class of linear and non-linear optimal control problems by homotopy perturbationmethodrdquo IMA Journal of Mathematical Control and Informa-tion vol 28 no 4 pp 539ndash553 2011

[37] D C Montgomery Applied Statistics and Probability for Engi-neers John Wiley amp Sons Toronto Canada 1996

[38] S Garcıa D Molina M Lozano and H Francisco ldquoA study onthe use of non-parametric tests for analyzing the evolutionaryalgorithmsrsquo behaviour a case study on the CECrsquo2005 SpecialSession on Real Parameter Optimizationrdquo Journal of Heuristicsvol 15 no 6 pp 617ndash644 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


et al [12] developed an ecologically inspired optimizationtechnique called invasive weed optimization (IWO) forsolving optimal control problems The other well-knownmetaheuristic algorithms used for solving NOCPs are geneticprogramming (GP) [13] particle swarm optimization (PSO)[14 15] ant colony optimization (ACO) [16] and DE [17 18]

To increase the quality of solutions and decrease therunning time hybrid methods were introduced which useda local search in the implementation of a population-basedmetaheuristics [19] Modares and Naghibi-Sistani [20] pro-posed a hybrid algorithm by integrating an improved PSOwith successive quadratic programming (SQP) for solvingNOCPs Recently Sun et al [21] proposed a hybrid improvedGA which used simplex method (SM) to perform a localsearch for solving NOCPs and applied it for chemicalprocesses

Based on the success of the hybrid methods for solvingNOCPs mentioned above we here use a modified hybridgenetic algorithm (MHGA) which combines GA with SQPsee [22] as a local search SQP is an iterative algorithm forsolving nonlinear programming (NLP) problems which usesgradient information It can moreover be used for solvingNOCPs see [23ndash25] For decreasing the running time in theearly generations (iterations) of MHGA a less number ofiterations for SQP was used and then when the promisingregion of search space was found we increase the number ofiterations of SQP gradually

To perform MHGA for solving an NOCP the timeinterval is uniformly divided by using a constant number oftime nodes Next in each of these time nodes the controlvariable is approximated by a scaler vector of control inputvalues Thus an infinite dimensional NOCP is changed toa finite dimensional NLP Now we encounter two conflictsituations the quality of the global solution and the neededcomputational time In other words when the number oftime nodes is increased then we expect that the quality ofthe global solution is also increased but we know that in thissituation the computational time is increased dramatically Inother situation if we consider less number of time nodesthen the computational time is decreased but we may finda poor local solution To conquer these problems MHGAperforms in two phases In the first phase (explorationphase) to decrease the computational time and to find apromising region of search space MHGA uses a less numberof time nodes After phase 1 to increase the quality ofsolutions obtained from phase 1 the number of time nodesis increased Using the population obtained in phase 1 thevalues of the new control inputs are estimated by Linear orSpline interpolations Next in the second phase (exploitationphase) MHGA uses the solutions constructed by the aboveprocedure as an initial population

The paper is organized as follows in Section 2 theformulation of the problem is introduced In Section 3 theproposed MHGA is presented In Section 4 we introduceour algorithm for solving NOCP In Section 5 we provide20 numerical benchmark examples to compare the proposedalgorithm with the other recently proposed algorithmsIn Section 6 we consider two statistical approaches for

the comparison of the LI and SI methods and investigation ofsensitivity analysis of the algorithm parameters The impactof SQP as local search in the proposed algorithm is surveyedin Section 7 We conclude in Section 8

2 Formulation of Problem

The bounded continuous-time NOCP is considered as find-ing the control input 119906(119905) isin R119898 over the planning horizon[1199050 119905119891] which minimizes the cost functional

119869 = 120601 (119909 (119905119891) 119905119891) + int

119905119891

1199050

119892 (119909 (119905) 119906 (119905) 119905) 119889119905 (1)

subject to

(119905) = 119891 (119909 (119905) 119906 (119905) 119905) (2)

119888 (119909 119906 119905) = 0 (3)

119889 (119909 119906 119905) le 0 (4)

120595 (119909 (119905119891) 119905119891) = 0 (5)

119909 (1199050) = 1199090 (6)

where 119909(119905) isin R119899 denotes the state vector for the system and1199090isin R119899 is the initial state The functions 119891 R119899 timesR119898 timesR rarr

R119899 119892 R119899 timesR119898 timesR rarr R 119888 R119899 timesR119898 timesR rarr R119899119888 119889 R119899 times

R119898timesR rarr R119899119889 120595 R119899timesR rarr R119899120595 and 120601 R119899timesR rarr R areassumed to be sufficiently smooth on appropriate open setsCost function (1) must be minimized subject to dynamic (2)control and state equality constraints (3) and control and stateinequality constraints (4) the final state constraints (5) andthe initial condition (6) A special case of the NOCPs is thelinear quadratic regulator (LQR) problemwhere the dynamicequations are linear and the objective function is a quadraticfunction of 119909 and 119906 The minimum time problems trackingproblem terminal control problem andminimum energy areanother special case of NOCPs

3 Modified Hybrid Genetic Algorithm

In this section first MHGA as a subprocedure for the mainalgorithm is introduced To perform MHGA the controlvariables are discretized Next NOCP is changed into a finitedimensional NLP see [21 26] Now we can imply a GA tofind the global solution of the corresponding NLP In thefollowing we introduce GA operators

31 Underlying GA GAs introduced by Holland in 1975 areheuristics and probabilistic methods [27] These algorithmsstart with an initial population of solutions This populationis evaluated by using genetic operators that include selectioncrossover andmutation Here inMHGA the underlying GAhas the following steps

InitializationThe time interval is divided into119873119905minus1 subinter-

vals using time nodes 1199050 119905

119873119905minus1and then the control input





1198962) then

Let 119896 = 1198961

elseLet 119896 = 119896

2

end ifend if



until 1198961

= 1198962




(1) Let 119905119895= 1199050+ 119895ℎ where ℎ = (119905

119891minus 1199050)(119873119905minus 1) 119895 =


0and 119905119891are the







119894 = 1 2 119898 119895 = 0 1 119873119905minus 1

(7)



119905matrix as 119880 = (119906

119894119895)119898times119873119905

= [119906119895]119873119905minus1

119895=0

Next we let 119880(119896) = (119906119894119895)119898times119873119905

119896 = 1 2 119873119901as 119896th


the population



119905



) to 119880(119896) as the


119868 (119880(119896)

) = 119869 +

119899119889

sum

119897=1

119873119905minus1

sum

119895=0

1198721119897max 0 119889

119897(119909119895 119906119895 119905119895)

+

119899119888

sum

ℎ=1

119873119905minus1

sum

119895=0

1198722ℎ1198882

ℎ(119909119895 119906119895 119905119895)

+

119899120595

sum

119894=119901

11987231199011205952

119901(119909119873119905minus1

119905119873119905minus1

)

(8)

where 1198721= [119872

11 119872

1119899119889]119879 1198722= [119872

21 119872

2119899119888]119879 and

1198723

= [11987231 119872



119888 119889119897(sdot sdot) 119897 = 1 2 119899

119889

and 120595119901(sdot sdot) 119901 = 1 2 119899


respectively




1205821isin [0 1] 120582

2isin [minus120582max 0] 120582

3isin [1 1 + 120582max]


(2) Let

of119896 = 120582119896119880(1)

+ (1 minus 120582119896) 119880(2)

119896 = 1 2 3 (10)



and 119895 = 1 119873119905 if (of119896)


119896

)119894119895=

119906right(119894) and if (of119896


119896

)119894119895= 119906left(119894)


constructed by (10)


(of)119894119895= (of)

119894119895+ 119903119894119895sdot 120572 119894 = 1 2 119898 119895 = 1 2 119873

119905

(11)



gt


119894119895lt 119906left(119894) then

let (of)119894119895= 119906left(119894)


1le119894le119873119901119868(119880(119894)




(119894)

)

10038161003816100381610038161003816gt 120576

10038171003817100381710038171003817of minus 119880

(119894)10038171003817100381710038171003817gt 120576

(12)






120593119891=100381710038171003817100381712059510038171003817100381710038172

lt 120576 (13)

where120595 = [1205951 1205952 120595




0 1199061 119906



min 119868 (119906) = 119868 (1199060 1199061 119906

119873119905minus1) (14)


119905minus 1 (15)



119896




119879


min 1

2

119889119879

119867(119906119896

) 119889 + nabla119868 (119906119896

)

119879

119889 (16)

nablaℎ (119906119896

)

119879

119889 + ℎ (119906119896

) le 0 (17)


120597119868

120597119906119895

=


120575

119895 = 0 1 119873119905minus 1 (18)


0 120597119868120597119906



119894 119894 = 1 119898 is an estimate






+ 119889119896 where 119889119896






Initialization

sqpmaxiter



using (8)




Stoppingconditions




(12))



Input NtNpNiNg Pm

120576Mi i = 1 2 3 and









1199051in phase 1 to 119873






119891 defined in




and 1199041199021199011198981198861199091198941199051198901199031






and 1199041199021199011198981198861199091198941199051198901199032




119869 of the


119864119869=

10038161003816100381610038161003816100381610038161003816


119869lowast

10038161003816100381610038161003816100381610038161003816

(19)


119870120595= 119864119869+ 120593119891 (20)





119905 119873119901 119873119894 119873119892 119875119898and 119904119902119901119898119886119909119894119905119890119903




119894= 4

and 119875119898119894


= 1198731198922

1198731198941

= 1198731198942 1198731199011

= 9 and 1198731199012














=

311198731199052= 71119873

119892= 300 and119873




1199051= 31 119873

1199052=

71 119873119892





119891= 757 times 10



minus9





120593119891

119870120595

Time

VDPCGA 17404 00912 267 times 10

minus11

00912 50128LI 15949 0 108 times 10



CRPCGA 163 times 10




599 times 10minus9

12405

FFRPCGA 8363 13256 465 times 10

minus3

13302 1413LI 3596 0 122 times 10



minus4 134340









Number 7HPM 02353 01677 420 times 10

minus6 01677 NRLI 02015 0 235 times 10

minus9



Number 8






minus15


Number 9


Number 10



Number 11


minus9



Number 12





285 times 10minus8

54771


Number 13




minus2

70 times 10minus3

632 times 10minus9

70 times 10minus3

49027

Number 14


minus4

276 times 10minus6

267 times 10minus4

115144


Table 1 Continued


120593119891

119870120595

Time

Number 15




minus4

520 times 10minus3

164 times 10minus9

520 times 10minus3

34014

Number 16


minus12

394 times 10minus12

46982


Number 17





minus10

279 times 10minus10

30959

Number 18



Number 19


minus3

01919 164509


Number 20


minus5

225 times 10minus4



aNot reported



1199051= 31 119873

1199052= 71 119873

119892= 300 and

119873119894


1minus 4 119909

2 1199093minus 4 119909



120595for LI



2(119905) + 05 minus 8(119905 minus 05)

2


1199051=

31 1198731199052

= 91 119873119892= 100 and 119873







1(119905) and 119909

2(119905)


1199051=

31 1198731199052

= 51 119873119892= 100 and 119873



lowast






119906right 2 2 10 20 5 2 1 2 1 20119872119894

103


2 mdash 102

120576 10minus12

10minus10


minus9



119906right 5 1 2 120587 1 3 30 1 10 10119872119894

10 102 mdash 10

2

102

102 mdash 10 70 10

minus3

120576 10minus9

10minus11

10minus11 mdash 10

minus10

10minus10

10minus10 mdash 10

minus3

10minus4

0 5 10 15 20 25 30 35 40 45 50017

0175

018

0185

019

0195

02

0205

Iter

Perfo

rman

ce in

dex

(a)

0 01 02 03 04 05 06 07 08 09 1minus25

minus2

minus15

minus1

minus05

0

Time

Ineq

ualit

y co

nstr

aint

(b)



1199051= 31 119873

1199052= 51 119873

119892= 100 119873

119894= 50 with



1 119905) = minus6 minus 119909







119891= 282 times 10



2 4 6 8 10 12 14minus5353

minus5352

minus5351

minus535

minus5349

minus5348

minus5347

minus5346

minus5345

minus5344

minus5343

Iter

Perfo

rman

ce in

dex

(a)

0 05 1 15 2 25 3minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Time

Ineq

ualit

y co

nstr

aint

(b)







120595





1 15 2 25 3 35 40

002

004

006

008

01

012

Iter

Perfo

rman

ce in

dex









5 10 15 20 25165

17

175

18

185

Iter

Perfo

rman

ce in

dex

(a)

0 05 1 15 2 25 3 35 4 45 50

005

01

015

02

025

TimeIn

equa

lity

cons

trai

nt(b)


5 10 15 20 25 30 35 40 45 500155

016

0165

017

0175

018

0185

019

0195

02

0205

Iter

Perfo

rman

ce in

dex

(a)

0 01 02 03 04 05 06 07 08 09 1minus25

minus2

minus15

minus1

minus05

0

Time

Ineq

ualit

y co

nstr

aint

(b)









119869for LI SI










5 10 15 20 25minus025

minus0245

minus024

minus0235

minus023

minus0225

Iter

Perfo

rman

ce in

dex

(a)

1 15 2 25 3 35 4 45 5 55 60

02

04

06

08

1

12

14

16

IterFi

nal c

ondi

tion

times10minus5

(b)


2 4 6 8 10 12 14minus025

minus0245

minus024

minus0235

minus023

minus0225

minus022

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

14

16

18

2

Iter

Fina

l con

ditio

n

times10minus5

(b)






1 2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

07

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

IterFi

nal c

ondi

tion

times10minus3

(b)


5 10 15 20 25 30

35

4

45

5

55

6

65

7

75

8

85

9

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

Iter

Fina

l con

ditio

n

(b)









1199052 the


and 1198731199012 the



5 10 15 20 25 30 35 40 45 50

00955

0096

00965

0097

00975

0098

00985

0099

00995

Iter

Perfo

rman

ce in

dex

(a)

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

35

4

45

IterFi

nal c

ondi

tion

times10minus3

(b)


5 10 15 20 25

21

22

23

24

25

26

27

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 80

1

2

3

4

5

6

Iter

Fina

l con

ditio

n

times10minus3

(b)







1 2 3 4 5 6 7minus887

minus886

minus885

minus884

minus883

minus882

minus881

minus88

minus879

minus878

minus877

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 70

01

02

03

04

05

06

07

08

09

1

IterFi

nal c

ondi

tion

times10minus3

(b)


1 15 2 25 3 35 4 45 5 55 600331003320033200332003320033200333003330033300333

Iter

Perfo

rman

ce in

dex










1199051


119869119870120595




119869 120593119891119870120595 are independent


119891is





119869and


119892 and other



1199011


119892





1 15 2 25 3 35 40

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 80

200

400

600

800

1000

1200

1400

1600

IterFi

nal c

ondi

tion

(b)


2 4 6 8 10 12 14 16 18 20

50

100

150

200

250

300

350

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 10 11 120

05

1

15

2

25

3

Iter

Fina

l con

ditio

n

(b)



1199011=

91198731199012

= 121198731199051= 31119873

1199052= 91119873

119892= 100 and119873

119894= 50 with








Output 1198731199051


119869

11 35 15333 00069 15286 1559821 35 15328 00097 15285 1584931 35 15329 00076 15285 1569941 35 15455 00607 15285 18449

Total 140 15357 00294 15285 18449

120593119891


952 times 10minus8

1 times 10minus12

452 times 10minus7


235 times 10minus7

1 times 10minus12

141 times 10minus6


491 times 10minus8

1 times 10minus12

221 times 10minus7


285 times 10minus7

1 times 10minus12

126 times 10minus6


Time

11 35 897838 272081 456614 155439421 35 1098147 460264 604192 284031031 35 1163591 300227 772673 179510341 35 1448290 393091 810425 2849514

Total 140 1142131 410891 456615 2849514

119864119869

11 35 00031 00045 0 0020421 35 00028 00064 0 0036931 35 00029 00050 0 0027141 35 00111 00397 0 02070

Total 140 00047 00192 0 02070

119870120595

11 35 00031 00045 475 times 10minus9

00204

21 35 00027 00064 550 times 10minus10

00369

31 35 00029 00050 154 times 10minus13

00271

41 35 00111 00397 193 times 10minus10

02070



Parameters 119869 119864119869

120593119891

Time 119870120595


1198731199051

1294 1296 0624 1079 12961198731199052

110572 393 0868 5939 3931198731199011

1945 1835 1271 4317 18351198731199012

2740 2478 1011 2302 2478119873119892

3541 3591 0491 0890 3591119873119894

0495 0309 0301 5849 1046

119875 value

1198731199051

0280 0279 0601 0 02791198731199052

0 0005 0489 0 00051198731199011

0125 0143 0286 0006 01431198731199012

0021 0034 0413 0 0034119873119892

0009 0008 0743 0472 0008119873119894

0739 0819 0877 0 0386



7 The Impact of SQP






119891119870120595

Time 119869 120593119891

119870120595


minus9


minus8




minus8




minus9


minus9




minus9


minus9




minus9



00252 57797 00151 139 times 10minus8

00472 66098Number 14 34761 672 times 10

minus6

00237 104585 34290 757 times 10minus6



111 times 10minus8


minus4

704 times 10minus9




minus10




minus8















Problem no1 2 3 4 5 6 7 8 9 10


120601119891

330 times 10minus9



11 12 13 14 15 16 17 18 19 20119869 48816 minus01273 00153 45727 979 times 10

minus4 23182 minus90882 00336 2513 20274120601119891

167 times 10minus5


minus4

594 times 10minus3


8 Conclusions


Appendix

Test Problems


(1) (VDP) [9] 119892 = (12)(1199092

1+ 1199092

2+ 1199062

) 1199050= 0 119905119891= 5 119891 =

[1199092 minus1199092+(1minus119909

2

1)1199092+119906]119879 1199090= [1 0]

119879120595 = 1199091minus1199092+1

(2) (CRP) [9] 119892 = (12)(1199092

1+1199092

2+01119906

2

) 1199050= 0 119905119891= 078

119891 = [1199091minus2(1199091+025)+(119909

2+05) exp(25119909

1(1199091+2))minus

(1199091+025)119906 05minus119909

2minus(1199092+05) exp(25119909

1(1199091+2))]119879

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(3) (FFRP) [9] 119892 = (12)(1199062

1+ 1199062

2+ 1199062

3+ 1199062

4) 1199050

=

0 119905119891

= 5 119891 = [1199092 (110)((119906

1+ 1199062) cos119909

5minus

(1199062+ 1199064) sin119909

5) 1199094 (110)((119906

1+ 1199063) sin119909

5+ (1199062+

1199064) cos119909

5) 1199096 (512)((119906

1+ 1199063) minus (119906

2+ 1199064))]119879 1199090=

[0 0 0 0 0 0]119879 120595 = [119909

1minus 4 1199092 1199093minus 4 1199094 1199095 1199096]119879

(4) (MSNIC) [20] 120601 = 1199093 1199050= 0 119905119891= 1 119891 = [119909

2 minus1199092+

119906 1199092

1+ 1199092

2+ 0005119906

2

]119879 119889 = [minus(119906 minus 20)(119906 + 20) 119909

2+

05 minus 8(119905 minus 05)2

]119879 1199090= [0 minus1 0]

119879

(5) (CSTCR) [20] 119892 = 1199092

1+ 1199092

2+ 01119906

2 1199050= 0 119905119891= 078

119891 = [minus(2 + 119906)(1199091+ 025) + (119909

2+ 05) exp(25119909

1(1199091+

2)) 05 minus 1199092minus (1199092+ 05) exp(25119909

1(1199091+ 2))]

119879 1199090=

[009 009]119879

(6) [34] 119892 = 21199091 1199050= 0 119905119891= 3 119891 = [119909

2 119906]119879 119889 = [minus(2 +

119906)(2 minus 119906) minus(6 + 1199091)]119879 1199090= [2 0]

119879(7) [36] 119892 = 119906

2 1199050= 0 119905

119891= 1 119891 = (12)119909

2 sin119909 + 1199061199090= 0 120595 = 119909 minus 05

(8) [26] 119892 = 1199092cos2119906 119905

0= 0 119905119891= 120587 119891 = sin(1199062) 119909

0=

1205872(9) [26] 119892 = (12)(119909

2

1+ 1199092

2+ 1199062

) 1199050= 0 119905

119891= 5 119891 =

[1199092 minus1199091+ (1 minus 119909

2

1)1199092+ 119906]119879 119889 = minus(119909

2+ 025) 119909

0=

[1 0]119879

(10) [26] 119892 = 1199092

1+ 1199092

2+ 0005119906

2

1199050= 0 119905

119891= 1 119891 =

[1199092 minus1199092+119906]119879


05) minus 05 minus 1199092)]119879

1199090= [0 minus1]

119879(11) [26] 119892 = (12)119906

2 1199050= 0 119905

119891= 2 119891 = [119909

2 119906]119879 1199090=

[1 1]119879 120595 = [119909

1 1199092]119879

(12) [26] 119892 = minus1199092 1199050= 0 119905

119891= 1 119891 = [119909

2 119906]119879 119889 =

minus(1 minus 119906)(1 + 119906) 1199090= [0 0]

119879 120595 = 1199092

(13) [26] 119892 = (12)(1199092

1+ 1199092

2+ 01119906

2

) 1199050= 0 119905

119891= 078

119891 = [minus2(1199091+ 025) + (119909

2+ 05) exp(25119909

1(1199091+ 2)) minus

(1199091+025)119906 05minus119909

2minus(1199092+05) exp(25119909

1(1199091+2))]119879

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(14) [26] 119892 = (12)1199062 1199050= 0 119905

119891= 10 119891 = [cos 119906 minus

1199092 sin 119906]119879 119889 = minus(120587 + 119906)(120587 minus 119906) 119909

0= [366 minus186]

119879120595 = [119909

1 1199092]119879

(15) [26]119892 = (12)(1199092

1+1199092

2) 1199050= 0 119905119891= 078119891 = [minus2(119909

1+

025)+(1199092+05) exp(25119909

1(1199091+2))minus(119909

1+025)119906 05minus

1199092minus(1199092+05) exp(25119909

1(1199091+2))]119879 119889 = minus(1minus119906)(1+119906)

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(16) [26] 120601 = 1199093 1199050= 0 119905

119891= 1 119891 = [119909

2 119906 (12)119906

2

]119879

119889 = 1199091minus 19 119909

0= [0 0 0]

119879 120595 = [1199091 1199092+ 1]119879

(17) [26] 120601 = minus1199093 1199050= 0 119905

119891= 5 119891 = [119909

2 minus2 + 119906119909

3


0= [10 minus2 10]

119879120595 = [119909

1 1199092]119879

(18) [26] 120601 = (1199091minus1)2

+1199092

2+1199092

3 119892 = (12)119906

2 1199050= 0 119905119891= 5

119891 = [1199093cos 119906 119909

3sin 119906 sin 119906]119879 119909

0= [0 0 0]

119879


(19) [26] 119892 = 45(1199092

3+ 1199092

6) + 05(119906

2

1+ 1199062

2) 1199050= 0 119905

119891= 1

119891 = [91199094 91199095 91199096 9(1199061+ 1725119909

3) 91199062 minus(9119909

2)(1199061minus

270751199093+ 211990951199096)]119879 1199090= [0 22 0 0 minus1 0]

119879 120595 =

[1199091minus 10 119909

2minus 14 119909

3 1199094minus 25 119909

5 1199096]119879


2 1199093minus

4 1199094 1199095minus 1205874 119909

6]119879



References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













1198962) then

Let 119896 = 1198961

elseLet 119896 = 119896

2

end ifend if



until 1198961

= 1198962




(1) Let 119905119895= 1199050+ 119895ℎ where ℎ = (119905

119891minus 1199050)(119873119905minus 1) 119895 =


0and 119905119891are the







119894 = 1 2 119898 119895 = 0 1 119873119905minus 1

(7)



119905matrix as 119880 = (119906

119894119895)119898times119873119905

= [119906119895]119873119905minus1

119895=0

Next we let 119880(119896) = (119906119894119895)119898times119873119905

119896 = 1 2 119873119901as 119896th


the population



119905



) to 119880(119896) as the


119868 (119880(119896)

) = 119869 +

119899119889

sum

119897=1

119873119905minus1

sum

119895=0

1198721119897max 0 119889

119897(119909119895 119906119895 119905119895)

+

119899119888

sum

ℎ=1

119873119905minus1

sum

119895=0

1198722ℎ1198882

ℎ(119909119895 119906119895 119905119895)

+

119899120595

sum

119894=119901

11987231199011205952

119901(119909119873119905minus1

119905119873119905minus1

)

(8)

where 1198721= [119872

11 119872

1119899119889]119879 1198722= [119872

21 119872

2119899119888]119879 and

1198723

= [11987231 119872



119888 119889119897(sdot sdot) 119897 = 1 2 119899

119889

and 120595119901(sdot sdot) 119901 = 1 2 119899


respectively




1205821isin [0 1] 120582

2isin [minus120582max 0] 120582

3isin [1 1 + 120582max]


(2) Let

of119896 = 120582119896119880(1)

+ (1 minus 120582119896) 119880(2)

119896 = 1 2 3 (10)



and 119895 = 1 119873119905 if (of119896)


119896

)119894119895=

119906right(119894) and if (of119896


119896

)119894119895= 119906left(119894)


constructed by (10)


(of)119894119895= (of)

119894119895+ 119903119894119895sdot 120572 119894 = 1 2 119898 119895 = 1 2 119873

119905

(11)



gt


119894119895lt 119906left(119894) then

let (of)119894119895= 119906left(119894)


1le119894le119873119901119868(119880(119894)




(119894)

)

10038161003816100381610038161003816gt 120576

10038171003817100381710038171003817of minus 119880

(119894)10038171003817100381710038171003817gt 120576

(12)






120593119891=100381710038171003817100381712059510038171003817100381710038172

lt 120576 (13)

where120595 = [1205951 1205952 120595




0 1199061 119906



min 119868 (119906) = 119868 (1199060 1199061 119906

119873119905minus1) (14)


119905minus 1 (15)



119896




119879


min 1

2

119889119879

119867(119906119896

) 119889 + nabla119868 (119906119896

)

119879

119889 (16)

nablaℎ (119906119896

)

119879

119889 + ℎ (119906119896

) le 0 (17)


120597119868

120597119906119895

=


120575

119895 = 0 1 119873119905minus 1 (18)


0 120597119868120597119906



119894 119894 = 1 119898 is an estimate






+ 119889119896 where 119889119896






Initialization

sqpmaxiter



using (8)




Stoppingconditions




(12))



Input NtNpNiNg Pm

120576Mi i = 1 2 3 and









1199051in phase 1 to 119873






119891 defined in




and 1199041199021199011198981198861199091198941199051198901199031






and 1199041199021199011198981198861199091198941199051198901199032




119869 of the


119864119869=

10038161003816100381610038161003816100381610038161003816


119869lowast

10038161003816100381610038161003816100381610038161003816

(19)


119870120595= 119864119869+ 120593119891 (20)





119905 119873119901 119873119894 119873119892 119875119898and 119904119902119901119898119886119909119894119905119890119903




119894= 4

and 119875119898119894


= 1198731198922

1198731198941

= 1198731198942 1198731199011

= 9 and 1198731199012














=

311198731199052= 71119873

119892= 300 and119873




1199051= 31 119873

1199052=

71 119873119892





119891= 757 times 10



minus9





120593119891

119870120595

Time

VDPCGA 17404 00912 267 times 10

minus11

00912 50128LI 15949 0 108 times 10



CRPCGA 163 times 10




599 times 10minus9

12405

FFRPCGA 8363 13256 465 times 10

minus3

13302 1413LI 3596 0 122 times 10



minus4 134340









Number 7HPM 02353 01677 420 times 10

minus6 01677 NRLI 02015 0 235 times 10

minus9



Number 8






minus15


Number 9


Number 10



Number 11


minus9



Number 12





285 times 10minus8

54771


Number 13




minus2

70 times 10minus3

632 times 10minus9

70 times 10minus3

49027

Number 14


minus4

276 times 10minus6

267 times 10minus4

115144


Table 1 Continued


120593119891

119870120595

Time

Number 15




minus4

520 times 10minus3

164 times 10minus9

520 times 10minus3

34014

Number 16


minus12

394 times 10minus12

46982


Number 17





minus10

279 times 10minus10

30959

Number 18



Number 19


minus3

01919 164509


Number 20


minus5

225 times 10minus4



aNot reported



1199051= 31 119873

1199052= 71 119873

119892= 300 and

119873119894


1minus 4 119909

2 1199093minus 4 119909



120595for LI



2(119905) + 05 minus 8(119905 minus 05)

2


1199051=

31 1198731199052

= 91 119873119892= 100 and 119873







1(119905) and 119909

2(119905)


1199051=

31 1198731199052

= 51 119873119892= 100 and 119873



lowast






119906right 2 2 10 20 5 2 1 2 1 20119872119894

103


2 mdash 102

120576 10minus12

10minus10


minus9



119906right 5 1 2 120587 1 3 30 1 10 10119872119894

10 102 mdash 10

2

102

102 mdash 10 70 10

minus3

120576 10minus9

10minus11

10minus11 mdash 10

minus10

10minus10

10minus10 mdash 10

minus3

10minus4

0 5 10 15 20 25 30 35 40 45 50017

0175

018

0185

019

0195

02

0205

Iter

Perfo

rman

ce in

dex

(a)

0 01 02 03 04 05 06 07 08 09 1minus25

minus2

minus15

minus1

minus05

0

Time

Ineq

ualit

y co

nstr

aint

(b)



1199051= 31 119873

1199052= 51 119873

119892= 100 119873

119894= 50 with



1 119905) = minus6 minus 119909







119891= 282 times 10



2 4 6 8 10 12 14minus5353

minus5352

minus5351

minus535

minus5349

minus5348

minus5347

minus5346

minus5345

minus5344

minus5343

Iter

Perfo

rman

ce in

dex

(a)

0 05 1 15 2 25 3minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Time

Ineq

ualit

y co

nstr

aint

(b)







120595





1 15 2 25 3 35 40

002

004

006

008

01

012

Iter

Perfo

rman

ce in

dex









5 10 15 20 25165

17

175

18

185

Iter

Perfo

rman

ce in

dex

(a)

0 05 1 15 2 25 3 35 4 45 50

005

01

015

02

025

TimeIn

equa

lity

cons

trai

nt(b)


5 10 15 20 25 30 35 40 45 500155

016

0165

017

0175

018

0185

019

0195

02

0205

Iter

Perfo

rman

ce in

dex

(a)

0 01 02 03 04 05 06 07 08 09 1minus25

minus2

minus15

minus1

minus05

0

Time

Ineq

ualit

y co

nstr

aint

(b)









119869for LI SI










5 10 15 20 25minus025

minus0245

minus024

minus0235

minus023

minus0225

Iter

Perfo

rman

ce in

dex

(a)

1 15 2 25 3 35 4 45 5 55 60

02

04

06

08

1

12

14

16

IterFi

nal c

ondi

tion

times10minus5

(b)


2 4 6 8 10 12 14minus025

minus0245

minus024

minus0235

minus023

minus0225

minus022

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

14

16

18

2

Iter

Fina

l con

ditio

n

times10minus5

(b)






1 2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

07

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

IterFi

nal c

ondi

tion

times10minus3

(b)


5 10 15 20 25 30

35

4

45

5

55

6

65

7

75

8

85

9

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

Iter

Fina

l con

ditio

n

(b)









1199052 the


and 1198731199012 the



5 10 15 20 25 30 35 40 45 50

00955

0096

00965

0097

00975

0098

00985

0099

00995

Iter

Perfo

rman

ce in

dex

(a)

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

35

4

45

IterFi

nal c

ondi

tion

times10minus3

(b)


5 10 15 20 25

21

22

23

24

25

26

27

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 80

1

2

3

4

5

6

Iter

Fina

l con

ditio

n

times10minus3

(b)







1 2 3 4 5 6 7minus887

minus886

minus885

minus884

minus883

minus882

minus881

minus88

minus879

minus878

minus877

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 70

01

02

03

04

05

06

07

08

09

1

IterFi

nal c

ondi

tion

times10minus3

(b)


1 15 2 25 3 35 4 45 5 55 600331003320033200332003320033200333003330033300333

Iter

Perfo

rman

ce in

dex










1199051


119869119870120595




119869 120593119891119870120595 are independent


119891is





119869and


119892 and other



1199011


119892





1 15 2 25 3 35 40

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 80

200

400

600

800

1000

1200

1400

1600

IterFi

nal c

ondi

tion

(b)


2 4 6 8 10 12 14 16 18 20

50

100

150

200

250

300

350

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 10 11 120

05

1

15

2

25

3

Iter

Fina

l con

ditio

n

(b)



1199011=

91198731199012

= 121198731199051= 31119873

1199052= 91119873

119892= 100 and119873

119894= 50 with








Output 1198731199051


119869

11 35 15333 00069 15286 1559821 35 15328 00097 15285 1584931 35 15329 00076 15285 1569941 35 15455 00607 15285 18449

Total 140 15357 00294 15285 18449

120593119891


952 times 10minus8

1 times 10minus12

452 times 10minus7


235 times 10minus7

1 times 10minus12

141 times 10minus6


491 times 10minus8

1 times 10minus12

221 times 10minus7


285 times 10minus7

1 times 10minus12

126 times 10minus6


Time

11 35 897838 272081 456614 155439421 35 1098147 460264 604192 284031031 35 1163591 300227 772673 179510341 35 1448290 393091 810425 2849514

Total 140 1142131 410891 456615 2849514

119864119869

11 35 00031 00045 0 0020421 35 00028 00064 0 0036931 35 00029 00050 0 0027141 35 00111 00397 0 02070

Total 140 00047 00192 0 02070

119870120595

11 35 00031 00045 475 times 10minus9

00204

21 35 00027 00064 550 times 10minus10

00369

31 35 00029 00050 154 times 10minus13

00271

41 35 00111 00397 193 times 10minus10

02070



Parameters 119869 119864119869

120593119891

Time 119870120595


1198731199051

1294 1296 0624 1079 12961198731199052

110572 393 0868 5939 3931198731199011

1945 1835 1271 4317 18351198731199012

2740 2478 1011 2302 2478119873119892

3541 3591 0491 0890 3591119873119894

0495 0309 0301 5849 1046

119875 value

1198731199051

0280 0279 0601 0 02791198731199052

0 0005 0489 0 00051198731199011

0125 0143 0286 0006 01431198731199012

0021 0034 0413 0 0034119873119892

0009 0008 0743 0472 0008119873119894

0739 0819 0877 0 0386



7 The Impact of SQP






119891119870120595

Time 119869 120593119891

119870120595


minus9


minus8




minus8




minus9


minus9




minus9


minus9




minus9



00252 57797 00151 139 times 10minus8

00472 66098Number 14 34761 672 times 10

minus6

00237 104585 34290 757 times 10minus6



111 times 10minus8


minus4

704 times 10minus9




minus10




minus8















Problem no1 2 3 4 5 6 7 8 9 10


120601119891

330 times 10minus9



11 12 13 14 15 16 17 18 19 20119869 48816 minus01273 00153 45727 979 times 10

minus4 23182 minus90882 00336 2513 20274120601119891

167 times 10minus5


minus4

594 times 10minus3


8 Conclusions


Appendix

Test Problems


(1) (VDP) [9] 119892 = (12)(1199092

1+ 1199092

2+ 1199062

) 1199050= 0 119905119891= 5 119891 =

[1199092 minus1199092+(1minus119909

2

1)1199092+119906]119879 1199090= [1 0]

119879120595 = 1199091minus1199092+1

(2) (CRP) [9] 119892 = (12)(1199092

1+1199092

2+01119906

2

) 1199050= 0 119905119891= 078

119891 = [1199091minus2(1199091+025)+(119909

2+05) exp(25119909

1(1199091+2))minus

(1199091+025)119906 05minus119909

2minus(1199092+05) exp(25119909

1(1199091+2))]119879

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(3) (FFRP) [9] 119892 = (12)(1199062

1+ 1199062

2+ 1199062

3+ 1199062

4) 1199050

=

0 119905119891

= 5 119891 = [1199092 (110)((119906

1+ 1199062) cos119909

5minus

(1199062+ 1199064) sin119909

5) 1199094 (110)((119906

1+ 1199063) sin119909

5+ (1199062+

1199064) cos119909

5) 1199096 (512)((119906

1+ 1199063) minus (119906

2+ 1199064))]119879 1199090=

[0 0 0 0 0 0]119879 120595 = [119909

1minus 4 1199092 1199093minus 4 1199094 1199095 1199096]119879

(4) (MSNIC) [20] 120601 = 1199093 1199050= 0 119905119891= 1 119891 = [119909

2 minus1199092+

119906 1199092

1+ 1199092

2+ 0005119906

2

]119879 119889 = [minus(119906 minus 20)(119906 + 20) 119909

2+

05 minus 8(119905 minus 05)2

]119879 1199090= [0 minus1 0]

119879

(5) (CSTCR) [20] 119892 = 1199092

1+ 1199092

2+ 01119906

2 1199050= 0 119905119891= 078

119891 = [minus(2 + 119906)(1199091+ 025) + (119909

2+ 05) exp(25119909

1(1199091+

2)) 05 minus 1199092minus (1199092+ 05) exp(25119909

1(1199091+ 2))]

119879 1199090=

[009 009]119879

(6) [34] 119892 = 21199091 1199050= 0 119905119891= 3 119891 = [119909

2 119906]119879 119889 = [minus(2 +

119906)(2 minus 119906) minus(6 + 1199091)]119879 1199090= [2 0]

119879(7) [36] 119892 = 119906

2 1199050= 0 119905

119891= 1 119891 = (12)119909

2 sin119909 + 1199061199090= 0 120595 = 119909 minus 05

(8) [26] 119892 = 1199092cos2119906 119905

0= 0 119905119891= 120587 119891 = sin(1199062) 119909

0=

1205872(9) [26] 119892 = (12)(119909

2

1+ 1199092

2+ 1199062

) 1199050= 0 119905

119891= 5 119891 =

[1199092 minus1199091+ (1 minus 119909

2

1)1199092+ 119906]119879 119889 = minus(119909

2+ 025) 119909

0=

[1 0]119879

(10) [26] 119892 = 1199092

1+ 1199092

2+ 0005119906

2

1199050= 0 119905

119891= 1 119891 =

[1199092 minus1199092+119906]119879


05) minus 05 minus 1199092)]119879

1199090= [0 minus1]

119879(11) [26] 119892 = (12)119906

2 1199050= 0 119905

119891= 2 119891 = [119909

2 119906]119879 1199090=

[1 1]119879 120595 = [119909

1 1199092]119879

(12) [26] 119892 = minus1199092 1199050= 0 119905

119891= 1 119891 = [119909

2 119906]119879 119889 =

minus(1 minus 119906)(1 + 119906) 1199090= [0 0]

119879 120595 = 1199092

(13) [26] 119892 = (12)(1199092

1+ 1199092

2+ 01119906

2

) 1199050= 0 119905

119891= 078

119891 = [minus2(1199091+ 025) + (119909

2+ 05) exp(25119909

1(1199091+ 2)) minus

(1199091+025)119906 05minus119909

2minus(1199092+05) exp(25119909

1(1199091+2))]119879

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(14) [26] 119892 = (12)1199062 1199050= 0 119905

119891= 10 119891 = [cos 119906 minus

1199092 sin 119906]119879 119889 = minus(120587 + 119906)(120587 minus 119906) 119909

0= [366 minus186]

119879120595 = [119909

1 1199092]119879

(15) [26]119892 = (12)(1199092

1+1199092

2) 1199050= 0 119905119891= 078119891 = [minus2(119909

1+

025)+(1199092+05) exp(25119909

1(1199091+2))minus(119909

1+025)119906 05minus

1199092minus(1199092+05) exp(25119909

1(1199091+2))]119879 119889 = minus(1minus119906)(1+119906)

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(16) [26] 120601 = 1199093 1199050= 0 119905

119891= 1 119891 = [119909

2 119906 (12)119906

2

]119879

119889 = 1199091minus 19 119909

0= [0 0 0]

119879 120595 = [1199091 1199092+ 1]119879

(17) [26] 120601 = minus1199093 1199050= 0 119905

119891= 5 119891 = [119909

2 minus2 + 119906119909

3


0= [10 minus2 10]

119879120595 = [119909

1 1199092]119879

(18) [26] 120601 = (1199091minus1)2

+1199092

2+1199092

3 119892 = (12)119906

2 1199050= 0 119905119891= 5

119891 = [1199093cos 119906 119909

3sin 119906 sin 119906]119879 119909

0= [0 0 0]

119879


(19) [26] 119892 = 45(1199092

3+ 1199092

6) + 05(119906

2

1+ 1199062

2) 1199050= 0 119905

119891= 1

119891 = [91199094 91199095 91199096 9(1199061+ 1725119909

3) 91199062 minus(9119909

2)(1199061minus

270751199093+ 211990951199096)]119879 1199090= [0 22 0 0 minus1 0]

119879 120595 =

[1199091minus 10 119909

2minus 14 119909

3 1199094minus 25 119909

5 1199096]119879


2 1199093minus

4 1199094 1199095minus 1205874 119909

6]119879



References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











and 119895 = 1 119873119905 if (of119896)


119896

)119894119895=

119906right(119894) and if (of119896


119896

)119894119895= 119906left(119894)


constructed by (10)


(of)119894119895= (of)

119894119895+ 119903119894119895sdot 120572 119894 = 1 2 119898 119895 = 1 2 119873

119905

(11)



gt


119894119895lt 119906left(119894) then

let (of)119894119895= 119906left(119894)


1le119894le119873119901119868(119880(119894)




(119894)

)

10038161003816100381610038161003816gt 120576

10038171003817100381710038171003817of minus 119880

(119894)10038171003817100381710038171003817gt 120576

(12)






120593119891=100381710038171003817100381712059510038171003817100381710038172

lt 120576 (13)

where120595 = [1205951 1205952 120595




0 1199061 119906



min 119868 (119906) = 119868 (1199060 1199061 119906

119873119905minus1) (14)


119905minus 1 (15)



119896




119879


min 1

2

119889119879

119867(119906119896

) 119889 + nabla119868 (119906119896

)

119879

119889 (16)

nablaℎ (119906119896

)

119879

119889 + ℎ (119906119896

) le 0 (17)


120597119868

120597119906119895

=


120575

119895 = 0 1 119873119905minus 1 (18)


0 120597119868120597119906



119894 119894 = 1 119898 is an estimate






+ 119889119896 where 119889119896






Initialization

sqpmaxiter



using (8)




Stoppingconditions




(12))



Input NtNpNiNg Pm

120576Mi i = 1 2 3 and









1199051in phase 1 to 119873






119891 defined in




and 1199041199021199011198981198861199091198941199051198901199031






and 1199041199021199011198981198861199091198941199051198901199032




119869 of the


119864119869=

10038161003816100381610038161003816100381610038161003816


119869lowast

10038161003816100381610038161003816100381610038161003816

(19)


119870120595= 119864119869+ 120593119891 (20)





119905 119873119901 119873119894 119873119892 119875119898and 119904119902119901119898119886119909119894119905119890119903




119894= 4

and 119875119898119894


= 1198731198922

1198731198941

= 1198731198942 1198731199011

= 9 and 1198731199012














=

311198731199052= 71119873

119892= 300 and119873




1199051= 31 119873

1199052=

71 119873119892





119891= 757 times 10



minus9





120593119891

119870120595

Time

VDPCGA 17404 00912 267 times 10

minus11

00912 50128LI 15949 0 108 times 10



CRPCGA 163 times 10




599 times 10minus9

12405

FFRPCGA 8363 13256 465 times 10

minus3

13302 1413LI 3596 0 122 times 10



minus4 134340









Number 7HPM 02353 01677 420 times 10

minus6 01677 NRLI 02015 0 235 times 10

minus9



Number 8






minus15


Number 9


Number 10



Number 11


minus9



Number 12





285 times 10minus8

54771


Number 13




minus2

70 times 10minus3

632 times 10minus9

70 times 10minus3

49027

Number 14


minus4

276 times 10minus6

267 times 10minus4

115144


Table 1 Continued


120593119891

119870120595

Time

Number 15




minus4

520 times 10minus3

164 times 10minus9

520 times 10minus3

34014

Number 16


minus12

394 times 10minus12

46982


Number 17





minus10

279 times 10minus10

30959

Number 18



Number 19


minus3

01919 164509


Number 20


minus5

225 times 10minus4



aNot reported



1199051= 31 119873

1199052= 71 119873

119892= 300 and

119873119894


1minus 4 119909

2 1199093minus 4 119909



120595for LI



2(119905) + 05 minus 8(119905 minus 05)

2


1199051=

31 1198731199052

= 91 119873119892= 100 and 119873







1(119905) and 119909

2(119905)


1199051=

31 1198731199052

= 51 119873119892= 100 and 119873



lowast






119906right 2 2 10 20 5 2 1 2 1 20119872119894

103


2 mdash 102

120576 10minus12

10minus10


minus9



119906right 5 1 2 120587 1 3 30 1 10 10119872119894

10 102 mdash 10

2

102

102 mdash 10 70 10

minus3

120576 10minus9

10minus11

10minus11 mdash 10

minus10

10minus10

10minus10 mdash 10

minus3

10minus4

0 5 10 15 20 25 30 35 40 45 50017

0175

018

0185

019

0195

02

0205

Iter

Perfo

rman

ce in

dex

(a)

0 01 02 03 04 05 06 07 08 09 1minus25

minus2

minus15

minus1

minus05

0

Time

Ineq

ualit

y co

nstr

aint

(b)



1199051= 31 119873

1199052= 51 119873

119892= 100 119873

119894= 50 with



1 119905) = minus6 minus 119909







119891= 282 times 10



2 4 6 8 10 12 14minus5353

minus5352

minus5351

minus535

minus5349

minus5348

minus5347

minus5346

minus5345

minus5344

minus5343

Iter

Perfo

rman

ce in

dex

(a)

0 05 1 15 2 25 3minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Time

Ineq

ualit

y co

nstr

aint

(b)







120595





1 15 2 25 3 35 40

002

004

006

008

01

012

Iter

Perfo

rman

ce in

dex









5 10 15 20 25165

17

175

18

185

Iter

Perfo

rman

ce in

dex

(a)

0 05 1 15 2 25 3 35 4 45 50

005

01

015

02

025

TimeIn

equa

lity

cons

trai

nt(b)


5 10 15 20 25 30 35 40 45 500155

016

0165

017

0175

018

0185

019

0195

02

0205

Iter

Perfo

rman

ce in

dex

(a)

0 01 02 03 04 05 06 07 08 09 1minus25

minus2

minus15

minus1

minus05

0

Time

Ineq

ualit

y co

nstr

aint

(b)









119869for LI SI










5 10 15 20 25minus025

minus0245

minus024

minus0235

minus023

minus0225

Iter

Perfo

rman

ce in

dex

(a)

1 15 2 25 3 35 4 45 5 55 60

02

04

06

08

1

12

14

16

IterFi

nal c

ondi

tion

times10minus5

(b)


2 4 6 8 10 12 14minus025

minus0245

minus024

minus0235

minus023

minus0225

minus022

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

14

16

18

2

Iter

Fina

l con

ditio

n

times10minus5

(b)






1 2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

07

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

IterFi

nal c

ondi

tion

times10minus3

(b)


5 10 15 20 25 30

35

4

45

5

55

6

65

7

75

8

85

9

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

Iter

Fina

l con

ditio

n

(b)









1199052 the


and 1198731199012 the



5 10 15 20 25 30 35 40 45 50

00955

0096

00965

0097

00975

0098

00985

0099

00995

Iter

Perfo

rman

ce in

dex

(a)

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

35

4

45

IterFi

nal c

ondi

tion

times10minus3

(b)


5 10 15 20 25

21

22

23

24

25

26

27

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 80

1

2

3

4

5

6

Iter

Fina

l con

ditio

n

times10minus3

(b)







1 2 3 4 5 6 7minus887

minus886

minus885

minus884

minus883

minus882

minus881

minus88

minus879

minus878

minus877

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 70

01

02

03

04

05

06

07

08

09

1

IterFi

nal c

ondi

tion

times10minus3

(b)


1 15 2 25 3 35 4 45 5 55 600331003320033200332003320033200333003330033300333

Iter

Perfo

rman

ce in

dex










1199051


119869119870120595




119869 120593119891119870120595 are independent


119891is





119869and


119892 and other



1199011


119892





1 15 2 25 3 35 40

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 80

200

400

600

800

1000

1200

1400

1600

IterFi

nal c

ondi

tion

(b)


2 4 6 8 10 12 14 16 18 20

50

100

150

200

250

300

350

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 10 11 120

05

1

15

2

25

3

Iter

Fina

l con

ditio

n

(b)



1199011=

91198731199012

= 121198731199051= 31119873

1199052= 91119873

119892= 100 and119873

119894= 50 with








Output 1198731199051


119869

11 35 15333 00069 15286 1559821 35 15328 00097 15285 1584931 35 15329 00076 15285 1569941 35 15455 00607 15285 18449

Total 140 15357 00294 15285 18449

120593119891


952 times 10minus8

1 times 10minus12

452 times 10minus7


235 times 10minus7

1 times 10minus12

141 times 10minus6


491 times 10minus8

1 times 10minus12

221 times 10minus7


285 times 10minus7

1 times 10minus12

126 times 10minus6


Time

11 35 897838 272081 456614 155439421 35 1098147 460264 604192 284031031 35 1163591 300227 772673 179510341 35 1448290 393091 810425 2849514

Total 140 1142131 410891 456615 2849514

119864119869

11 35 00031 00045 0 0020421 35 00028 00064 0 0036931 35 00029 00050 0 0027141 35 00111 00397 0 02070

Total 140 00047 00192 0 02070

119870120595

11 35 00031 00045 475 times 10minus9

00204

21 35 00027 00064 550 times 10minus10

00369

31 35 00029 00050 154 times 10minus13

00271

41 35 00111 00397 193 times 10minus10

02070



Parameters 119869 119864119869

120593119891

Time 119870120595


1198731199051

1294 1296 0624 1079 12961198731199052

110572 393 0868 5939 3931198731199011

1945 1835 1271 4317 18351198731199012

2740 2478 1011 2302 2478119873119892

3541 3591 0491 0890 3591119873119894

0495 0309 0301 5849 1046

119875 value

1198731199051

0280 0279 0601 0 02791198731199052

0 0005 0489 0 00051198731199011

0125 0143 0286 0006 01431198731199012

0021 0034 0413 0 0034119873119892

0009 0008 0743 0472 0008119873119894

0739 0819 0877 0 0386



7 The Impact of SQP






119891119870120595

Time 119869 120593119891

119870120595


minus9


minus8




minus8




minus9


minus9




minus9


minus9




minus9



00252 57797 00151 139 times 10minus8

00472 66098Number 14 34761 672 times 10

minus6

00237 104585 34290 757 times 10minus6



111 times 10minus8


minus4

704 times 10minus9




minus10




minus8















Problem no1 2 3 4 5 6 7 8 9 10


120601119891

330 times 10minus9



11 12 13 14 15 16 17 18 19 20119869 48816 minus01273 00153 45727 979 times 10

minus4 23182 minus90882 00336 2513 20274120601119891

167 times 10minus5


minus4

594 times 10minus3


8 Conclusions


Appendix

Test Problems


(1) (VDP) [9] 119892 = (12)(1199092

1+ 1199092

2+ 1199062

) 1199050= 0 119905119891= 5 119891 =

[1199092 minus1199092+(1minus119909

2

1)1199092+119906]119879 1199090= [1 0]

119879120595 = 1199091minus1199092+1

(2) (CRP) [9] 119892 = (12)(1199092

1+1199092

2+01119906

2

) 1199050= 0 119905119891= 078

119891 = [1199091minus2(1199091+025)+(119909

2+05) exp(25119909

1(1199091+2))minus

(1199091+025)119906 05minus119909

2minus(1199092+05) exp(25119909

1(1199091+2))]119879

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(3) (FFRP) [9] 119892 = (12)(1199062

1+ 1199062

2+ 1199062

3+ 1199062

4) 1199050

=

0 119905119891

= 5 119891 = [1199092 (110)((119906

1+ 1199062) cos119909

5minus

(1199062+ 1199064) sin119909

5) 1199094 (110)((119906

1+ 1199063) sin119909

5+ (1199062+

1199064) cos119909

5) 1199096 (512)((119906

1+ 1199063) minus (119906

2+ 1199064))]119879 1199090=

[0 0 0 0 0 0]119879 120595 = [119909

1minus 4 1199092 1199093minus 4 1199094 1199095 1199096]119879

(4) (MSNIC) [20] 120601 = 1199093 1199050= 0 119905119891= 1 119891 = [119909

2 minus1199092+

119906 1199092

1+ 1199092

2+ 0005119906

2

]119879 119889 = [minus(119906 minus 20)(119906 + 20) 119909

2+

05 minus 8(119905 minus 05)2

]119879 1199090= [0 minus1 0]

119879

(5) (CSTCR) [20] 119892 = 1199092

1+ 1199092

2+ 01119906

2 1199050= 0 119905119891= 078

119891 = [minus(2 + 119906)(1199091+ 025) + (119909

2+ 05) exp(25119909

1(1199091+

2)) 05 minus 1199092minus (1199092+ 05) exp(25119909

1(1199091+ 2))]

119879 1199090=

[009 009]119879

(6) [34] 119892 = 21199091 1199050= 0 119905119891= 3 119891 = [119909

2 119906]119879 119889 = [minus(2 +

119906)(2 minus 119906) minus(6 + 1199091)]119879 1199090= [2 0]

119879(7) [36] 119892 = 119906

2 1199050= 0 119905

119891= 1 119891 = (12)119909

2 sin119909 + 1199061199090= 0 120595 = 119909 minus 05

(8) [26] 119892 = 1199092cos2119906 119905

0= 0 119905119891= 120587 119891 = sin(1199062) 119909

0=

1205872(9) [26] 119892 = (12)(119909

2

1+ 1199092

2+ 1199062

) 1199050= 0 119905

119891= 5 119891 =

[1199092 minus1199091+ (1 minus 119909

2

1)1199092+ 119906]119879 119889 = minus(119909

2+ 025) 119909

0=

[1 0]119879

(10) [26] 119892 = 1199092

1+ 1199092

2+ 0005119906

2

1199050= 0 119905

119891= 1 119891 =

[1199092 minus1199092+119906]119879


05) minus 05 minus 1199092)]119879

1199090= [0 minus1]

119879(11) [26] 119892 = (12)119906

2 1199050= 0 119905

119891= 2 119891 = [119909

2 119906]119879 1199090=

[1 1]119879 120595 = [119909

1 1199092]119879

(12) [26] 119892 = minus1199092 1199050= 0 119905

119891= 1 119891 = [119909

2 119906]119879 119889 =

minus(1 minus 119906)(1 + 119906) 1199090= [0 0]

119879 120595 = 1199092

(13) [26] 119892 = (12)(1199092

1+ 1199092

2+ 01119906

2

) 1199050= 0 119905

119891= 078

119891 = [minus2(1199091+ 025) + (119909

2+ 05) exp(25119909

1(1199091+ 2)) minus

(1199091+025)119906 05minus119909

2minus(1199092+05) exp(25119909

1(1199091+2))]119879

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(14) [26] 119892 = (12)1199062 1199050= 0 119905

119891= 10 119891 = [cos 119906 minus

1199092 sin 119906]119879 119889 = minus(120587 + 119906)(120587 minus 119906) 119909

0= [366 minus186]

119879120595 = [119909

1 1199092]119879

(15) [26]119892 = (12)(1199092

1+1199092

2) 1199050= 0 119905119891= 078119891 = [minus2(119909

1+

025)+(1199092+05) exp(25119909

1(1199091+2))minus(119909

1+025)119906 05minus

1199092minus(1199092+05) exp(25119909

1(1199091+2))]119879 119889 = minus(1minus119906)(1+119906)

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(16) [26] 120601 = 1199093 1199050= 0 119905

119891= 1 119891 = [119909

2 119906 (12)119906

2

]119879

119889 = 1199091minus 19 119909

0= [0 0 0]

119879 120595 = [1199091 1199092+ 1]119879

(17) [26] 120601 = minus1199093 1199050= 0 119905

119891= 5 119891 = [119909

2 minus2 + 119906119909

3


0= [10 minus2 10]

119879120595 = [119909

1 1199092]119879

(18) [26] 120601 = (1199091minus1)2

+1199092

2+1199092

3 119892 = (12)119906

2 1199050= 0 119905119891= 5

119891 = [1199093cos 119906 119909

3sin 119906 sin 119906]119879 119909

0= [0 0 0]

119879


(19) [26] 119892 = 45(1199092

3+ 1199092

6) + 05(119906

2

1+ 1199062

2) 1199050= 0 119905

119891= 1

119891 = [91199094 91199095 91199096 9(1199061+ 1725119909

3) 91199062 minus(9119909

2)(1199061minus

270751199093+ 211990951199096)]119879 1199090= [0 22 0 0 minus1 0]

119879 120595 =

[1199091minus 10 119909

2minus 14 119909

3 1199094minus 25 119909

5 1199096]119879


2 1199093minus

4 1199094 1199095minus 1205874 119909

6]119879



References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Initialization

sqpmaxiter



using (8)




Stoppingconditions




(12))



Input NtNpNiNg Pm

120576Mi i = 1 2 3 and









1199051in phase 1 to 119873






119891 defined in




and 1199041199021199011198981198861199091198941199051198901199031






and 1199041199021199011198981198861199091198941199051198901199032




119869 of the


119864119869=

10038161003816100381610038161003816100381610038161003816


119869lowast

10038161003816100381610038161003816100381610038161003816

(19)


119870120595= 119864119869+ 120593119891 (20)





119905 119873119901 119873119894 119873119892 119875119898and 119904119902119901119898119886119909119894119905119890119903




119894= 4

and 119875119898119894


= 1198731198922

1198731198941

= 1198731198942 1198731199011

= 9 and 1198731199012














=

311198731199052= 71119873

119892= 300 and119873




1199051= 31 119873

1199052=

71 119873119892





119891= 757 times 10



minus9





120593119891

119870120595

Time

VDPCGA 17404 00912 267 times 10

minus11

00912 50128LI 15949 0 108 times 10



CRPCGA 163 times 10




599 times 10minus9

12405

FFRPCGA 8363 13256 465 times 10

minus3

13302 1413LI 3596 0 122 times 10



minus4 134340









Number 7HPM 02353 01677 420 times 10

minus6 01677 NRLI 02015 0 235 times 10

minus9



Number 8






minus15


Number 9


Number 10



Number 11


minus9



Number 12





285 times 10minus8

54771


Number 13




minus2

70 times 10minus3

632 times 10minus9

70 times 10minus3

49027

Number 14


minus4

276 times 10minus6

267 times 10minus4

115144


Table 1 Continued


120593119891

119870120595

Time

Number 15




minus4

520 times 10minus3

164 times 10minus9

520 times 10minus3

34014

Number 16


minus12

394 times 10minus12

46982


Number 17





minus10

279 times 10minus10

30959

Number 18



Number 19


minus3

01919 164509


Number 20


minus5

225 times 10minus4



aNot reported



1199051= 31 119873

1199052= 71 119873

119892= 300 and

119873119894


1minus 4 119909

2 1199093minus 4 119909



120595for LI



2(119905) + 05 minus 8(119905 minus 05)

2


1199051=

31 1198731199052

= 91 119873119892= 100 and 119873







1(119905) and 119909

2(119905)


1199051=

31 1198731199052

= 51 119873119892= 100 and 119873



lowast






119906right 2 2 10 20 5 2 1 2 1 20119872119894

103


2 mdash 102

120576 10minus12

10minus10


minus9



119906right 5 1 2 120587 1 3 30 1 10 10119872119894

10 102 mdash 10

2

102

102 mdash 10 70 10

minus3

120576 10minus9

10minus11

10minus11 mdash 10

minus10

10minus10

10minus10 mdash 10

minus3

10minus4

0 5 10 15 20 25 30 35 40 45 50017

0175

018

0185

019

0195

02

0205

Iter

Perfo

rman

ce in

dex

(a)

0 01 02 03 04 05 06 07 08 09 1minus25

minus2

minus15

minus1

minus05

0

Time

Ineq

ualit

y co

nstr

aint

(b)



1199051= 31 119873

1199052= 51 119873

119892= 100 119873

119894= 50 with



1 119905) = minus6 minus 119909







119891= 282 times 10



2 4 6 8 10 12 14minus5353

minus5352

minus5351

minus535

minus5349

minus5348

minus5347

minus5346

minus5345

minus5344

minus5343

Iter

Perfo

rman

ce in

dex

(a)

0 05 1 15 2 25 3minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Time

Ineq

ualit

y co

nstr

aint

(b)







120595





1 15 2 25 3 35 40

002

004

006

008

01

012

Iter

Perfo

rman

ce in

dex









5 10 15 20 25165

17

175

18

185

Iter

Perfo

rman

ce in

dex

(a)

0 05 1 15 2 25 3 35 4 45 50

005

01

015

02

025

TimeIn

equa

lity

cons

trai

nt(b)


5 10 15 20 25 30 35 40 45 500155

016

0165

017

0175

018

0185

019

0195

02

0205

Iter

Perfo

rman

ce in

dex

(a)

0 01 02 03 04 05 06 07 08 09 1minus25

minus2

minus15

minus1

minus05

0

Time

Ineq

ualit

y co

nstr

aint

(b)









119869for LI SI










5 10 15 20 25minus025

minus0245

minus024

minus0235

minus023

minus0225

Iter

Perfo

rman

ce in

dex

(a)

1 15 2 25 3 35 4 45 5 55 60

02

04

06

08

1

12

14

16

IterFi

nal c

ondi

tion

times10minus5

(b)


2 4 6 8 10 12 14minus025

minus0245

minus024

minus0235

minus023

minus0225

minus022

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

14

16

18

2

Iter

Fina

l con

ditio

n

times10minus5

(b)






1 2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

07

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

IterFi

nal c

ondi

tion

times10minus3

(b)


5 10 15 20 25 30

35

4

45

5

55

6

65

7

75

8

85

9

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

Iter

Fina

l con

ditio

n

(b)









1199052 the


and 1198731199012 the



5 10 15 20 25 30 35 40 45 50

00955

0096

00965

0097

00975

0098

00985

0099

00995

Iter

Perfo

rman

ce in

dex

(a)

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

35

4

45

IterFi

nal c

ondi

tion

times10minus3

(b)


5 10 15 20 25

21

22

23

24

25

26

27

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 80

1

2

3

4

5

6

Iter

Fina

l con

ditio

n

times10minus3

(b)







1 2 3 4 5 6 7minus887

minus886

minus885

minus884

minus883

minus882

minus881

minus88

minus879

minus878

minus877

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 70

01

02

03

04

05

06

07

08

09

1

IterFi

nal c

ondi

tion

times10minus3

(b)


1 15 2 25 3 35 4 45 5 55 600331003320033200332003320033200333003330033300333

Iter

Perfo

rman

ce in

dex










1199051


119869119870120595




119869 120593119891119870120595 are independent


119891is





119869and


119892 and other



1199011


119892





1 15 2 25 3 35 40

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 80

200

400

600

800

1000

1200

1400

1600

IterFi

nal c

ondi

tion

(b)


2 4 6 8 10 12 14 16 18 20

50

100

150

200

250

300

350

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 10 11 120

05

1

15

2

25

3

Iter

Fina

l con

ditio

n

(b)



1199011=

91198731199012

= 121198731199051= 31119873

1199052= 91119873

119892= 100 and119873

119894= 50 with








Output 1198731199051


119869

11 35 15333 00069 15286 1559821 35 15328 00097 15285 1584931 35 15329 00076 15285 1569941 35 15455 00607 15285 18449

Total 140 15357 00294 15285 18449

120593119891


952 times 10minus8

1 times 10minus12

452 times 10minus7


235 times 10minus7

1 times 10minus12

141 times 10minus6


491 times 10minus8

1 times 10minus12

221 times 10minus7


285 times 10minus7

1 times 10minus12

126 times 10minus6


Time

11 35 897838 272081 456614 155439421 35 1098147 460264 604192 284031031 35 1163591 300227 772673 179510341 35 1448290 393091 810425 2849514

Total 140 1142131 410891 456615 2849514

119864119869

11 35 00031 00045 0 0020421 35 00028 00064 0 0036931 35 00029 00050 0 0027141 35 00111 00397 0 02070

Total 140 00047 00192 0 02070

119870120595

11 35 00031 00045 475 times 10minus9

00204

21 35 00027 00064 550 times 10minus10

00369

31 35 00029 00050 154 times 10minus13

00271

41 35 00111 00397 193 times 10minus10

02070



Parameters 119869 119864119869

120593119891

Time 119870120595


1198731199051

1294 1296 0624 1079 12961198731199052

110572 393 0868 5939 3931198731199011

1945 1835 1271 4317 18351198731199012

2740 2478 1011 2302 2478119873119892

3541 3591 0491 0890 3591119873119894

0495 0309 0301 5849 1046

119875 value

1198731199051

0280 0279 0601 0 02791198731199052

0 0005 0489 0 00051198731199011

0125 0143 0286 0006 01431198731199012

0021 0034 0413 0 0034119873119892

0009 0008 0743 0472 0008119873119894

0739 0819 0877 0 0386



7 The Impact of SQP






119891119870120595

Time 119869 120593119891

119870120595


minus9


minus8




minus8




minus9


minus9




minus9


minus9




minus9



00252 57797 00151 139 times 10minus8

00472 66098Number 14 34761 672 times 10

minus6

00237 104585 34290 757 times 10minus6



111 times 10minus8


minus4

704 times 10minus9




minus10




minus8















Problem no1 2 3 4 5 6 7 8 9 10


120601119891

330 times 10minus9



11 12 13 14 15 16 17 18 19 20119869 48816 minus01273 00153 45727 979 times 10

minus4 23182 minus90882 00336 2513 20274120601119891

167 times 10minus5


minus4

594 times 10minus3


8 Conclusions


Appendix

Test Problems


(1) (VDP) [9] 119892 = (12)(1199092

1+ 1199092

2+ 1199062

) 1199050= 0 119905119891= 5 119891 =

[1199092 minus1199092+(1minus119909

2

1)1199092+119906]119879 1199090= [1 0]

119879120595 = 1199091minus1199092+1

(2) (CRP) [9] 119892 = (12)(1199092

1+1199092

2+01119906

2

) 1199050= 0 119905119891= 078

119891 = [1199091minus2(1199091+025)+(119909

2+05) exp(25119909

1(1199091+2))minus

(1199091+025)119906 05minus119909

2minus(1199092+05) exp(25119909

1(1199091+2))]119879

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(3) (FFRP) [9] 119892 = (12)(1199062

1+ 1199062

2+ 1199062

3+ 1199062

4) 1199050

=

0 119905119891

= 5 119891 = [1199092 (110)((119906

1+ 1199062) cos119909

5minus

(1199062+ 1199064) sin119909

5) 1199094 (110)((119906

1+ 1199063) sin119909

5+ (1199062+

1199064) cos119909

5) 1199096 (512)((119906

1+ 1199063) minus (119906

2+ 1199064))]119879 1199090=

[0 0 0 0 0 0]119879 120595 = [119909

1minus 4 1199092 1199093minus 4 1199094 1199095 1199096]119879

(4) (MSNIC) [20] 120601 = 1199093 1199050= 0 119905119891= 1 119891 = [119909

2 minus1199092+

119906 1199092

1+ 1199092

2+ 0005119906

2

]119879 119889 = [minus(119906 minus 20)(119906 + 20) 119909

2+

05 minus 8(119905 minus 05)2

]119879 1199090= [0 minus1 0]

119879

(5) (CSTCR) [20] 119892 = 1199092

1+ 1199092

2+ 01119906

2 1199050= 0 119905119891= 078

119891 = [minus(2 + 119906)(1199091+ 025) + (119909

2+ 05) exp(25119909

1(1199091+

2)) 05 minus 1199092minus (1199092+ 05) exp(25119909

1(1199091+ 2))]

119879 1199090=

[009 009]119879

(6) [34] 119892 = 21199091 1199050= 0 119905119891= 3 119891 = [119909

2 119906]119879 119889 = [minus(2 +

119906)(2 minus 119906) minus(6 + 1199091)]119879 1199090= [2 0]

119879(7) [36] 119892 = 119906

2 1199050= 0 119905

119891= 1 119891 = (12)119909

2 sin119909 + 1199061199090= 0 120595 = 119909 minus 05

(8) [26] 119892 = 1199092cos2119906 119905

0= 0 119905119891= 120587 119891 = sin(1199062) 119909

0=

1205872(9) [26] 119892 = (12)(119909

2

1+ 1199092

2+ 1199062

) 1199050= 0 119905

119891= 5 119891 =

[1199092 minus1199091+ (1 minus 119909

2

1)1199092+ 119906]119879 119889 = minus(119909

2+ 025) 119909

0=

[1 0]119879

(10) [26] 119892 = 1199092

1+ 1199092

2+ 0005119906

2

1199050= 0 119905

119891= 1 119891 =

[1199092 minus1199092+119906]119879


05) minus 05 minus 1199092)]119879

1199090= [0 minus1]

119879(11) [26] 119892 = (12)119906

2 1199050= 0 119905

119891= 2 119891 = [119909

2 119906]119879 1199090=

[1 1]119879 120595 = [119909

1 1199092]119879

(12) [26] 119892 = minus1199092 1199050= 0 119905

119891= 1 119891 = [119909

2 119906]119879 119889 =

minus(1 minus 119906)(1 + 119906) 1199090= [0 0]

119879 120595 = 1199092

(13) [26] 119892 = (12)(1199092

1+ 1199092

2+ 01119906

2

) 1199050= 0 119905

119891= 078

119891 = [minus2(1199091+ 025) + (119909

2+ 05) exp(25119909

1(1199091+ 2)) minus

(1199091+025)119906 05minus119909

2minus(1199092+05) exp(25119909

1(1199091+2))]119879

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(14) [26] 119892 = (12)1199062 1199050= 0 119905

119891= 10 119891 = [cos 119906 minus

1199092 sin 119906]119879 119889 = minus(120587 + 119906)(120587 minus 119906) 119909

0= [366 minus186]

119879120595 = [119909

1 1199092]119879

(15) [26]119892 = (12)(1199092

1+1199092

2) 1199050= 0 119905119891= 078119891 = [minus2(119909

1+

025)+(1199092+05) exp(25119909

1(1199091+2))minus(119909

1+025)119906 05minus

1199092minus(1199092+05) exp(25119909

1(1199091+2))]119879 119889 = minus(1minus119906)(1+119906)

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(16) [26] 120601 = 1199093 1199050= 0 119905

119891= 1 119891 = [119909

2 119906 (12)119906

2

]119879

119889 = 1199091minus 19 119909

0= [0 0 0]

119879 120595 = [1199091 1199092+ 1]119879

(17) [26] 120601 = minus1199093 1199050= 0 119905

119891= 5 119891 = [119909

2 minus2 + 119906119909

3


0= [10 minus2 10]

119879120595 = [119909

1 1199092]119879

(18) [26] 120601 = (1199091minus1)2

+1199092

2+1199092

3 119892 = (12)119906

2 1199050= 0 119905119891= 5

119891 = [1199093cos 119906 119909

3sin 119906 sin 119906]119879 119909

0= [0 0 0]

119879


(19) [26] 119892 = 45(1199092

3+ 1199092

6) + 05(119906

2

1+ 1199062

2) 1199050= 0 119905

119891= 1

119891 = [91199094 91199095 91199096 9(1199061+ 1725119909

3) 91199062 minus(9119909

2)(1199061minus

270751199093+ 211990951199096)]119879 1199090= [0 22 0 0 minus1 0]

119879 120595 =

[1199091minus 10 119909

2minus 14 119909

3 1199094minus 25 119909

5 1199096]119879


2 1199093minus

4 1199094 1199095minus 1205874 119909

6]119879



References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












and 1199041199021199011198981198861199091198941199051198901199031






and 1199041199021199011198981198861199091198941199051198901199032




119869 of the


119864119869=

10038161003816100381610038161003816100381610038161003816


119869lowast

10038161003816100381610038161003816100381610038161003816

(19)


119870120595= 119864119869+ 120593119891 (20)





119905 119873119901 119873119894 119873119892 119875119898and 119904119902119901119898119886119909119894119905119890119903




119894= 4

and 119875119898119894


= 1198731198922

1198731198941

= 1198731198942 1198731199011

= 9 and 1198731199012














=

311198731199052= 71119873

119892= 300 and119873




1199051= 31 119873

1199052=

71 119873119892





119891= 757 times 10



minus9





120593119891

119870120595

Time

VDPCGA 17404 00912 267 times 10

minus11

00912 50128LI 15949 0 108 times 10



CRPCGA 163 times 10




599 times 10minus9

12405

FFRPCGA 8363 13256 465 times 10

minus3

13302 1413LI 3596 0 122 times 10



minus4 134340









Number 7HPM 02353 01677 420 times 10

minus6 01677 NRLI 02015 0 235 times 10

minus9



Number 8






minus15


Number 9


Number 10



Number 11


minus9



Number 12





285 times 10minus8

54771


Number 13




minus2

70 times 10minus3

632 times 10minus9

70 times 10minus3

49027

Number 14


minus4

276 times 10minus6

267 times 10minus4

115144


Table 1 Continued


120593119891

119870120595

Time

Number 15




minus4

520 times 10minus3

164 times 10minus9

520 times 10minus3

34014

Number 16


minus12

394 times 10minus12

46982


Number 17





minus10

279 times 10minus10

30959

Number 18



Number 19


minus3

01919 164509


Number 20


minus5

225 times 10minus4



aNot reported



1199051= 31 119873

1199052= 71 119873

119892= 300 and

119873119894


1minus 4 119909

2 1199093minus 4 119909



120595for LI



2(119905) + 05 minus 8(119905 minus 05)

2


1199051=

31 1198731199052

= 91 119873119892= 100 and 119873







1(119905) and 119909

2(119905)


1199051=

31 1198731199052

= 51 119873119892= 100 and 119873



lowast






119906right 2 2 10 20 5 2 1 2 1 20119872119894

103


2 mdash 102

120576 10minus12

10minus10


minus9



119906right 5 1 2 120587 1 3 30 1 10 10119872119894

10 102 mdash 10

2

102

102 mdash 10 70 10

minus3

120576 10minus9

10minus11

10minus11 mdash 10

minus10

10minus10

10minus10 mdash 10

minus3

10minus4

0 5 10 15 20 25 30 35 40 45 50017

0175

018

0185

019

0195

02

0205

Iter

Perfo

rman

ce in

dex

(a)

0 01 02 03 04 05 06 07 08 09 1minus25

minus2

minus15

minus1

minus05

0

Time

Ineq

ualit

y co

nstr

aint

(b)



1199051= 31 119873

1199052= 51 119873

119892= 100 119873

119894= 50 with



1 119905) = minus6 minus 119909







119891= 282 times 10



2 4 6 8 10 12 14minus5353

minus5352

minus5351

minus535

minus5349

minus5348

minus5347

minus5346

minus5345

minus5344

minus5343

Iter

Perfo

rman

ce in

dex

(a)

0 05 1 15 2 25 3minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Time

Ineq

ualit

y co

nstr

aint

(b)







120595





1 15 2 25 3 35 40

002

004

006

008

01

012

Iter

Perfo

rman

ce in

dex









5 10 15 20 25165

17

175

18

185

Iter

Perfo

rman

ce in

dex

(a)

0 05 1 15 2 25 3 35 4 45 50

005

01

015

02

025

TimeIn

equa

lity

cons

trai

nt(b)


5 10 15 20 25 30 35 40 45 500155

016

0165

017

0175

018

0185

019

0195

02

0205

Iter

Perfo

rman

ce in

dex

(a)

0 01 02 03 04 05 06 07 08 09 1minus25

minus2

minus15

minus1

minus05

0

Time

Ineq

ualit

y co

nstr

aint

(b)









119869for LI SI










5 10 15 20 25minus025

minus0245

minus024

minus0235

minus023

minus0225

Iter

Perfo

rman

ce in

dex

(a)

1 15 2 25 3 35 4 45 5 55 60

02

04

06

08

1

12

14

16

IterFi

nal c

ondi

tion

times10minus5

(b)


2 4 6 8 10 12 14minus025

minus0245

minus024

minus0235

minus023

minus0225

minus022

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

14

16

18

2

Iter

Fina

l con

ditio

n

times10minus5

(b)






1 2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

07

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

IterFi

nal c

ondi

tion

times10minus3

(b)


5 10 15 20 25 30

35

4

45

5

55

6

65

7

75

8

85

9

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

Iter

Fina

l con

ditio

n

(b)









1199052 the


and 1198731199012 the



5 10 15 20 25 30 35 40 45 50

00955

0096

00965

0097

00975

0098

00985

0099

00995

Iter

Perfo

rman

ce in

dex

(a)

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

35

4

45

IterFi

nal c

ondi

tion

times10minus3

(b)


5 10 15 20 25

21

22

23

24

25

26

27

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 80

1

2

3

4

5

6

Iter

Fina

l con

ditio

n

times10minus3

(b)







1 2 3 4 5 6 7minus887

minus886

minus885

minus884

minus883

minus882

minus881

minus88

minus879

minus878

minus877

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 70

01

02

03

04

05

06

07

08

09

1

IterFi

nal c

ondi

tion

times10minus3

(b)


1 15 2 25 3 35 4 45 5 55 600331003320033200332003320033200333003330033300333

Iter

Perfo

rman

ce in

dex










1199051


119869119870120595




119869 120593119891119870120595 are independent


119891is





119869and


119892 and other



1199011


119892





1 15 2 25 3 35 40

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 80

200

400

600

800

1000

1200

1400

1600

IterFi

nal c

ondi

tion

(b)


2 4 6 8 10 12 14 16 18 20

50

100

150

200

250

300

350

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 10 11 120

05

1

15

2

25

3

Iter

Fina

l con

ditio

n

(b)



1199011=

91198731199012

= 121198731199051= 31119873

1199052= 91119873

119892= 100 and119873

119894= 50 with








Output 1198731199051


119869

11 35 15333 00069 15286 1559821 35 15328 00097 15285 1584931 35 15329 00076 15285 1569941 35 15455 00607 15285 18449

Total 140 15357 00294 15285 18449

120593119891


952 times 10minus8

1 times 10minus12

452 times 10minus7


235 times 10minus7

1 times 10minus12

141 times 10minus6


491 times 10minus8

1 times 10minus12

221 times 10minus7


285 times 10minus7

1 times 10minus12

126 times 10minus6


Time

11 35 897838 272081 456614 155439421 35 1098147 460264 604192 284031031 35 1163591 300227 772673 179510341 35 1448290 393091 810425 2849514

Total 140 1142131 410891 456615 2849514

119864119869

11 35 00031 00045 0 0020421 35 00028 00064 0 0036931 35 00029 00050 0 0027141 35 00111 00397 0 02070

Total 140 00047 00192 0 02070

119870120595

11 35 00031 00045 475 times 10minus9

00204

21 35 00027 00064 550 times 10minus10

00369

31 35 00029 00050 154 times 10minus13

00271

41 35 00111 00397 193 times 10minus10

02070



Parameters 119869 119864119869

120593119891

Time 119870120595


1198731199051

1294 1296 0624 1079 12961198731199052

110572 393 0868 5939 3931198731199011

1945 1835 1271 4317 18351198731199012

2740 2478 1011 2302 2478119873119892

3541 3591 0491 0890 3591119873119894

0495 0309 0301 5849 1046

119875 value

1198731199051

0280 0279 0601 0 02791198731199052

0 0005 0489 0 00051198731199011

0125 0143 0286 0006 01431198731199012

0021 0034 0413 0 0034119873119892

0009 0008 0743 0472 0008119873119894

0739 0819 0877 0 0386



7 The Impact of SQP






119891119870120595

Time 119869 120593119891

119870120595


minus9


minus8




minus8




minus9


minus9




minus9


minus9




minus9



00252 57797 00151 139 times 10minus8

00472 66098Number 14 34761 672 times 10

minus6

00237 104585 34290 757 times 10minus6



111 times 10minus8


minus4

704 times 10minus9




minus10




minus8















Problem no1 2 3 4 5 6 7 8 9 10


120601119891

330 times 10minus9



11 12 13 14 15 16 17 18 19 20119869 48816 minus01273 00153 45727 979 times 10

minus4 23182 minus90882 00336 2513 20274120601119891

167 times 10minus5


minus4

594 times 10minus3


8 Conclusions


Appendix

Test Problems


(1) (VDP) [9] 119892 = (12)(1199092

1+ 1199092

2+ 1199062

) 1199050= 0 119905119891= 5 119891 =

[1199092 minus1199092+(1minus119909

2

1)1199092+119906]119879 1199090= [1 0]

119879120595 = 1199091minus1199092+1

(2) (CRP) [9] 119892 = (12)(1199092

1+1199092

2+01119906

2

) 1199050= 0 119905119891= 078

119891 = [1199091minus2(1199091+025)+(119909

2+05) exp(25119909

1(1199091+2))minus

(1199091+025)119906 05minus119909

2minus(1199092+05) exp(25119909

1(1199091+2))]119879

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(3) (FFRP) [9] 119892 = (12)(1199062

1+ 1199062

2+ 1199062

3+ 1199062

4) 1199050

=

0 119905119891

= 5 119891 = [1199092 (110)((119906

1+ 1199062) cos119909

5minus

(1199062+ 1199064) sin119909

5) 1199094 (110)((119906

1+ 1199063) sin119909

5+ (1199062+

1199064) cos119909

5) 1199096 (512)((119906

1+ 1199063) minus (119906

2+ 1199064))]119879 1199090=

[0 0 0 0 0 0]119879 120595 = [119909

1minus 4 1199092 1199093minus 4 1199094 1199095 1199096]119879

(4) (MSNIC) [20] 120601 = 1199093 1199050= 0 119905119891= 1 119891 = [119909

2 minus1199092+

119906 1199092

1+ 1199092

2+ 0005119906

2

]119879 119889 = [minus(119906 minus 20)(119906 + 20) 119909

2+

05 minus 8(119905 minus 05)2

]119879 1199090= [0 minus1 0]

119879

(5) (CSTCR) [20] 119892 = 1199092

1+ 1199092

2+ 01119906

2 1199050= 0 119905119891= 078

119891 = [minus(2 + 119906)(1199091+ 025) + (119909

2+ 05) exp(25119909

1(1199091+

2)) 05 minus 1199092minus (1199092+ 05) exp(25119909

1(1199091+ 2))]

119879 1199090=

[009 009]119879

(6) [34] 119892 = 21199091 1199050= 0 119905119891= 3 119891 = [119909

2 119906]119879 119889 = [minus(2 +

119906)(2 minus 119906) minus(6 + 1199091)]119879 1199090= [2 0]

119879(7) [36] 119892 = 119906

2 1199050= 0 119905

119891= 1 119891 = (12)119909

2 sin119909 + 1199061199090= 0 120595 = 119909 minus 05

(8) [26] 119892 = 1199092cos2119906 119905

0= 0 119905119891= 120587 119891 = sin(1199062) 119909

0=

1205872(9) [26] 119892 = (12)(119909

2

1+ 1199092

2+ 1199062

) 1199050= 0 119905

119891= 5 119891 =

[1199092 minus1199091+ (1 minus 119909

2

1)1199092+ 119906]119879 119889 = minus(119909

2+ 025) 119909

0=

[1 0]119879

(10) [26] 119892 = 1199092

1+ 1199092

2+ 0005119906

2

1199050= 0 119905

119891= 1 119891 =

[1199092 minus1199092+119906]119879


05) minus 05 minus 1199092)]119879

1199090= [0 minus1]

119879(11) [26] 119892 = (12)119906

2 1199050= 0 119905

119891= 2 119891 = [119909

2 119906]119879 1199090=

[1 1]119879 120595 = [119909

1 1199092]119879

(12) [26] 119892 = minus1199092 1199050= 0 119905

119891= 1 119891 = [119909

2 119906]119879 119889 =

minus(1 minus 119906)(1 + 119906) 1199090= [0 0]

119879 120595 = 1199092

(13) [26] 119892 = (12)(1199092

1+ 1199092

2+ 01119906

2

) 1199050= 0 119905

119891= 078

119891 = [minus2(1199091+ 025) + (119909

2+ 05) exp(25119909

1(1199091+ 2)) minus

(1199091+025)119906 05minus119909

2minus(1199092+05) exp(25119909

1(1199091+2))]119879

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(14) [26] 119892 = (12)1199062 1199050= 0 119905

119891= 10 119891 = [cos 119906 minus

1199092 sin 119906]119879 119889 = minus(120587 + 119906)(120587 minus 119906) 119909

0= [366 minus186]

119879120595 = [119909

1 1199092]119879

(15) [26]119892 = (12)(1199092

1+1199092

2) 1199050= 0 119905119891= 078119891 = [minus2(119909

1+

025)+(1199092+05) exp(25119909

1(1199091+2))minus(119909

1+025)119906 05minus

1199092minus(1199092+05) exp(25119909

1(1199091+2))]119879 119889 = minus(1minus119906)(1+119906)

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(16) [26] 120601 = 1199093 1199050= 0 119905

119891= 1 119891 = [119909

2 119906 (12)119906

2

]119879

119889 = 1199091minus 19 119909

0= [0 0 0]

119879 120595 = [1199091 1199092+ 1]119879

(17) [26] 120601 = minus1199093 1199050= 0 119905

119891= 5 119891 = [119909

2 minus2 + 119906119909

3


0= [10 minus2 10]

119879120595 = [119909

1 1199092]119879

(18) [26] 120601 = (1199091minus1)2

+1199092

2+1199092

3 119892 = (12)119906

2 1199050= 0 119905119891= 5

119891 = [1199093cos 119906 119909

3sin 119906 sin 119906]119879 119909

0= [0 0 0]

119879


(19) [26] 119892 = 45(1199092

3+ 1199092

6) + 05(119906

2

1+ 1199062

2) 1199050= 0 119905

119891= 1

119891 = [91199094 91199095 91199096 9(1199061+ 1725119909

3) 91199062 minus(9119909

2)(1199061minus

270751199093+ 211990951199096)]119879 1199090= [0 22 0 0 minus1 0]

119879 120595 =

[1199091minus 10 119909

2minus 14 119909

3 1199094minus 25 119909

5 1199096]119879


2 1199093minus

4 1199094 1199095minus 1205874 119909

6]119879



References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












120593119891

119870120595

Time

VDPCGA 17404 00912 267 times 10

minus11

00912 50128LI 15949 0 108 times 10



CRPCGA 163 times 10




599 times 10minus9

12405

FFRPCGA 8363 13256 465 times 10

minus3

13302 1413LI 3596 0 122 times 10



minus4 134340









Number 7HPM 02353 01677 420 times 10

minus6 01677 NRLI 02015 0 235 times 10

minus9



Number 8






minus15


Number 9


Number 10



Number 11


minus9



Number 12





285 times 10minus8

54771


Number 13




minus2

70 times 10minus3

632 times 10minus9

70 times 10minus3

49027

Number 14


minus4

276 times 10minus6

267 times 10minus4

115144


Table 1 Continued


120593119891

119870120595

Time

Number 15




minus4

520 times 10minus3

164 times 10minus9

520 times 10minus3

34014

Number 16


minus12

394 times 10minus12

46982


Number 17





minus10

279 times 10minus10

30959

Number 18



Number 19


minus3

01919 164509


Number 20


minus5

225 times 10minus4



aNot reported



1199051= 31 119873

1199052= 71 119873

119892= 300 and

119873119894


1minus 4 119909

2 1199093minus 4 119909



120595for LI



2(119905) + 05 minus 8(119905 minus 05)

2


1199051=

31 1198731199052

= 91 119873119892= 100 and 119873







1(119905) and 119909

2(119905)


1199051=

31 1198731199052

= 51 119873119892= 100 and 119873



lowast






119906right 2 2 10 20 5 2 1 2 1 20119872119894

103


2 mdash 102

120576 10minus12

10minus10


minus9



119906right 5 1 2 120587 1 3 30 1 10 10119872119894

10 102 mdash 10

2

102

102 mdash 10 70 10

minus3

120576 10minus9

10minus11

10minus11 mdash 10

minus10

10minus10

10minus10 mdash 10

minus3

10minus4

0 5 10 15 20 25 30 35 40 45 50017

0175

018

0185

019

0195

02

0205

Iter

Perfo

rman

ce in

dex

(a)

0 01 02 03 04 05 06 07 08 09 1minus25

minus2

minus15

minus1

minus05

0

Time

Ineq

ualit

y co

nstr

aint

(b)



1199051= 31 119873

1199052= 51 119873

119892= 100 119873

119894= 50 with



1 119905) = minus6 minus 119909







119891= 282 times 10



2 4 6 8 10 12 14minus5353

minus5352

minus5351

minus535

minus5349

minus5348

minus5347

minus5346

minus5345

minus5344

minus5343

Iter

Perfo

rman

ce in

dex

(a)

0 05 1 15 2 25 3minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Time

Ineq

ualit

y co

nstr

aint

(b)







120595





1 15 2 25 3 35 40

002

004

006

008

01

012

Iter

Perfo

rman

ce in

dex









5 10 15 20 25165

17

175

18

185

Iter

Perfo

rman

ce in

dex

(a)

0 05 1 15 2 25 3 35 4 45 50

005

01

015

02

025

TimeIn

equa

lity

cons

trai

nt(b)


5 10 15 20 25 30 35 40 45 500155

016

0165

017

0175

018

0185

019

0195

02

0205

Iter

Perfo

rman

ce in

dex

(a)

0 01 02 03 04 05 06 07 08 09 1minus25

minus2

minus15

minus1

minus05

0

Time

Ineq

ualit

y co

nstr

aint

(b)









119869for LI SI










5 10 15 20 25minus025

minus0245

minus024

minus0235

minus023

minus0225

Iter

Perfo

rman

ce in

dex

(a)

1 15 2 25 3 35 4 45 5 55 60

02

04

06

08

1

12

14

16

IterFi

nal c

ondi

tion

times10minus5

(b)


2 4 6 8 10 12 14minus025

minus0245

minus024

minus0235

minus023

minus0225

minus022

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

14

16

18

2

Iter

Fina

l con

ditio

n

times10minus5

(b)






1 2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

07

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

IterFi

nal c

ondi

tion

times10minus3

(b)


5 10 15 20 25 30

35

4

45

5

55

6

65

7

75

8

85

9

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

Iter

Fina

l con

ditio

n

(b)









1199052 the


and 1198731199012 the



5 10 15 20 25 30 35 40 45 50

00955

0096

00965

0097

00975

0098

00985

0099

00995

Iter

Perfo

rman

ce in

dex

(a)

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

35

4

45

IterFi

nal c

ondi

tion

times10minus3

(b)


5 10 15 20 25

21

22

23

24

25

26

27

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 80

1

2

3

4

5

6

Iter

Fina

l con

ditio

n

times10minus3

(b)







1 2 3 4 5 6 7minus887

minus886

minus885

minus884

minus883

minus882

minus881

minus88

minus879

minus878

minus877

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 70

01

02

03

04

05

06

07

08

09

1

IterFi

nal c

ondi

tion

times10minus3

(b)


1 15 2 25 3 35 4 45 5 55 600331003320033200332003320033200333003330033300333

Iter

Perfo

rman

ce in

dex










1199051


119869119870120595




119869 120593119891119870120595 are independent


119891is





119869and


119892 and other



1199011


119892





1 15 2 25 3 35 40

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 80

200

400

600

800

1000

1200

1400

1600

IterFi

nal c

ondi

tion

(b)


2 4 6 8 10 12 14 16 18 20

50

100

150

200

250

300

350

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 10 11 120

05

1

15

2

25

3

Iter

Fina

l con

ditio

n

(b)



1199011=

91198731199012

= 121198731199051= 31119873

1199052= 91119873

119892= 100 and119873

119894= 50 with








Output 1198731199051


119869

11 35 15333 00069 15286 1559821 35 15328 00097 15285 1584931 35 15329 00076 15285 1569941 35 15455 00607 15285 18449

Total 140 15357 00294 15285 18449

120593119891


952 times 10minus8

1 times 10minus12

452 times 10minus7


235 times 10minus7

1 times 10minus12

141 times 10minus6


491 times 10minus8

1 times 10minus12

221 times 10minus7


285 times 10minus7

1 times 10minus12

126 times 10minus6


Time

11 35 897838 272081 456614 155439421 35 1098147 460264 604192 284031031 35 1163591 300227 772673 179510341 35 1448290 393091 810425 2849514

Total 140 1142131 410891 456615 2849514

119864119869

11 35 00031 00045 0 0020421 35 00028 00064 0 0036931 35 00029 00050 0 0027141 35 00111 00397 0 02070

Total 140 00047 00192 0 02070

119870120595

11 35 00031 00045 475 times 10minus9

00204

21 35 00027 00064 550 times 10minus10

00369

31 35 00029 00050 154 times 10minus13

00271

41 35 00111 00397 193 times 10minus10

02070



Parameters 119869 119864119869

120593119891

Time 119870120595


1198731199051

1294 1296 0624 1079 12961198731199052

110572 393 0868 5939 3931198731199011

1945 1835 1271 4317 18351198731199012

2740 2478 1011 2302 2478119873119892

3541 3591 0491 0890 3591119873119894

0495 0309 0301 5849 1046

119875 value

1198731199051

0280 0279 0601 0 02791198731199052

0 0005 0489 0 00051198731199011

0125 0143 0286 0006 01431198731199012

0021 0034 0413 0 0034119873119892

0009 0008 0743 0472 0008119873119894

0739 0819 0877 0 0386



7 The Impact of SQP






119891119870120595

Time 119869 120593119891

119870120595


minus9


minus8




minus8




minus9


minus9




minus9


minus9




minus9



00252 57797 00151 139 times 10minus8

00472 66098Number 14 34761 672 times 10

minus6

00237 104585 34290 757 times 10minus6



111 times 10minus8


minus4

704 times 10minus9




minus10




minus8















Problem no1 2 3 4 5 6 7 8 9 10


120601119891

330 times 10minus9



11 12 13 14 15 16 17 18 19 20119869 48816 minus01273 00153 45727 979 times 10

minus4 23182 minus90882 00336 2513 20274120601119891

167 times 10minus5


minus4

594 times 10minus3


8 Conclusions


Appendix

Test Problems


(1) (VDP) [9] 119892 = (12)(1199092

1+ 1199092

2+ 1199062

) 1199050= 0 119905119891= 5 119891 =

[1199092 minus1199092+(1minus119909

2

1)1199092+119906]119879 1199090= [1 0]

119879120595 = 1199091minus1199092+1

(2) (CRP) [9] 119892 = (12)(1199092

1+1199092

2+01119906

2

) 1199050= 0 119905119891= 078

119891 = [1199091minus2(1199091+025)+(119909

2+05) exp(25119909

1(1199091+2))minus

(1199091+025)119906 05minus119909

2minus(1199092+05) exp(25119909

1(1199091+2))]119879

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(3) (FFRP) [9] 119892 = (12)(1199062

1+ 1199062

2+ 1199062

3+ 1199062

4) 1199050

=

0 119905119891

= 5 119891 = [1199092 (110)((119906

1+ 1199062) cos119909

5minus

(1199062+ 1199064) sin119909

5) 1199094 (110)((119906

1+ 1199063) sin119909

5+ (1199062+

1199064) cos119909

5) 1199096 (512)((119906

1+ 1199063) minus (119906

2+ 1199064))]119879 1199090=

[0 0 0 0 0 0]119879 120595 = [119909

1minus 4 1199092 1199093minus 4 1199094 1199095 1199096]119879

(4) (MSNIC) [20] 120601 = 1199093 1199050= 0 119905119891= 1 119891 = [119909

2 minus1199092+

119906 1199092

1+ 1199092

2+ 0005119906

2

]119879 119889 = [minus(119906 minus 20)(119906 + 20) 119909

2+

05 minus 8(119905 minus 05)2

]119879 1199090= [0 minus1 0]

119879

(5) (CSTCR) [20] 119892 = 1199092

1+ 1199092

2+ 01119906

2 1199050= 0 119905119891= 078

119891 = [minus(2 + 119906)(1199091+ 025) + (119909

2+ 05) exp(25119909

1(1199091+

2)) 05 minus 1199092minus (1199092+ 05) exp(25119909

1(1199091+ 2))]

119879 1199090=

[009 009]119879

(6) [34] 119892 = 21199091 1199050= 0 119905119891= 3 119891 = [119909

2 119906]119879 119889 = [minus(2 +

119906)(2 minus 119906) minus(6 + 1199091)]119879 1199090= [2 0]

119879(7) [36] 119892 = 119906

2 1199050= 0 119905

119891= 1 119891 = (12)119909

2 sin119909 + 1199061199090= 0 120595 = 119909 minus 05

(8) [26] 119892 = 1199092cos2119906 119905

0= 0 119905119891= 120587 119891 = sin(1199062) 119909

0=

1205872(9) [26] 119892 = (12)(119909

2

1+ 1199092

2+ 1199062

) 1199050= 0 119905

119891= 5 119891 =

[1199092 minus1199091+ (1 minus 119909

2

1)1199092+ 119906]119879 119889 = minus(119909

2+ 025) 119909

0=

[1 0]119879

(10) [26] 119892 = 1199092

1+ 1199092

2+ 0005119906

2

1199050= 0 119905

119891= 1 119891 =

[1199092 minus1199092+119906]119879


05) minus 05 minus 1199092)]119879

1199090= [0 minus1]

119879(11) [26] 119892 = (12)119906

2 1199050= 0 119905

119891= 2 119891 = [119909

2 119906]119879 1199090=

[1 1]119879 120595 = [119909

1 1199092]119879

(12) [26] 119892 = minus1199092 1199050= 0 119905

119891= 1 119891 = [119909

2 119906]119879 119889 =

minus(1 minus 119906)(1 + 119906) 1199090= [0 0]

119879 120595 = 1199092

(13) [26] 119892 = (12)(1199092

1+ 1199092

2+ 01119906

2

) 1199050= 0 119905

119891= 078

119891 = [minus2(1199091+ 025) + (119909

2+ 05) exp(25119909

1(1199091+ 2)) minus

(1199091+025)119906 05minus119909

2minus(1199092+05) exp(25119909

1(1199091+2))]119879

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(14) [26] 119892 = (12)1199062 1199050= 0 119905

119891= 10 119891 = [cos 119906 minus

1199092 sin 119906]119879 119889 = minus(120587 + 119906)(120587 minus 119906) 119909

0= [366 minus186]

119879120595 = [119909

1 1199092]119879

(15) [26]119892 = (12)(1199092

1+1199092

2) 1199050= 0 119905119891= 078119891 = [minus2(119909

1+

025)+(1199092+05) exp(25119909

1(1199091+2))minus(119909

1+025)119906 05minus

1199092minus(1199092+05) exp(25119909

1(1199091+2))]119879 119889 = minus(1minus119906)(1+119906)

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(16) [26] 120601 = 1199093 1199050= 0 119905

119891= 1 119891 = [119909

2 119906 (12)119906

2

]119879

119889 = 1199091minus 19 119909

0= [0 0 0]

119879 120595 = [1199091 1199092+ 1]119879

(17) [26] 120601 = minus1199093 1199050= 0 119905

119891= 5 119891 = [119909

2 minus2 + 119906119909

3


0= [10 minus2 10]

119879120595 = [119909

1 1199092]119879

(18) [26] 120601 = (1199091minus1)2

+1199092

2+1199092

3 119892 = (12)119906

2 1199050= 0 119905119891= 5

119891 = [1199093cos 119906 119909

3sin 119906 sin 119906]119879 119909

0= [0 0 0]

119879


(19) [26] 119892 = 45(1199092

3+ 1199092

6) + 05(119906

2

1+ 1199062

2) 1199050= 0 119905

119891= 1

119891 = [91199094 91199095 91199096 9(1199061+ 1725119909

3) 91199062 minus(9119909

2)(1199061minus

270751199093+ 211990951199096)]119879 1199090= [0 22 0 0 minus1 0]

119879 120595 =

[1199091minus 10 119909

2minus 14 119909

3 1199094minus 25 119909

5 1199096]119879


2 1199093minus

4 1199094 1199095minus 1205874 119909

6]119879



References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table 1 Continued


120593119891

119870120595

Time

Number 15




minus4

520 times 10minus3

164 times 10minus9

520 times 10minus3

34014

Number 16


minus12

394 times 10minus12

46982


Number 17





minus10

279 times 10minus10

30959

Number 18



Number 19


minus3

01919 164509


Number 20


minus5

225 times 10minus4



aNot reported



1199051= 31 119873

1199052= 71 119873

119892= 300 and

119873119894


1minus 4 119909

2 1199093minus 4 119909



120595for LI



2(119905) + 05 minus 8(119905 minus 05)

2


1199051=

31 1198731199052

= 91 119873119892= 100 and 119873







1(119905) and 119909

2(119905)


1199051=

31 1198731199052

= 51 119873119892= 100 and 119873



lowast






119906right 2 2 10 20 5 2 1 2 1 20119872119894

103


2 mdash 102

120576 10minus12

10minus10


minus9



119906right 5 1 2 120587 1 3 30 1 10 10119872119894

10 102 mdash 10

2

102

102 mdash 10 70 10

minus3

120576 10minus9

10minus11

10minus11 mdash 10

minus10

10minus10

10minus10 mdash 10

minus3

10minus4

0 5 10 15 20 25 30 35 40 45 50017

0175

018

0185

019

0195

02

0205

Iter

Perfo

rman

ce in

dex

(a)

0 01 02 03 04 05 06 07 08 09 1minus25

minus2

minus15

minus1

minus05

0

Time

Ineq

ualit

y co

nstr

aint

(b)



1199051= 31 119873

1199052= 51 119873

119892= 100 119873

119894= 50 with



1 119905) = minus6 minus 119909







119891= 282 times 10



2 4 6 8 10 12 14minus5353

minus5352

minus5351

minus535

minus5349

minus5348

minus5347

minus5346

minus5345

minus5344

minus5343

Iter

Perfo

rman

ce in

dex

(a)

0 05 1 15 2 25 3minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Time

Ineq

ualit

y co

nstr

aint

(b)







120595





1 15 2 25 3 35 40

002

004

006

008

01

012

Iter

Perfo

rman

ce in

dex









5 10 15 20 25165

17

175

18

185

Iter

Perfo

rman

ce in

dex

(a)

0 05 1 15 2 25 3 35 4 45 50

005

01

015

02

025

TimeIn

equa

lity

cons

trai

nt(b)


5 10 15 20 25 30 35 40 45 500155

016

0165

017

0175

018

0185

019

0195

02

0205

Iter

Perfo

rman

ce in

dex

(a)

0 01 02 03 04 05 06 07 08 09 1minus25

minus2

minus15

minus1

minus05

0

Time

Ineq

ualit

y co

nstr

aint

(b)









119869for LI SI










5 10 15 20 25minus025

minus0245

minus024

minus0235

minus023

minus0225

Iter

Perfo

rman

ce in

dex

(a)

1 15 2 25 3 35 4 45 5 55 60

02

04

06

08

1

12

14

16

IterFi

nal c

ondi

tion

times10minus5

(b)


2 4 6 8 10 12 14minus025

minus0245

minus024

minus0235

minus023

minus0225

minus022

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

14

16

18

2

Iter

Fina

l con

ditio

n

times10minus5

(b)






1 2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

07

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

IterFi

nal c

ondi

tion

times10minus3

(b)


5 10 15 20 25 30

35

4

45

5

55

6

65

7

75

8

85

9

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

Iter

Fina

l con

ditio

n

(b)









1199052 the


and 1198731199012 the



5 10 15 20 25 30 35 40 45 50

00955

0096

00965

0097

00975

0098

00985

0099

00995

Iter

Perfo

rman

ce in

dex

(a)

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

35

4

45

IterFi

nal c

ondi

tion

times10minus3

(b)


5 10 15 20 25

21

22

23

24

25

26

27

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 80

1

2

3

4

5

6

Iter

Fina

l con

ditio

n

times10minus3

(b)







1 2 3 4 5 6 7minus887

minus886

minus885

minus884

minus883

minus882

minus881

minus88

minus879

minus878

minus877

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 70

01

02

03

04

05

06

07

08

09

1

IterFi

nal c

ondi

tion

times10minus3

(b)


1 15 2 25 3 35 4 45 5 55 600331003320033200332003320033200333003330033300333

Iter

Perfo

rman

ce in

dex










1199051


119869119870120595




119869 120593119891119870120595 are independent


119891is





119869and


119892 and other



1199011


119892





1 15 2 25 3 35 40

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 80

200

400

600

800

1000

1200

1400

1600

IterFi

nal c

ondi

tion

(b)


2 4 6 8 10 12 14 16 18 20

50

100

150

200

250

300

350

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 10 11 120

05

1

15

2

25

3

Iter

Fina

l con

ditio

n

(b)



1199011=

91198731199012

= 121198731199051= 31119873

1199052= 91119873

119892= 100 and119873

119894= 50 with








Output 1198731199051


119869

11 35 15333 00069 15286 1559821 35 15328 00097 15285 1584931 35 15329 00076 15285 1569941 35 15455 00607 15285 18449

Total 140 15357 00294 15285 18449

120593119891


952 times 10minus8

1 times 10minus12

452 times 10minus7


235 times 10minus7

1 times 10minus12

141 times 10minus6


491 times 10minus8

1 times 10minus12

221 times 10minus7


285 times 10minus7

1 times 10minus12

126 times 10minus6


Time

11 35 897838 272081 456614 155439421 35 1098147 460264 604192 284031031 35 1163591 300227 772673 179510341 35 1448290 393091 810425 2849514

Total 140 1142131 410891 456615 2849514

119864119869

11 35 00031 00045 0 0020421 35 00028 00064 0 0036931 35 00029 00050 0 0027141 35 00111 00397 0 02070

Total 140 00047 00192 0 02070

119870120595

11 35 00031 00045 475 times 10minus9

00204

21 35 00027 00064 550 times 10minus10

00369

31 35 00029 00050 154 times 10minus13

00271

41 35 00111 00397 193 times 10minus10

02070



Parameters 119869 119864119869

120593119891

Time 119870120595


1198731199051

1294 1296 0624 1079 12961198731199052

110572 393 0868 5939 3931198731199011

1945 1835 1271 4317 18351198731199012

2740 2478 1011 2302 2478119873119892

3541 3591 0491 0890 3591119873119894

0495 0309 0301 5849 1046

119875 value

1198731199051

0280 0279 0601 0 02791198731199052

0 0005 0489 0 00051198731199011

0125 0143 0286 0006 01431198731199012

0021 0034 0413 0 0034119873119892

0009 0008 0743 0472 0008119873119894

0739 0819 0877 0 0386



7 The Impact of SQP






119891119870120595

Time 119869 120593119891

119870120595


minus9


minus8




minus8




minus9


minus9




minus9


minus9




minus9



00252 57797 00151 139 times 10minus8

00472 66098Number 14 34761 672 times 10

minus6

00237 104585 34290 757 times 10minus6



111 times 10minus8


minus4

704 times 10minus9




minus10




minus8















Problem no1 2 3 4 5 6 7 8 9 10


120601119891

330 times 10minus9



11 12 13 14 15 16 17 18 19 20119869 48816 minus01273 00153 45727 979 times 10

minus4 23182 minus90882 00336 2513 20274120601119891

167 times 10minus5


minus4

594 times 10minus3


8 Conclusions


Appendix

Test Problems


(1) (VDP) [9] 119892 = (12)(1199092

1+ 1199092

2+ 1199062

) 1199050= 0 119905119891= 5 119891 =

[1199092 minus1199092+(1minus119909

2

1)1199092+119906]119879 1199090= [1 0]

119879120595 = 1199091minus1199092+1

(2) (CRP) [9] 119892 = (12)(1199092

1+1199092

2+01119906

2

) 1199050= 0 119905119891= 078

119891 = [1199091minus2(1199091+025)+(119909

2+05) exp(25119909

1(1199091+2))minus

(1199091+025)119906 05minus119909

2minus(1199092+05) exp(25119909

1(1199091+2))]119879

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(3) (FFRP) [9] 119892 = (12)(1199062

1+ 1199062

2+ 1199062

3+ 1199062

4) 1199050

=

0 119905119891

= 5 119891 = [1199092 (110)((119906

1+ 1199062) cos119909

5minus

(1199062+ 1199064) sin119909

5) 1199094 (110)((119906

1+ 1199063) sin119909

5+ (1199062+

1199064) cos119909

5) 1199096 (512)((119906

1+ 1199063) minus (119906

2+ 1199064))]119879 1199090=

[0 0 0 0 0 0]119879 120595 = [119909

1minus 4 1199092 1199093minus 4 1199094 1199095 1199096]119879

(4) (MSNIC) [20] 120601 = 1199093 1199050= 0 119905119891= 1 119891 = [119909

2 minus1199092+

119906 1199092

1+ 1199092

2+ 0005119906

2

]119879 119889 = [minus(119906 minus 20)(119906 + 20) 119909

2+

05 minus 8(119905 minus 05)2

]119879 1199090= [0 minus1 0]

119879

(5) (CSTCR) [20] 119892 = 1199092

1+ 1199092

2+ 01119906

2 1199050= 0 119905119891= 078

119891 = [minus(2 + 119906)(1199091+ 025) + (119909

2+ 05) exp(25119909

1(1199091+

2)) 05 minus 1199092minus (1199092+ 05) exp(25119909

1(1199091+ 2))]

119879 1199090=

[009 009]119879

(6) [34] 119892 = 21199091 1199050= 0 119905119891= 3 119891 = [119909

2 119906]119879 119889 = [minus(2 +

119906)(2 minus 119906) minus(6 + 1199091)]119879 1199090= [2 0]

119879(7) [36] 119892 = 119906

2 1199050= 0 119905

119891= 1 119891 = (12)119909

2 sin119909 + 1199061199090= 0 120595 = 119909 minus 05

(8) [26] 119892 = 1199092cos2119906 119905

0= 0 119905119891= 120587 119891 = sin(1199062) 119909

0=

1205872(9) [26] 119892 = (12)(119909

2

1+ 1199092

2+ 1199062

) 1199050= 0 119905

119891= 5 119891 =

[1199092 minus1199091+ (1 minus 119909

2

1)1199092+ 119906]119879 119889 = minus(119909

2+ 025) 119909

0=

[1 0]119879

(10) [26] 119892 = 1199092

1+ 1199092

2+ 0005119906

2

1199050= 0 119905

119891= 1 119891 =

[1199092 minus1199092+119906]119879


05) minus 05 minus 1199092)]119879

1199090= [0 minus1]

119879(11) [26] 119892 = (12)119906

2 1199050= 0 119905

119891= 2 119891 = [119909

2 119906]119879 1199090=

[1 1]119879 120595 = [119909

1 1199092]119879

(12) [26] 119892 = minus1199092 1199050= 0 119905

119891= 1 119891 = [119909

2 119906]119879 119889 =

minus(1 minus 119906)(1 + 119906) 1199090= [0 0]

119879 120595 = 1199092

(13) [26] 119892 = (12)(1199092

1+ 1199092

2+ 01119906

2

) 1199050= 0 119905

119891= 078

119891 = [minus2(1199091+ 025) + (119909

2+ 05) exp(25119909

1(1199091+ 2)) minus

(1199091+025)119906 05minus119909

2minus(1199092+05) exp(25119909

1(1199091+2))]119879

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(14) [26] 119892 = (12)1199062 1199050= 0 119905

119891= 10 119891 = [cos 119906 minus

1199092 sin 119906]119879 119889 = minus(120587 + 119906)(120587 minus 119906) 119909

0= [366 minus186]

119879120595 = [119909

1 1199092]119879

(15) [26]119892 = (12)(1199092

1+1199092

2) 1199050= 0 119905119891= 078119891 = [minus2(119909

1+

025)+(1199092+05) exp(25119909

1(1199091+2))minus(119909

1+025)119906 05minus

1199092minus(1199092+05) exp(25119909

1(1199091+2))]119879 119889 = minus(1minus119906)(1+119906)

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(16) [26] 120601 = 1199093 1199050= 0 119905

119891= 1 119891 = [119909

2 119906 (12)119906

2

]119879

119889 = 1199091minus 19 119909

0= [0 0 0]

119879 120595 = [1199091 1199092+ 1]119879

(17) [26] 120601 = minus1199093 1199050= 0 119905

119891= 5 119891 = [119909

2 minus2 + 119906119909

3


0= [10 minus2 10]

119879120595 = [119909

1 1199092]119879

(18) [26] 120601 = (1199091minus1)2

+1199092

2+1199092

3 119892 = (12)119906

2 1199050= 0 119905119891= 5

119891 = [1199093cos 119906 119909

3sin 119906 sin 119906]119879 119909

0= [0 0 0]

119879


(19) [26] 119892 = 45(1199092

3+ 1199092

6) + 05(119906

2

1+ 1199062

2) 1199050= 0 119905

119891= 1

119891 = [91199094 91199095 91199096 9(1199061+ 1725119909

3) 91199062 minus(9119909

2)(1199061minus

270751199093+ 211990951199096)]119879 1199090= [0 22 0 0 minus1 0]

119879 120595 =

[1199091minus 10 119909

2minus 14 119909

3 1199094minus 25 119909

5 1199096]119879


2 1199093minus

4 1199094 1199095minus 1205874 119909

6]119879



References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119906right 2 2 10 20 5 2 1 2 1 20119872119894

103


2 mdash 102

120576 10minus12

10minus10


minus9



119906right 5 1 2 120587 1 3 30 1 10 10119872119894

10 102 mdash 10

2

102

102 mdash 10 70 10

minus3

120576 10minus9

10minus11

10minus11 mdash 10

minus10

10minus10

10minus10 mdash 10

minus3

10minus4

0 5 10 15 20 25 30 35 40 45 50017

0175

018

0185

019

0195

02

0205

Iter

Perfo

rman

ce in

dex

(a)

0 01 02 03 04 05 06 07 08 09 1minus25

minus2

minus15

minus1

minus05

0

Time

Ineq

ualit

y co

nstr

aint

(b)



1199051= 31 119873

1199052= 51 119873

119892= 100 119873

119894= 50 with



1 119905) = minus6 minus 119909







119891= 282 times 10



2 4 6 8 10 12 14minus5353

minus5352

minus5351

minus535

minus5349

minus5348

minus5347

minus5346

minus5345

minus5344

minus5343

Iter

Perfo

rman

ce in

dex

(a)

0 05 1 15 2 25 3minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Time

Ineq

ualit

y co

nstr

aint

(b)







120595





1 15 2 25 3 35 40

002

004

006

008

01

012

Iter

Perfo

rman

ce in

dex









5 10 15 20 25165

17

175

18

185

Iter

Perfo

rman

ce in

dex

(a)

0 05 1 15 2 25 3 35 4 45 50

005

01

015

02

025

TimeIn

equa

lity

cons

trai

nt(b)


5 10 15 20 25 30 35 40 45 500155

016

0165

017

0175

018

0185

019

0195

02

0205

Iter

Perfo

rman

ce in

dex

(a)

0 01 02 03 04 05 06 07 08 09 1minus25

minus2

minus15

minus1

minus05

0

Time

Ineq

ualit

y co

nstr

aint

(b)









119869for LI SI










5 10 15 20 25minus025

minus0245

minus024

minus0235

minus023

minus0225

Iter

Perfo

rman

ce in

dex

(a)

1 15 2 25 3 35 4 45 5 55 60

02

04

06

08

1

12

14

16

IterFi

nal c

ondi

tion

times10minus5

(b)


2 4 6 8 10 12 14minus025

minus0245

minus024

minus0235

minus023

minus0225

minus022

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

14

16

18

2

Iter

Fina

l con

ditio

n

times10minus5

(b)






1 2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

07

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

IterFi

nal c

ondi

tion

times10minus3

(b)


5 10 15 20 25 30

35

4

45

5

55

6

65

7

75

8

85

9

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

Iter

Fina

l con

ditio

n

(b)









1199052 the


and 1198731199012 the



5 10 15 20 25 30 35 40 45 50

00955

0096

00965

0097

00975

0098

00985

0099

00995

Iter

Perfo

rman

ce in

dex

(a)

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

35

4

45

IterFi

nal c

ondi

tion

times10minus3

(b)


5 10 15 20 25

21

22

23

24

25

26

27

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 80

1

2

3

4

5

6

Iter

Fina

l con

ditio

n

times10minus3

(b)







1 2 3 4 5 6 7minus887

minus886

minus885

minus884

minus883

minus882

minus881

minus88

minus879

minus878

minus877

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 70

01

02

03

04

05

06

07

08

09

1

IterFi

nal c

ondi

tion

times10minus3

(b)


1 15 2 25 3 35 4 45 5 55 600331003320033200332003320033200333003330033300333

Iter

Perfo

rman

ce in

dex










1199051


119869119870120595




119869 120593119891119870120595 are independent


119891is





119869and


119892 and other



1199011


119892





1 15 2 25 3 35 40

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 80

200

400

600

800

1000

1200

1400

1600

IterFi

nal c

ondi

tion

(b)


2 4 6 8 10 12 14 16 18 20

50

100

150

200

250

300

350

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 10 11 120

05

1

15

2

25

3

Iter

Fina

l con

ditio

n

(b)



1199011=

91198731199012

= 121198731199051= 31119873

1199052= 91119873

119892= 100 and119873

119894= 50 with








Output 1198731199051


119869

11 35 15333 00069 15286 1559821 35 15328 00097 15285 1584931 35 15329 00076 15285 1569941 35 15455 00607 15285 18449

Total 140 15357 00294 15285 18449

120593119891


952 times 10minus8

1 times 10minus12

452 times 10minus7


235 times 10minus7

1 times 10minus12

141 times 10minus6


491 times 10minus8

1 times 10minus12

221 times 10minus7


285 times 10minus7

1 times 10minus12

126 times 10minus6


Time

11 35 897838 272081 456614 155439421 35 1098147 460264 604192 284031031 35 1163591 300227 772673 179510341 35 1448290 393091 810425 2849514

Total 140 1142131 410891 456615 2849514

119864119869

11 35 00031 00045 0 0020421 35 00028 00064 0 0036931 35 00029 00050 0 0027141 35 00111 00397 0 02070

Total 140 00047 00192 0 02070

119870120595

11 35 00031 00045 475 times 10minus9

00204

21 35 00027 00064 550 times 10minus10

00369

31 35 00029 00050 154 times 10minus13

00271

41 35 00111 00397 193 times 10minus10

02070



Parameters 119869 119864119869

120593119891

Time 119870120595


1198731199051

1294 1296 0624 1079 12961198731199052

110572 393 0868 5939 3931198731199011

1945 1835 1271 4317 18351198731199012

2740 2478 1011 2302 2478119873119892

3541 3591 0491 0890 3591119873119894

0495 0309 0301 5849 1046

119875 value

1198731199051

0280 0279 0601 0 02791198731199052

0 0005 0489 0 00051198731199011

0125 0143 0286 0006 01431198731199012

0021 0034 0413 0 0034119873119892

0009 0008 0743 0472 0008119873119894

0739 0819 0877 0 0386



7 The Impact of SQP






119891119870120595

Time 119869 120593119891

119870120595


minus9


minus8




minus8




minus9


minus9




minus9


minus9




minus9



00252 57797 00151 139 times 10minus8

00472 66098Number 14 34761 672 times 10

minus6

00237 104585 34290 757 times 10minus6



111 times 10minus8


minus4

704 times 10minus9




minus10




minus8















Problem no1 2 3 4 5 6 7 8 9 10


120601119891

330 times 10minus9



11 12 13 14 15 16 17 18 19 20119869 48816 minus01273 00153 45727 979 times 10

minus4 23182 minus90882 00336 2513 20274120601119891

167 times 10minus5


minus4

594 times 10minus3


8 Conclusions


Appendix

Test Problems


(1) (VDP) [9] 119892 = (12)(1199092

1+ 1199092

2+ 1199062

) 1199050= 0 119905119891= 5 119891 =

[1199092 minus1199092+(1minus119909

2

1)1199092+119906]119879 1199090= [1 0]

119879120595 = 1199091minus1199092+1

(2) (CRP) [9] 119892 = (12)(1199092

1+1199092

2+01119906

2

) 1199050= 0 119905119891= 078

119891 = [1199091minus2(1199091+025)+(119909

2+05) exp(25119909

1(1199091+2))minus

(1199091+025)119906 05minus119909

2minus(1199092+05) exp(25119909

1(1199091+2))]119879

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(3) (FFRP) [9] 119892 = (12)(1199062

1+ 1199062

2+ 1199062

3+ 1199062

4) 1199050

=

0 119905119891

= 5 119891 = [1199092 (110)((119906

1+ 1199062) cos119909

5minus

(1199062+ 1199064) sin119909

5) 1199094 (110)((119906

1+ 1199063) sin119909

5+ (1199062+

1199064) cos119909

5) 1199096 (512)((119906

1+ 1199063) minus (119906

2+ 1199064))]119879 1199090=

[0 0 0 0 0 0]119879 120595 = [119909

1minus 4 1199092 1199093minus 4 1199094 1199095 1199096]119879

(4) (MSNIC) [20] 120601 = 1199093 1199050= 0 119905119891= 1 119891 = [119909

2 minus1199092+

119906 1199092

1+ 1199092

2+ 0005119906

2

]119879 119889 = [minus(119906 minus 20)(119906 + 20) 119909

2+

05 minus 8(119905 minus 05)2

]119879 1199090= [0 minus1 0]

119879

(5) (CSTCR) [20] 119892 = 1199092

1+ 1199092

2+ 01119906

2 1199050= 0 119905119891= 078

119891 = [minus(2 + 119906)(1199091+ 025) + (119909

2+ 05) exp(25119909

1(1199091+

2)) 05 minus 1199092minus (1199092+ 05) exp(25119909

1(1199091+ 2))]

119879 1199090=

[009 009]119879

(6) [34] 119892 = 21199091 1199050= 0 119905119891= 3 119891 = [119909

2 119906]119879 119889 = [minus(2 +

119906)(2 minus 119906) minus(6 + 1199091)]119879 1199090= [2 0]

119879(7) [36] 119892 = 119906

2 1199050= 0 119905

119891= 1 119891 = (12)119909

2 sin119909 + 1199061199090= 0 120595 = 119909 minus 05

(8) [26] 119892 = 1199092cos2119906 119905

0= 0 119905119891= 120587 119891 = sin(1199062) 119909

0=

1205872(9) [26] 119892 = (12)(119909

2

1+ 1199092

2+ 1199062

) 1199050= 0 119905

119891= 5 119891 =

[1199092 minus1199091+ (1 minus 119909

2

1)1199092+ 119906]119879 119889 = minus(119909

2+ 025) 119909

0=

[1 0]119879

(10) [26] 119892 = 1199092

1+ 1199092

2+ 0005119906

2

1199050= 0 119905

119891= 1 119891 =

[1199092 minus1199092+119906]119879


05) minus 05 minus 1199092)]119879

1199090= [0 minus1]

119879(11) [26] 119892 = (12)119906

2 1199050= 0 119905

119891= 2 119891 = [119909

2 119906]119879 1199090=

[1 1]119879 120595 = [119909

1 1199092]119879

(12) [26] 119892 = minus1199092 1199050= 0 119905

119891= 1 119891 = [119909

2 119906]119879 119889 =

minus(1 minus 119906)(1 + 119906) 1199090= [0 0]

119879 120595 = 1199092

(13) [26] 119892 = (12)(1199092

1+ 1199092

2+ 01119906

2

) 1199050= 0 119905

119891= 078

119891 = [minus2(1199091+ 025) + (119909

2+ 05) exp(25119909

1(1199091+ 2)) minus

(1199091+025)119906 05minus119909

2minus(1199092+05) exp(25119909

1(1199091+2))]119879

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(14) [26] 119892 = (12)1199062 1199050= 0 119905

119891= 10 119891 = [cos 119906 minus

1199092 sin 119906]119879 119889 = minus(120587 + 119906)(120587 minus 119906) 119909

0= [366 minus186]

119879120595 = [119909

1 1199092]119879

(15) [26]119892 = (12)(1199092

1+1199092

2) 1199050= 0 119905119891= 078119891 = [minus2(119909

1+

025)+(1199092+05) exp(25119909

1(1199091+2))minus(119909

1+025)119906 05minus

1199092minus(1199092+05) exp(25119909

1(1199091+2))]119879 119889 = minus(1minus119906)(1+119906)

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(16) [26] 120601 = 1199093 1199050= 0 119905

119891= 1 119891 = [119909

2 119906 (12)119906

2

]119879

119889 = 1199091minus 19 119909

0= [0 0 0]

119879 120595 = [1199091 1199092+ 1]119879

(17) [26] 120601 = minus1199093 1199050= 0 119905

119891= 5 119891 = [119909

2 minus2 + 119906119909

3


0= [10 minus2 10]

119879120595 = [119909

1 1199092]119879

(18) [26] 120601 = (1199091minus1)2

+1199092

2+1199092

3 119892 = (12)119906

2 1199050= 0 119905119891= 5

119891 = [1199093cos 119906 119909

3sin 119906 sin 119906]119879 119909

0= [0 0 0]

119879


(19) [26] 119892 = 45(1199092

3+ 1199092

6) + 05(119906

2

1+ 1199062

2) 1199050= 0 119905

119891= 1

119891 = [91199094 91199095 91199096 9(1199061+ 1725119909

3) 91199062 minus(9119909

2)(1199061minus

270751199093+ 211990951199096)]119879 1199090= [0 22 0 0 minus1 0]

119879 120595 =

[1199091minus 10 119909

2minus 14 119909

3 1199094minus 25 119909

5 1199096]119879


2 1199093minus

4 1199094 1199095minus 1205874 119909

6]119879



References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










2 4 6 8 10 12 14minus5353

minus5352

minus5351

minus535

minus5349

minus5348

minus5347

minus5346

minus5345

minus5344

minus5343

Iter

Perfo

rman

ce in

dex

(a)

0 05 1 15 2 25 3minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Time

Ineq

ualit

y co

nstr

aint

(b)







120595





1 15 2 25 3 35 40

002

004

006

008

01

012

Iter

Perfo

rman

ce in

dex









5 10 15 20 25165

17

175

18

185

Iter

Perfo

rman

ce in

dex

(a)

0 05 1 15 2 25 3 35 4 45 50

005

01

015

02

025

TimeIn

equa

lity

cons

trai

nt(b)


5 10 15 20 25 30 35 40 45 500155

016

0165

017

0175

018

0185

019

0195

02

0205

Iter

Perfo

rman

ce in

dex

(a)

0 01 02 03 04 05 06 07 08 09 1minus25

minus2

minus15

minus1

minus05

0

Time

Ineq

ualit

y co

nstr

aint

(b)









119869for LI SI










5 10 15 20 25minus025

minus0245

minus024

minus0235

minus023

minus0225

Iter

Perfo

rman

ce in

dex

(a)

1 15 2 25 3 35 4 45 5 55 60

02

04

06

08

1

12

14

16

IterFi

nal c

ondi

tion

times10minus5

(b)


2 4 6 8 10 12 14minus025

minus0245

minus024

minus0235

minus023

minus0225

minus022

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

14

16

18

2

Iter

Fina

l con

ditio

n

times10minus5

(b)






1 2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

07

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

IterFi

nal c

ondi

tion

times10minus3

(b)


5 10 15 20 25 30

35

4

45

5

55

6

65

7

75

8

85

9

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

Iter

Fina

l con

ditio

n

(b)









1199052 the


and 1198731199012 the



5 10 15 20 25 30 35 40 45 50

00955

0096

00965

0097

00975

0098

00985

0099

00995

Iter

Perfo

rman

ce in

dex

(a)

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

35

4

45

IterFi

nal c

ondi

tion

times10minus3

(b)


5 10 15 20 25

21

22

23

24

25

26

27

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 80

1

2

3

4

5

6

Iter

Fina

l con

ditio

n

times10minus3

(b)







1 2 3 4 5 6 7minus887

minus886

minus885

minus884

minus883

minus882

minus881

minus88

minus879

minus878

minus877

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 70

01

02

03

04

05

06

07

08

09

1

IterFi

nal c

ondi

tion

times10minus3

(b)


1 15 2 25 3 35 4 45 5 55 600331003320033200332003320033200333003330033300333

Iter

Perfo

rman

ce in

dex










1199051


119869119870120595




119869 120593119891119870120595 are independent


119891is





119869and


119892 and other



1199011


119892





1 15 2 25 3 35 40

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 80

200

400

600

800

1000

1200

1400

1600

IterFi

nal c

ondi

tion

(b)


2 4 6 8 10 12 14 16 18 20

50

100

150

200

250

300

350

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 10 11 120

05

1

15

2

25

3

Iter

Fina

l con

ditio

n

(b)



1199011=

91198731199012

= 121198731199051= 31119873

1199052= 91119873

119892= 100 and119873

119894= 50 with








Output 1198731199051


119869

11 35 15333 00069 15286 1559821 35 15328 00097 15285 1584931 35 15329 00076 15285 1569941 35 15455 00607 15285 18449

Total 140 15357 00294 15285 18449

120593119891


952 times 10minus8

1 times 10minus12

452 times 10minus7


235 times 10minus7

1 times 10minus12

141 times 10minus6


491 times 10minus8

1 times 10minus12

221 times 10minus7


285 times 10minus7

1 times 10minus12

126 times 10minus6


Time

11 35 897838 272081 456614 155439421 35 1098147 460264 604192 284031031 35 1163591 300227 772673 179510341 35 1448290 393091 810425 2849514

Total 140 1142131 410891 456615 2849514

119864119869

11 35 00031 00045 0 0020421 35 00028 00064 0 0036931 35 00029 00050 0 0027141 35 00111 00397 0 02070

Total 140 00047 00192 0 02070

119870120595

11 35 00031 00045 475 times 10minus9

00204

21 35 00027 00064 550 times 10minus10

00369

31 35 00029 00050 154 times 10minus13

00271

41 35 00111 00397 193 times 10minus10

02070



Parameters 119869 119864119869

120593119891

Time 119870120595


1198731199051

1294 1296 0624 1079 12961198731199052

110572 393 0868 5939 3931198731199011

1945 1835 1271 4317 18351198731199012

2740 2478 1011 2302 2478119873119892

3541 3591 0491 0890 3591119873119894

0495 0309 0301 5849 1046

119875 value

1198731199051

0280 0279 0601 0 02791198731199052

0 0005 0489 0 00051198731199011

0125 0143 0286 0006 01431198731199012

0021 0034 0413 0 0034119873119892

0009 0008 0743 0472 0008119873119894

0739 0819 0877 0 0386



7 The Impact of SQP






119891119870120595

Time 119869 120593119891

119870120595


minus9


minus8




minus8




minus9


minus9




minus9


minus9




minus9



00252 57797 00151 139 times 10minus8

00472 66098Number 14 34761 672 times 10

minus6

00237 104585 34290 757 times 10minus6



111 times 10minus8


minus4

704 times 10minus9




minus10




minus8















Problem no1 2 3 4 5 6 7 8 9 10


120601119891

330 times 10minus9



11 12 13 14 15 16 17 18 19 20119869 48816 minus01273 00153 45727 979 times 10

minus4 23182 minus90882 00336 2513 20274120601119891

167 times 10minus5


minus4

594 times 10minus3


8 Conclusions


Appendix

Test Problems


(1) (VDP) [9] 119892 = (12)(1199092

1+ 1199092

2+ 1199062

) 1199050= 0 119905119891= 5 119891 =

[1199092 minus1199092+(1minus119909

2

1)1199092+119906]119879 1199090= [1 0]

119879120595 = 1199091minus1199092+1

(2) (CRP) [9] 119892 = (12)(1199092

1+1199092

2+01119906

2

) 1199050= 0 119905119891= 078

119891 = [1199091minus2(1199091+025)+(119909

2+05) exp(25119909

1(1199091+2))minus

(1199091+025)119906 05minus119909

2minus(1199092+05) exp(25119909

1(1199091+2))]119879

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(3) (FFRP) [9] 119892 = (12)(1199062

1+ 1199062

2+ 1199062

3+ 1199062

4) 1199050

=

0 119905119891

= 5 119891 = [1199092 (110)((119906

1+ 1199062) cos119909

5minus

(1199062+ 1199064) sin119909

5) 1199094 (110)((119906

1+ 1199063) sin119909

5+ (1199062+

1199064) cos119909

5) 1199096 (512)((119906

1+ 1199063) minus (119906

2+ 1199064))]119879 1199090=

[0 0 0 0 0 0]119879 120595 = [119909

1minus 4 1199092 1199093minus 4 1199094 1199095 1199096]119879

(4) (MSNIC) [20] 120601 = 1199093 1199050= 0 119905119891= 1 119891 = [119909

2 minus1199092+

119906 1199092

1+ 1199092

2+ 0005119906

2

]119879 119889 = [minus(119906 minus 20)(119906 + 20) 119909

2+

05 minus 8(119905 minus 05)2

]119879 1199090= [0 minus1 0]

119879

(5) (CSTCR) [20] 119892 = 1199092

1+ 1199092

2+ 01119906

2 1199050= 0 119905119891= 078

119891 = [minus(2 + 119906)(1199091+ 025) + (119909

2+ 05) exp(25119909

1(1199091+

2)) 05 minus 1199092minus (1199092+ 05) exp(25119909

1(1199091+ 2))]

119879 1199090=

[009 009]119879

(6) [34] 119892 = 21199091 1199050= 0 119905119891= 3 119891 = [119909

2 119906]119879 119889 = [minus(2 +

119906)(2 minus 119906) minus(6 + 1199091)]119879 1199090= [2 0]

119879(7) [36] 119892 = 119906

2 1199050= 0 119905

119891= 1 119891 = (12)119909

2 sin119909 + 1199061199090= 0 120595 = 119909 minus 05

(8) [26] 119892 = 1199092cos2119906 119905

0= 0 119905119891= 120587 119891 = sin(1199062) 119909

0=

1205872(9) [26] 119892 = (12)(119909

2

1+ 1199092

2+ 1199062

) 1199050= 0 119905

119891= 5 119891 =

[1199092 minus1199091+ (1 minus 119909

2

1)1199092+ 119906]119879 119889 = minus(119909

2+ 025) 119909

0=

[1 0]119879

(10) [26] 119892 = 1199092

1+ 1199092

2+ 0005119906

2

1199050= 0 119905

119891= 1 119891 =

[1199092 minus1199092+119906]119879


05) minus 05 minus 1199092)]119879

1199090= [0 minus1]

119879(11) [26] 119892 = (12)119906

2 1199050= 0 119905

119891= 2 119891 = [119909

2 119906]119879 1199090=

[1 1]119879 120595 = [119909

1 1199092]119879

(12) [26] 119892 = minus1199092 1199050= 0 119905

119891= 1 119891 = [119909

2 119906]119879 119889 =

minus(1 minus 119906)(1 + 119906) 1199090= [0 0]

119879 120595 = 1199092

(13) [26] 119892 = (12)(1199092

1+ 1199092

2+ 01119906

2

) 1199050= 0 119905

119891= 078

119891 = [minus2(1199091+ 025) + (119909

2+ 05) exp(25119909

1(1199091+ 2)) minus

(1199091+025)119906 05minus119909

2minus(1199092+05) exp(25119909

1(1199091+2))]119879

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(14) [26] 119892 = (12)1199062 1199050= 0 119905

119891= 10 119891 = [cos 119906 minus

1199092 sin 119906]119879 119889 = minus(120587 + 119906)(120587 minus 119906) 119909

0= [366 minus186]

119879120595 = [119909

1 1199092]119879

(15) [26]119892 = (12)(1199092

1+1199092

2) 1199050= 0 119905119891= 078119891 = [minus2(119909

1+

025)+(1199092+05) exp(25119909

1(1199091+2))minus(119909

1+025)119906 05minus

1199092minus(1199092+05) exp(25119909

1(1199091+2))]119879 119889 = minus(1minus119906)(1+119906)

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(16) [26] 120601 = 1199093 1199050= 0 119905

119891= 1 119891 = [119909

2 119906 (12)119906

2

]119879

119889 = 1199091minus 19 119909

0= [0 0 0]

119879 120595 = [1199091 1199092+ 1]119879

(17) [26] 120601 = minus1199093 1199050= 0 119905

119891= 5 119891 = [119909

2 minus2 + 119906119909

3


0= [10 minus2 10]

119879120595 = [119909

1 1199092]119879

(18) [26] 120601 = (1199091minus1)2

+1199092

2+1199092

3 119892 = (12)119906

2 1199050= 0 119905119891= 5

119891 = [1199093cos 119906 119909

3sin 119906 sin 119906]119879 119909

0= [0 0 0]

119879


(19) [26] 119892 = 45(1199092

3+ 1199092

6) + 05(119906

2

1+ 1199062

2) 1199050= 0 119905

119891= 1

119891 = [91199094 91199095 91199096 9(1199061+ 1725119909

3) 91199062 minus(9119909

2)(1199061minus

270751199093+ 211990951199096)]119879 1199090= [0 22 0 0 minus1 0]

119879 120595 =

[1199091minus 10 119909

2minus 14 119909

3 1199094minus 25 119909

5 1199096]119879


2 1199093minus

4 1199094 1199095minus 1205874 119909

6]119879



References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










5 10 15 20 25165

17

175

18

185

Iter

Perfo

rman

ce in

dex

(a)

0 05 1 15 2 25 3 35 4 45 50

005

01

015

02

025

TimeIn

equa

lity

cons

trai

nt(b)


5 10 15 20 25 30 35 40 45 500155

016

0165

017

0175

018

0185

019

0195

02

0205

Iter

Perfo

rman

ce in

dex

(a)

0 01 02 03 04 05 06 07 08 09 1minus25

minus2

minus15

minus1

minus05

0

Time

Ineq

ualit

y co

nstr

aint

(b)









119869for LI SI










5 10 15 20 25minus025

minus0245

minus024

minus0235

minus023

minus0225

Iter

Perfo

rman

ce in

dex

(a)

1 15 2 25 3 35 4 45 5 55 60

02

04

06

08

1

12

14

16

IterFi

nal c

ondi

tion

times10minus5

(b)


2 4 6 8 10 12 14minus025

minus0245

minus024

minus0235

minus023

minus0225

minus022

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

14

16

18

2

Iter

Fina

l con

ditio

n

times10minus5

(b)






1 2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

07

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

IterFi

nal c

ondi

tion

times10minus3

(b)


5 10 15 20 25 30

35

4

45

5

55

6

65

7

75

8

85

9

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

Iter

Fina

l con

ditio

n

(b)









1199052 the


and 1198731199012 the



5 10 15 20 25 30 35 40 45 50

00955

0096

00965

0097

00975

0098

00985

0099

00995

Iter

Perfo

rman

ce in

dex

(a)

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

35

4

45

IterFi

nal c

ondi

tion

times10minus3

(b)


5 10 15 20 25

21

22

23

24

25

26

27

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 80

1

2

3

4

5

6

Iter

Fina

l con

ditio

n

times10minus3

(b)







1 2 3 4 5 6 7minus887

minus886

minus885

minus884

minus883

minus882

minus881

minus88

minus879

minus878

minus877

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 70

01

02

03

04

05

06

07

08

09

1

IterFi

nal c

ondi

tion

times10minus3

(b)


1 15 2 25 3 35 4 45 5 55 600331003320033200332003320033200333003330033300333

Iter

Perfo

rman

ce in

dex










1199051


119869119870120595




119869 120593119891119870120595 are independent


119891is





119869and


119892 and other



1199011


119892





1 15 2 25 3 35 40

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 80

200

400

600

800

1000

1200

1400

1600

IterFi

nal c

ondi

tion

(b)


2 4 6 8 10 12 14 16 18 20

50

100

150

200

250

300

350

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 10 11 120

05

1

15

2

25

3

Iter

Fina

l con

ditio

n

(b)



1199011=

91198731199012

= 121198731199051= 31119873

1199052= 91119873

119892= 100 and119873

119894= 50 with








Output 1198731199051


119869

11 35 15333 00069 15286 1559821 35 15328 00097 15285 1584931 35 15329 00076 15285 1569941 35 15455 00607 15285 18449

Total 140 15357 00294 15285 18449

120593119891


952 times 10minus8

1 times 10minus12

452 times 10minus7


235 times 10minus7

1 times 10minus12

141 times 10minus6


491 times 10minus8

1 times 10minus12

221 times 10minus7


285 times 10minus7

1 times 10minus12

126 times 10minus6


Time

11 35 897838 272081 456614 155439421 35 1098147 460264 604192 284031031 35 1163591 300227 772673 179510341 35 1448290 393091 810425 2849514

Total 140 1142131 410891 456615 2849514

119864119869

11 35 00031 00045 0 0020421 35 00028 00064 0 0036931 35 00029 00050 0 0027141 35 00111 00397 0 02070

Total 140 00047 00192 0 02070

119870120595

11 35 00031 00045 475 times 10minus9

00204

21 35 00027 00064 550 times 10minus10

00369

31 35 00029 00050 154 times 10minus13

00271

41 35 00111 00397 193 times 10minus10

02070



Parameters 119869 119864119869

120593119891

Time 119870120595


1198731199051

1294 1296 0624 1079 12961198731199052

110572 393 0868 5939 3931198731199011

1945 1835 1271 4317 18351198731199012

2740 2478 1011 2302 2478119873119892

3541 3591 0491 0890 3591119873119894

0495 0309 0301 5849 1046

119875 value

1198731199051

0280 0279 0601 0 02791198731199052

0 0005 0489 0 00051198731199011

0125 0143 0286 0006 01431198731199012

0021 0034 0413 0 0034119873119892

0009 0008 0743 0472 0008119873119894

0739 0819 0877 0 0386



7 The Impact of SQP






119891119870120595

Time 119869 120593119891

119870120595


minus9


minus8




minus8




minus9


minus9




minus9


minus9




minus9



00252 57797 00151 139 times 10minus8

00472 66098Number 14 34761 672 times 10

minus6

00237 104585 34290 757 times 10minus6



111 times 10minus8


minus4

704 times 10minus9




minus10




minus8















Problem no1 2 3 4 5 6 7 8 9 10


120601119891

330 times 10minus9



11 12 13 14 15 16 17 18 19 20119869 48816 minus01273 00153 45727 979 times 10

minus4 23182 minus90882 00336 2513 20274120601119891

167 times 10minus5


minus4

594 times 10minus3


8 Conclusions


Appendix

Test Problems


(1) (VDP) [9] 119892 = (12)(1199092

1+ 1199092

2+ 1199062

) 1199050= 0 119905119891= 5 119891 =

[1199092 minus1199092+(1minus119909

2

1)1199092+119906]119879 1199090= [1 0]

119879120595 = 1199091minus1199092+1

(2) (CRP) [9] 119892 = (12)(1199092

1+1199092

2+01119906

2

) 1199050= 0 119905119891= 078

119891 = [1199091minus2(1199091+025)+(119909

2+05) exp(25119909

1(1199091+2))minus

(1199091+025)119906 05minus119909

2minus(1199092+05) exp(25119909

1(1199091+2))]119879

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(3) (FFRP) [9] 119892 = (12)(1199062

1+ 1199062

2+ 1199062

3+ 1199062

4) 1199050

=

0 119905119891

= 5 119891 = [1199092 (110)((119906

1+ 1199062) cos119909

5minus

(1199062+ 1199064) sin119909

5) 1199094 (110)((119906

1+ 1199063) sin119909

5+ (1199062+

1199064) cos119909

5) 1199096 (512)((119906

1+ 1199063) minus (119906

2+ 1199064))]119879 1199090=

[0 0 0 0 0 0]119879 120595 = [119909

1minus 4 1199092 1199093minus 4 1199094 1199095 1199096]119879

(4) (MSNIC) [20] 120601 = 1199093 1199050= 0 119905119891= 1 119891 = [119909

2 minus1199092+

119906 1199092

1+ 1199092

2+ 0005119906

2

]119879 119889 = [minus(119906 minus 20)(119906 + 20) 119909

2+

05 minus 8(119905 minus 05)2

]119879 1199090= [0 minus1 0]

119879

(5) (CSTCR) [20] 119892 = 1199092

1+ 1199092

2+ 01119906

2 1199050= 0 119905119891= 078

119891 = [minus(2 + 119906)(1199091+ 025) + (119909

2+ 05) exp(25119909

1(1199091+

2)) 05 minus 1199092minus (1199092+ 05) exp(25119909

1(1199091+ 2))]

119879 1199090=

[009 009]119879

(6) [34] 119892 = 21199091 1199050= 0 119905119891= 3 119891 = [119909

2 119906]119879 119889 = [minus(2 +

119906)(2 minus 119906) minus(6 + 1199091)]119879 1199090= [2 0]

119879(7) [36] 119892 = 119906

2 1199050= 0 119905

119891= 1 119891 = (12)119909

2 sin119909 + 1199061199090= 0 120595 = 119909 minus 05

(8) [26] 119892 = 1199092cos2119906 119905

0= 0 119905119891= 120587 119891 = sin(1199062) 119909

0=

1205872(9) [26] 119892 = (12)(119909

2

1+ 1199092

2+ 1199062

) 1199050= 0 119905

119891= 5 119891 =

[1199092 minus1199091+ (1 minus 119909

2

1)1199092+ 119906]119879 119889 = minus(119909

2+ 025) 119909

0=

[1 0]119879

(10) [26] 119892 = 1199092

1+ 1199092

2+ 0005119906

2

1199050= 0 119905

119891= 1 119891 =

[1199092 minus1199092+119906]119879


05) minus 05 minus 1199092)]119879

1199090= [0 minus1]

119879(11) [26] 119892 = (12)119906

2 1199050= 0 119905

119891= 2 119891 = [119909

2 119906]119879 1199090=

[1 1]119879 120595 = [119909

1 1199092]119879

(12) [26] 119892 = minus1199092 1199050= 0 119905

119891= 1 119891 = [119909

2 119906]119879 119889 =

minus(1 minus 119906)(1 + 119906) 1199090= [0 0]

119879 120595 = 1199092

(13) [26] 119892 = (12)(1199092

1+ 1199092

2+ 01119906

2

) 1199050= 0 119905

119891= 078

119891 = [minus2(1199091+ 025) + (119909

2+ 05) exp(25119909

1(1199091+ 2)) minus

(1199091+025)119906 05minus119909

2minus(1199092+05) exp(25119909

1(1199091+2))]119879

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(14) [26] 119892 = (12)1199062 1199050= 0 119905

119891= 10 119891 = [cos 119906 minus

1199092 sin 119906]119879 119889 = minus(120587 + 119906)(120587 minus 119906) 119909

0= [366 minus186]

119879120595 = [119909

1 1199092]119879

(15) [26]119892 = (12)(1199092

1+1199092

2) 1199050= 0 119905119891= 078119891 = [minus2(119909

1+

025)+(1199092+05) exp(25119909

1(1199091+2))minus(119909

1+025)119906 05minus

1199092minus(1199092+05) exp(25119909

1(1199091+2))]119879 119889 = minus(1minus119906)(1+119906)

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(16) [26] 120601 = 1199093 1199050= 0 119905

119891= 1 119891 = [119909

2 119906 (12)119906

2

]119879

119889 = 1199091minus 19 119909

0= [0 0 0]

119879 120595 = [1199091 1199092+ 1]119879

(17) [26] 120601 = minus1199093 1199050= 0 119905

119891= 5 119891 = [119909

2 minus2 + 119906119909

3


0= [10 minus2 10]

119879120595 = [119909

1 1199092]119879

(18) [26] 120601 = (1199091minus1)2

+1199092

2+1199092

3 119892 = (12)119906

2 1199050= 0 119905119891= 5

119891 = [1199093cos 119906 119909

3sin 119906 sin 119906]119879 119909

0= [0 0 0]

119879


(19) [26] 119892 = 45(1199092

3+ 1199092

6) + 05(119906

2

1+ 1199062

2) 1199050= 0 119905

119891= 1

119891 = [91199094 91199095 91199096 9(1199061+ 1725119909

3) 91199062 minus(9119909

2)(1199061minus

270751199093+ 211990951199096)]119879 1199090= [0 22 0 0 minus1 0]

119879 120595 =

[1199091minus 10 119909

2minus 14 119909

3 1199094minus 25 119909

5 1199096]119879


2 1199093minus

4 1199094 1199095minus 1205874 119909

6]119879



References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










5 10 15 20 25minus025

minus0245

minus024

minus0235

minus023

minus0225

Iter

Perfo

rman

ce in

dex

(a)

1 15 2 25 3 35 4 45 5 55 60

02

04

06

08

1

12

14

16

IterFi

nal c

ondi

tion

times10minus5

(b)


2 4 6 8 10 12 14minus025

minus0245

minus024

minus0235

minus023

minus0225

minus022

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

14

16

18

2

Iter

Fina

l con

ditio

n

times10minus5

(b)






1 2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

07

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

IterFi

nal c

ondi

tion

times10minus3

(b)


5 10 15 20 25 30

35

4

45

5

55

6

65

7

75

8

85

9

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

Iter

Fina

l con

ditio

n

(b)









1199052 the


and 1198731199012 the



5 10 15 20 25 30 35 40 45 50

00955

0096

00965

0097

00975

0098

00985

0099

00995

Iter

Perfo

rman

ce in

dex

(a)

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

35

4

45

IterFi

nal c

ondi

tion

times10minus3

(b)


5 10 15 20 25

21

22

23

24

25

26

27

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 80

1

2

3

4

5

6

Iter

Fina

l con

ditio

n

times10minus3

(b)







1 2 3 4 5 6 7minus887

minus886

minus885

minus884

minus883

minus882

minus881

minus88

minus879

minus878

minus877

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 70

01

02

03

04

05

06

07

08

09

1

IterFi

nal c

ondi

tion

times10minus3

(b)


1 15 2 25 3 35 4 45 5 55 600331003320033200332003320033200333003330033300333

Iter

Perfo

rman

ce in

dex










1199051


119869119870120595




119869 120593119891119870120595 are independent


119891is





119869and


119892 and other



1199011


119892





1 15 2 25 3 35 40

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 80

200

400

600

800

1000

1200

1400

1600

IterFi

nal c

ondi

tion

(b)


2 4 6 8 10 12 14 16 18 20

50

100

150

200

250

300

350

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 10 11 120

05

1

15

2

25

3

Iter

Fina

l con

ditio

n

(b)



1199011=

91198731199012

= 121198731199051= 31119873

1199052= 91119873

119892= 100 and119873

119894= 50 with








Output 1198731199051


119869

11 35 15333 00069 15286 1559821 35 15328 00097 15285 1584931 35 15329 00076 15285 1569941 35 15455 00607 15285 18449

Total 140 15357 00294 15285 18449

120593119891


952 times 10minus8

1 times 10minus12

452 times 10minus7


235 times 10minus7

1 times 10minus12

141 times 10minus6


491 times 10minus8

1 times 10minus12

221 times 10minus7


285 times 10minus7

1 times 10minus12

126 times 10minus6


Time

11 35 897838 272081 456614 155439421 35 1098147 460264 604192 284031031 35 1163591 300227 772673 179510341 35 1448290 393091 810425 2849514

Total 140 1142131 410891 456615 2849514

119864119869

11 35 00031 00045 0 0020421 35 00028 00064 0 0036931 35 00029 00050 0 0027141 35 00111 00397 0 02070

Total 140 00047 00192 0 02070

119870120595

11 35 00031 00045 475 times 10minus9

00204

21 35 00027 00064 550 times 10minus10

00369

31 35 00029 00050 154 times 10minus13

00271

41 35 00111 00397 193 times 10minus10

02070



Parameters 119869 119864119869

120593119891

Time 119870120595


1198731199051

1294 1296 0624 1079 12961198731199052

110572 393 0868 5939 3931198731199011

1945 1835 1271 4317 18351198731199012

2740 2478 1011 2302 2478119873119892

3541 3591 0491 0890 3591119873119894

0495 0309 0301 5849 1046

119875 value

1198731199051

0280 0279 0601 0 02791198731199052

0 0005 0489 0 00051198731199011

0125 0143 0286 0006 01431198731199012

0021 0034 0413 0 0034119873119892

0009 0008 0743 0472 0008119873119894

0739 0819 0877 0 0386



7 The Impact of SQP






119891119870120595

Time 119869 120593119891

119870120595


minus9


minus8




minus8




minus9


minus9




minus9


minus9




minus9



00252 57797 00151 139 times 10minus8

00472 66098Number 14 34761 672 times 10

minus6

00237 104585 34290 757 times 10minus6



111 times 10minus8


minus4

704 times 10minus9




minus10




minus8















Problem no1 2 3 4 5 6 7 8 9 10


120601119891

330 times 10minus9



11 12 13 14 15 16 17 18 19 20119869 48816 minus01273 00153 45727 979 times 10

minus4 23182 minus90882 00336 2513 20274120601119891

167 times 10minus5


minus4

594 times 10minus3


8 Conclusions


Appendix

Test Problems


(1) (VDP) [9] 119892 = (12)(1199092

1+ 1199092

2+ 1199062

) 1199050= 0 119905119891= 5 119891 =

[1199092 minus1199092+(1minus119909

2

1)1199092+119906]119879 1199090= [1 0]

119879120595 = 1199091minus1199092+1

(2) (CRP) [9] 119892 = (12)(1199092

1+1199092

2+01119906

2

) 1199050= 0 119905119891= 078

119891 = [1199091minus2(1199091+025)+(119909

2+05) exp(25119909

1(1199091+2))minus

(1199091+025)119906 05minus119909

2minus(1199092+05) exp(25119909

1(1199091+2))]119879

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(3) (FFRP) [9] 119892 = (12)(1199062

1+ 1199062

2+ 1199062

3+ 1199062

4) 1199050

=

0 119905119891

= 5 119891 = [1199092 (110)((119906

1+ 1199062) cos119909

5minus

(1199062+ 1199064) sin119909

5) 1199094 (110)((119906

1+ 1199063) sin119909

5+ (1199062+

1199064) cos119909

5) 1199096 (512)((119906

1+ 1199063) minus (119906

2+ 1199064))]119879 1199090=

[0 0 0 0 0 0]119879 120595 = [119909

1minus 4 1199092 1199093minus 4 1199094 1199095 1199096]119879

(4) (MSNIC) [20] 120601 = 1199093 1199050= 0 119905119891= 1 119891 = [119909

2 minus1199092+

119906 1199092

1+ 1199092

2+ 0005119906

2

]119879 119889 = [minus(119906 minus 20)(119906 + 20) 119909

2+

05 minus 8(119905 minus 05)2

]119879 1199090= [0 minus1 0]

119879

(5) (CSTCR) [20] 119892 = 1199092

1+ 1199092

2+ 01119906

2 1199050= 0 119905119891= 078

119891 = [minus(2 + 119906)(1199091+ 025) + (119909

2+ 05) exp(25119909

1(1199091+

2)) 05 minus 1199092minus (1199092+ 05) exp(25119909

1(1199091+ 2))]

119879 1199090=

[009 009]119879

(6) [34] 119892 = 21199091 1199050= 0 119905119891= 3 119891 = [119909

2 119906]119879 119889 = [minus(2 +

119906)(2 minus 119906) minus(6 + 1199091)]119879 1199090= [2 0]

119879(7) [36] 119892 = 119906

2 1199050= 0 119905

119891= 1 119891 = (12)119909

2 sin119909 + 1199061199090= 0 120595 = 119909 minus 05

(8) [26] 119892 = 1199092cos2119906 119905

0= 0 119905119891= 120587 119891 = sin(1199062) 119909

0=

1205872(9) [26] 119892 = (12)(119909

2

1+ 1199092

2+ 1199062

) 1199050= 0 119905

119891= 5 119891 =

[1199092 minus1199091+ (1 minus 119909

2

1)1199092+ 119906]119879 119889 = minus(119909

2+ 025) 119909

0=

[1 0]119879

(10) [26] 119892 = 1199092

1+ 1199092

2+ 0005119906

2

1199050= 0 119905

119891= 1 119891 =

[1199092 minus1199092+119906]119879


05) minus 05 minus 1199092)]119879

1199090= [0 minus1]

119879(11) [26] 119892 = (12)119906

2 1199050= 0 119905

119891= 2 119891 = [119909

2 119906]119879 1199090=

[1 1]119879 120595 = [119909

1 1199092]119879

(12) [26] 119892 = minus1199092 1199050= 0 119905

119891= 1 119891 = [119909

2 119906]119879 119889 =

minus(1 minus 119906)(1 + 119906) 1199090= [0 0]

119879 120595 = 1199092

(13) [26] 119892 = (12)(1199092

1+ 1199092

2+ 01119906

2

) 1199050= 0 119905

119891= 078

119891 = [minus2(1199091+ 025) + (119909

2+ 05) exp(25119909

1(1199091+ 2)) minus

(1199091+025)119906 05minus119909

2minus(1199092+05) exp(25119909

1(1199091+2))]119879

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(14) [26] 119892 = (12)1199062 1199050= 0 119905

119891= 10 119891 = [cos 119906 minus

1199092 sin 119906]119879 119889 = minus(120587 + 119906)(120587 minus 119906) 119909

0= [366 minus186]

119879120595 = [119909

1 1199092]119879

(15) [26]119892 = (12)(1199092

1+1199092

2) 1199050= 0 119905119891= 078119891 = [minus2(119909

1+

025)+(1199092+05) exp(25119909

1(1199091+2))minus(119909

1+025)119906 05minus

1199092minus(1199092+05) exp(25119909

1(1199091+2))]119879 119889 = minus(1minus119906)(1+119906)

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(16) [26] 120601 = 1199093 1199050= 0 119905

119891= 1 119891 = [119909

2 119906 (12)119906

2

]119879

119889 = 1199091minus 19 119909

0= [0 0 0]

119879 120595 = [1199091 1199092+ 1]119879

(17) [26] 120601 = minus1199093 1199050= 0 119905

119891= 5 119891 = [119909

2 minus2 + 119906119909

3


0= [10 minus2 10]

119879120595 = [119909

1 1199092]119879

(18) [26] 120601 = (1199091minus1)2

+1199092

2+1199092

3 119892 = (12)119906

2 1199050= 0 119905119891= 5

119891 = [1199093cos 119906 119909

3sin 119906 sin 119906]119879 119909

0= [0 0 0]

119879


(19) [26] 119892 = 45(1199092

3+ 1199092

6) + 05(119906

2

1+ 1199062

2) 1199050= 0 119905

119891= 1

119891 = [91199094 91199095 91199096 9(1199061+ 1725119909

3) 91199062 minus(9119909

2)(1199061minus

270751199093+ 211990951199096)]119879 1199090= [0 22 0 0 minus1 0]

119879 120595 =

[1199091minus 10 119909

2minus 14 119909

3 1199094minus 25 119909

5 1199096]119879


2 1199093minus

4 1199094 1199095minus 1205874 119909

6]119879



References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1 2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

07

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

IterFi

nal c

ondi

tion

times10minus3

(b)


5 10 15 20 25 30

35

4

45

5

55

6

65

7

75

8

85

9

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

Iter

Fina

l con

ditio

n

(b)









1199052 the


and 1198731199012 the



5 10 15 20 25 30 35 40 45 50

00955

0096

00965

0097

00975

0098

00985

0099

00995

Iter

Perfo

rman

ce in

dex

(a)

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

35

4

45

IterFi

nal c

ondi

tion

times10minus3

(b)


5 10 15 20 25

21

22

23

24

25

26

27

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 80

1

2

3

4

5

6

Iter

Fina

l con

ditio

n

times10minus3

(b)







1 2 3 4 5 6 7minus887

minus886

minus885

minus884

minus883

minus882

minus881

minus88

minus879

minus878

minus877

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 70

01

02

03

04

05

06

07

08

09

1

IterFi

nal c

ondi

tion

times10minus3

(b)


1 15 2 25 3 35 4 45 5 55 600331003320033200332003320033200333003330033300333

Iter

Perfo

rman

ce in

dex










1199051


119869119870120595




119869 120593119891119870120595 are independent


119891is





119869and


119892 and other



1199011


119892





1 15 2 25 3 35 40

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 80

200

400

600

800

1000

1200

1400

1600

IterFi

nal c

ondi

tion

(b)


2 4 6 8 10 12 14 16 18 20

50

100

150

200

250

300

350

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 10 11 120

05

1

15

2

25

3

Iter

Fina

l con

ditio

n

(b)



1199011=

91198731199012

= 121198731199051= 31119873

1199052= 91119873

119892= 100 and119873

119894= 50 with








Output 1198731199051


119869

11 35 15333 00069 15286 1559821 35 15328 00097 15285 1584931 35 15329 00076 15285 1569941 35 15455 00607 15285 18449

Total 140 15357 00294 15285 18449

120593119891


952 times 10minus8

1 times 10minus12

452 times 10minus7


235 times 10minus7

1 times 10minus12

141 times 10minus6


491 times 10minus8

1 times 10minus12

221 times 10minus7


285 times 10minus7

1 times 10minus12

126 times 10minus6


Time

11 35 897838 272081 456614 155439421 35 1098147 460264 604192 284031031 35 1163591 300227 772673 179510341 35 1448290 393091 810425 2849514

Total 140 1142131 410891 456615 2849514

119864119869

11 35 00031 00045 0 0020421 35 00028 00064 0 0036931 35 00029 00050 0 0027141 35 00111 00397 0 02070

Total 140 00047 00192 0 02070

119870120595

11 35 00031 00045 475 times 10minus9

00204

21 35 00027 00064 550 times 10minus10

00369

31 35 00029 00050 154 times 10minus13

00271

41 35 00111 00397 193 times 10minus10

02070



Parameters 119869 119864119869

120593119891

Time 119870120595


1198731199051

1294 1296 0624 1079 12961198731199052

110572 393 0868 5939 3931198731199011

1945 1835 1271 4317 18351198731199012

2740 2478 1011 2302 2478119873119892

3541 3591 0491 0890 3591119873119894

0495 0309 0301 5849 1046

119875 value

1198731199051

0280 0279 0601 0 02791198731199052

0 0005 0489 0 00051198731199011

0125 0143 0286 0006 01431198731199012

0021 0034 0413 0 0034119873119892

0009 0008 0743 0472 0008119873119894

0739 0819 0877 0 0386



7 The Impact of SQP






119891119870120595

Time 119869 120593119891

119870120595


minus9


minus8




minus8




minus9


minus9




minus9


minus9




minus9



00252 57797 00151 139 times 10minus8

00472 66098Number 14 34761 672 times 10

minus6

00237 104585 34290 757 times 10minus6



111 times 10minus8


minus4

704 times 10minus9




minus10




minus8















Problem no1 2 3 4 5 6 7 8 9 10


120601119891

330 times 10minus9



11 12 13 14 15 16 17 18 19 20119869 48816 minus01273 00153 45727 979 times 10

minus4 23182 minus90882 00336 2513 20274120601119891

167 times 10minus5


minus4

594 times 10minus3


8 Conclusions


Appendix

Test Problems


(1) (VDP) [9] 119892 = (12)(1199092

1+ 1199092

2+ 1199062

) 1199050= 0 119905119891= 5 119891 =

[1199092 minus1199092+(1minus119909

2

1)1199092+119906]119879 1199090= [1 0]

119879120595 = 1199091minus1199092+1

(2) (CRP) [9] 119892 = (12)(1199092

1+1199092

2+01119906

2

) 1199050= 0 119905119891= 078

119891 = [1199091minus2(1199091+025)+(119909

2+05) exp(25119909

1(1199091+2))minus

(1199091+025)119906 05minus119909

2minus(1199092+05) exp(25119909

1(1199091+2))]119879

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(3) (FFRP) [9] 119892 = (12)(1199062

1+ 1199062

2+ 1199062

3+ 1199062

4) 1199050

=

0 119905119891

= 5 119891 = [1199092 (110)((119906

1+ 1199062) cos119909

5minus

(1199062+ 1199064) sin119909

5) 1199094 (110)((119906

1+ 1199063) sin119909

5+ (1199062+

1199064) cos119909

5) 1199096 (512)((119906

1+ 1199063) minus (119906

2+ 1199064))]119879 1199090=

[0 0 0 0 0 0]119879 120595 = [119909

1minus 4 1199092 1199093minus 4 1199094 1199095 1199096]119879

(4) (MSNIC) [20] 120601 = 1199093 1199050= 0 119905119891= 1 119891 = [119909

2 minus1199092+

119906 1199092

1+ 1199092

2+ 0005119906

2

]119879 119889 = [minus(119906 minus 20)(119906 + 20) 119909

2+

05 minus 8(119905 minus 05)2

]119879 1199090= [0 minus1 0]

119879

(5) (CSTCR) [20] 119892 = 1199092

1+ 1199092

2+ 01119906

2 1199050= 0 119905119891= 078

119891 = [minus(2 + 119906)(1199091+ 025) + (119909

2+ 05) exp(25119909

1(1199091+

2)) 05 minus 1199092minus (1199092+ 05) exp(25119909

1(1199091+ 2))]

119879 1199090=

[009 009]119879

(6) [34] 119892 = 21199091 1199050= 0 119905119891= 3 119891 = [119909

2 119906]119879 119889 = [minus(2 +

119906)(2 minus 119906) minus(6 + 1199091)]119879 1199090= [2 0]

119879(7) [36] 119892 = 119906

2 1199050= 0 119905

119891= 1 119891 = (12)119909

2 sin119909 + 1199061199090= 0 120595 = 119909 minus 05

(8) [26] 119892 = 1199092cos2119906 119905

0= 0 119905119891= 120587 119891 = sin(1199062) 119909

0=

1205872(9) [26] 119892 = (12)(119909

2

1+ 1199092

2+ 1199062

) 1199050= 0 119905

119891= 5 119891 =

[1199092 minus1199091+ (1 minus 119909

2

1)1199092+ 119906]119879 119889 = minus(119909

2+ 025) 119909

0=

[1 0]119879

(10) [26] 119892 = 1199092

1+ 1199092

2+ 0005119906

2

1199050= 0 119905

119891= 1 119891 =

[1199092 minus1199092+119906]119879


05) minus 05 minus 1199092)]119879

1199090= [0 minus1]

119879(11) [26] 119892 = (12)119906

2 1199050= 0 119905

119891= 2 119891 = [119909

2 119906]119879 1199090=

[1 1]119879 120595 = [119909

1 1199092]119879

(12) [26] 119892 = minus1199092 1199050= 0 119905

119891= 1 119891 = [119909

2 119906]119879 119889 =

minus(1 minus 119906)(1 + 119906) 1199090= [0 0]

119879 120595 = 1199092

(13) [26] 119892 = (12)(1199092

1+ 1199092

2+ 01119906

2

) 1199050= 0 119905

119891= 078

119891 = [minus2(1199091+ 025) + (119909

2+ 05) exp(25119909

1(1199091+ 2)) minus

(1199091+025)119906 05minus119909

2minus(1199092+05) exp(25119909

1(1199091+2))]119879

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(14) [26] 119892 = (12)1199062 1199050= 0 119905

119891= 10 119891 = [cos 119906 minus

1199092 sin 119906]119879 119889 = minus(120587 + 119906)(120587 minus 119906) 119909

0= [366 minus186]

119879120595 = [119909

1 1199092]119879

(15) [26]119892 = (12)(1199092

1+1199092

2) 1199050= 0 119905119891= 078119891 = [minus2(119909

1+

025)+(1199092+05) exp(25119909

1(1199091+2))minus(119909

1+025)119906 05minus

1199092minus(1199092+05) exp(25119909

1(1199091+2))]119879 119889 = minus(1minus119906)(1+119906)

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(16) [26] 120601 = 1199093 1199050= 0 119905

119891= 1 119891 = [119909

2 119906 (12)119906

2

]119879

119889 = 1199091minus 19 119909

0= [0 0 0]

119879 120595 = [1199091 1199092+ 1]119879

(17) [26] 120601 = minus1199093 1199050= 0 119905

119891= 5 119891 = [119909

2 minus2 + 119906119909

3


0= [10 minus2 10]

119879120595 = [119909

1 1199092]119879

(18) [26] 120601 = (1199091minus1)2

+1199092

2+1199092

3 119892 = (12)119906

2 1199050= 0 119905119891= 5

119891 = [1199093cos 119906 119909

3sin 119906 sin 119906]119879 119909

0= [0 0 0]

119879


(19) [26] 119892 = 45(1199092

3+ 1199092

6) + 05(119906

2

1+ 1199062

2) 1199050= 0 119905

119891= 1

119891 = [91199094 91199095 91199096 9(1199061+ 1725119909

3) 91199062 minus(9119909

2)(1199061minus

270751199093+ 211990951199096)]119879 1199090= [0 22 0 0 minus1 0]

119879 120595 =

[1199091minus 10 119909

2minus 14 119909

3 1199094minus 25 119909

5 1199096]119879


2 1199093minus

4 1199094 1199095minus 1205874 119909

6]119879



References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










5 10 15 20 25 30 35 40 45 50

00955

0096

00965

0097

00975

0098

00985

0099

00995

Iter

Perfo

rman

ce in

dex

(a)

2 4 6 8 10 12 14 16 18 200

05

1

15

2

25

3

35

4

45

IterFi

nal c

ondi

tion

times10minus3

(b)


5 10 15 20 25

21

22

23

24

25

26

27

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 80

1

2

3

4

5

6

Iter

Fina

l con

ditio

n

times10minus3

(b)







1 2 3 4 5 6 7minus887

minus886

minus885

minus884

minus883

minus882

minus881

minus88

minus879

minus878

minus877

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 70

01

02

03

04

05

06

07

08

09

1

IterFi

nal c

ondi

tion

times10minus3

(b)


1 15 2 25 3 35 4 45 5 55 600331003320033200332003320033200333003330033300333

Iter

Perfo

rman

ce in

dex










1199051


119869119870120595




119869 120593119891119870120595 are independent


119891is





119869and


119892 and other



1199011


119892





1 15 2 25 3 35 40

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 80

200

400

600

800

1000

1200

1400

1600

IterFi

nal c

ondi

tion

(b)


2 4 6 8 10 12 14 16 18 20

50

100

150

200

250

300

350

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 10 11 120

05

1

15

2

25

3

Iter

Fina

l con

ditio

n

(b)



1199011=

91198731199012

= 121198731199051= 31119873

1199052= 91119873

119892= 100 and119873

119894= 50 with








Output 1198731199051


119869

11 35 15333 00069 15286 1559821 35 15328 00097 15285 1584931 35 15329 00076 15285 1569941 35 15455 00607 15285 18449

Total 140 15357 00294 15285 18449

120593119891


952 times 10minus8

1 times 10minus12

452 times 10minus7


235 times 10minus7

1 times 10minus12

141 times 10minus6


491 times 10minus8

1 times 10minus12

221 times 10minus7


285 times 10minus7

1 times 10minus12

126 times 10minus6


Time

11 35 897838 272081 456614 155439421 35 1098147 460264 604192 284031031 35 1163591 300227 772673 179510341 35 1448290 393091 810425 2849514

Total 140 1142131 410891 456615 2849514

119864119869

11 35 00031 00045 0 0020421 35 00028 00064 0 0036931 35 00029 00050 0 0027141 35 00111 00397 0 02070

Total 140 00047 00192 0 02070

119870120595

11 35 00031 00045 475 times 10minus9

00204

21 35 00027 00064 550 times 10minus10

00369

31 35 00029 00050 154 times 10minus13

00271

41 35 00111 00397 193 times 10minus10

02070



Parameters 119869 119864119869

120593119891

Time 119870120595


1198731199051

1294 1296 0624 1079 12961198731199052

110572 393 0868 5939 3931198731199011

1945 1835 1271 4317 18351198731199012

2740 2478 1011 2302 2478119873119892

3541 3591 0491 0890 3591119873119894

0495 0309 0301 5849 1046

119875 value

1198731199051

0280 0279 0601 0 02791198731199052

0 0005 0489 0 00051198731199011

0125 0143 0286 0006 01431198731199012

0021 0034 0413 0 0034119873119892

0009 0008 0743 0472 0008119873119894

0739 0819 0877 0 0386



7 The Impact of SQP






119891119870120595

Time 119869 120593119891

119870120595


minus9


minus8




minus8




minus9


minus9




minus9


minus9




minus9



00252 57797 00151 139 times 10minus8

00472 66098Number 14 34761 672 times 10

minus6

00237 104585 34290 757 times 10minus6



111 times 10minus8


minus4

704 times 10minus9




minus10




minus8















Problem no1 2 3 4 5 6 7 8 9 10


120601119891

330 times 10minus9



11 12 13 14 15 16 17 18 19 20119869 48816 minus01273 00153 45727 979 times 10

minus4 23182 minus90882 00336 2513 20274120601119891

167 times 10minus5


minus4

594 times 10minus3


8 Conclusions


Appendix

Test Problems


(1) (VDP) [9] 119892 = (12)(1199092

1+ 1199092

2+ 1199062

) 1199050= 0 119905119891= 5 119891 =

[1199092 minus1199092+(1minus119909

2

1)1199092+119906]119879 1199090= [1 0]

119879120595 = 1199091minus1199092+1

(2) (CRP) [9] 119892 = (12)(1199092

1+1199092

2+01119906

2

) 1199050= 0 119905119891= 078

119891 = [1199091minus2(1199091+025)+(119909

2+05) exp(25119909

1(1199091+2))minus

(1199091+025)119906 05minus119909

2minus(1199092+05) exp(25119909

1(1199091+2))]119879

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(3) (FFRP) [9] 119892 = (12)(1199062

1+ 1199062

2+ 1199062

3+ 1199062

4) 1199050

=

0 119905119891

= 5 119891 = [1199092 (110)((119906

1+ 1199062) cos119909

5minus

(1199062+ 1199064) sin119909

5) 1199094 (110)((119906

1+ 1199063) sin119909

5+ (1199062+

1199064) cos119909

5) 1199096 (512)((119906

1+ 1199063) minus (119906

2+ 1199064))]119879 1199090=

[0 0 0 0 0 0]119879 120595 = [119909

1minus 4 1199092 1199093minus 4 1199094 1199095 1199096]119879

(4) (MSNIC) [20] 120601 = 1199093 1199050= 0 119905119891= 1 119891 = [119909

2 minus1199092+

119906 1199092

1+ 1199092

2+ 0005119906

2

]119879 119889 = [minus(119906 minus 20)(119906 + 20) 119909

2+

05 minus 8(119905 minus 05)2

]119879 1199090= [0 minus1 0]

119879

(5) (CSTCR) [20] 119892 = 1199092

1+ 1199092

2+ 01119906

2 1199050= 0 119905119891= 078

119891 = [minus(2 + 119906)(1199091+ 025) + (119909

2+ 05) exp(25119909

1(1199091+

2)) 05 minus 1199092minus (1199092+ 05) exp(25119909

1(1199091+ 2))]

119879 1199090=

[009 009]119879

(6) [34] 119892 = 21199091 1199050= 0 119905119891= 3 119891 = [119909

2 119906]119879 119889 = [minus(2 +

119906)(2 minus 119906) minus(6 + 1199091)]119879 1199090= [2 0]

119879(7) [36] 119892 = 119906

2 1199050= 0 119905

119891= 1 119891 = (12)119909

2 sin119909 + 1199061199090= 0 120595 = 119909 minus 05

(8) [26] 119892 = 1199092cos2119906 119905

0= 0 119905119891= 120587 119891 = sin(1199062) 119909

0=

1205872(9) [26] 119892 = (12)(119909

2

1+ 1199092

2+ 1199062

) 1199050= 0 119905

119891= 5 119891 =

[1199092 minus1199091+ (1 minus 119909

2

1)1199092+ 119906]119879 119889 = minus(119909

2+ 025) 119909

0=

[1 0]119879

(10) [26] 119892 = 1199092

1+ 1199092

2+ 0005119906

2

1199050= 0 119905

119891= 1 119891 =

[1199092 minus1199092+119906]119879


05) minus 05 minus 1199092)]119879

1199090= [0 minus1]

119879(11) [26] 119892 = (12)119906

2 1199050= 0 119905

119891= 2 119891 = [119909

2 119906]119879 1199090=

[1 1]119879 120595 = [119909

1 1199092]119879

(12) [26] 119892 = minus1199092 1199050= 0 119905

119891= 1 119891 = [119909

2 119906]119879 119889 =

minus(1 minus 119906)(1 + 119906) 1199090= [0 0]

119879 120595 = 1199092

(13) [26] 119892 = (12)(1199092

1+ 1199092

2+ 01119906

2

) 1199050= 0 119905

119891= 078

119891 = [minus2(1199091+ 025) + (119909

2+ 05) exp(25119909

1(1199091+ 2)) minus

(1199091+025)119906 05minus119909

2minus(1199092+05) exp(25119909

1(1199091+2))]119879

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(14) [26] 119892 = (12)1199062 1199050= 0 119905

119891= 10 119891 = [cos 119906 minus

1199092 sin 119906]119879 119889 = minus(120587 + 119906)(120587 minus 119906) 119909

0= [366 minus186]

119879120595 = [119909

1 1199092]119879

(15) [26]119892 = (12)(1199092

1+1199092

2) 1199050= 0 119905119891= 078119891 = [minus2(119909

1+

025)+(1199092+05) exp(25119909

1(1199091+2))minus(119909

1+025)119906 05minus

1199092minus(1199092+05) exp(25119909

1(1199091+2))]119879 119889 = minus(1minus119906)(1+119906)

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(16) [26] 120601 = 1199093 1199050= 0 119905

119891= 1 119891 = [119909

2 119906 (12)119906

2

]119879

119889 = 1199091minus 19 119909

0= [0 0 0]

119879 120595 = [1199091 1199092+ 1]119879

(17) [26] 120601 = minus1199093 1199050= 0 119905

119891= 5 119891 = [119909

2 minus2 + 119906119909

3


0= [10 minus2 10]

119879120595 = [119909

1 1199092]119879

(18) [26] 120601 = (1199091minus1)2

+1199092

2+1199092

3 119892 = (12)119906

2 1199050= 0 119905119891= 5

119891 = [1199093cos 119906 119909

3sin 119906 sin 119906]119879 119909

0= [0 0 0]

119879


(19) [26] 119892 = 45(1199092

3+ 1199092

6) + 05(119906

2

1+ 1199062

2) 1199050= 0 119905

119891= 1

119891 = [91199094 91199095 91199096 9(1199061+ 1725119909

3) 91199062 minus(9119909

2)(1199061minus

270751199093+ 211990951199096)]119879 1199090= [0 22 0 0 minus1 0]

119879 120595 =

[1199091minus 10 119909

2minus 14 119909

3 1199094minus 25 119909

5 1199096]119879


2 1199093minus

4 1199094 1199095minus 1205874 119909

6]119879



References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1 2 3 4 5 6 7minus887

minus886

minus885

minus884

minus883

minus882

minus881

minus88

minus879

minus878

minus877

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 70

01

02

03

04

05

06

07

08

09

1

IterFi

nal c

ondi

tion

times10minus3

(b)


1 15 2 25 3 35 4 45 5 55 600331003320033200332003320033200333003330033300333

Iter

Perfo

rman

ce in

dex










1199051


119869119870120595




119869 120593119891119870120595 are independent


119891is





119869and


119892 and other



1199011


119892





1 15 2 25 3 35 40

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 80

200

400

600

800

1000

1200

1400

1600

IterFi

nal c

ondi

tion

(b)


2 4 6 8 10 12 14 16 18 20

50

100

150

200

250

300

350

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 10 11 120

05

1

15

2

25

3

Iter

Fina

l con

ditio

n

(b)



1199011=

91198731199012

= 121198731199051= 31119873

1199052= 91119873

119892= 100 and119873

119894= 50 with








Output 1198731199051


119869

11 35 15333 00069 15286 1559821 35 15328 00097 15285 1584931 35 15329 00076 15285 1569941 35 15455 00607 15285 18449

Total 140 15357 00294 15285 18449

120593119891


952 times 10minus8

1 times 10minus12

452 times 10minus7


235 times 10minus7

1 times 10minus12

141 times 10minus6


491 times 10minus8

1 times 10minus12

221 times 10minus7


285 times 10minus7

1 times 10minus12

126 times 10minus6


Time

11 35 897838 272081 456614 155439421 35 1098147 460264 604192 284031031 35 1163591 300227 772673 179510341 35 1448290 393091 810425 2849514

Total 140 1142131 410891 456615 2849514

119864119869

11 35 00031 00045 0 0020421 35 00028 00064 0 0036931 35 00029 00050 0 0027141 35 00111 00397 0 02070

Total 140 00047 00192 0 02070

119870120595

11 35 00031 00045 475 times 10minus9

00204

21 35 00027 00064 550 times 10minus10

00369

31 35 00029 00050 154 times 10minus13

00271

41 35 00111 00397 193 times 10minus10

02070



Parameters 119869 119864119869

120593119891

Time 119870120595


1198731199051

1294 1296 0624 1079 12961198731199052

110572 393 0868 5939 3931198731199011

1945 1835 1271 4317 18351198731199012

2740 2478 1011 2302 2478119873119892

3541 3591 0491 0890 3591119873119894

0495 0309 0301 5849 1046

119875 value

1198731199051

0280 0279 0601 0 02791198731199052

0 0005 0489 0 00051198731199011

0125 0143 0286 0006 01431198731199012

0021 0034 0413 0 0034119873119892

0009 0008 0743 0472 0008119873119894

0739 0819 0877 0 0386



7 The Impact of SQP






119891119870120595

Time 119869 120593119891

119870120595


minus9


minus8




minus8




minus9


minus9




minus9


minus9




minus9



00252 57797 00151 139 times 10minus8

00472 66098Number 14 34761 672 times 10

minus6

00237 104585 34290 757 times 10minus6



111 times 10minus8


minus4

704 times 10minus9




minus10




minus8















Problem no1 2 3 4 5 6 7 8 9 10


120601119891

330 times 10minus9



11 12 13 14 15 16 17 18 19 20119869 48816 minus01273 00153 45727 979 times 10

minus4 23182 minus90882 00336 2513 20274120601119891

167 times 10minus5


minus4

594 times 10minus3


8 Conclusions


Appendix

Test Problems


(1) (VDP) [9] 119892 = (12)(1199092

1+ 1199092

2+ 1199062

) 1199050= 0 119905119891= 5 119891 =

[1199092 minus1199092+(1minus119909

2

1)1199092+119906]119879 1199090= [1 0]

119879120595 = 1199091minus1199092+1

(2) (CRP) [9] 119892 = (12)(1199092

1+1199092

2+01119906

2

) 1199050= 0 119905119891= 078

119891 = [1199091minus2(1199091+025)+(119909

2+05) exp(25119909

1(1199091+2))minus

(1199091+025)119906 05minus119909

2minus(1199092+05) exp(25119909

1(1199091+2))]119879

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(3) (FFRP) [9] 119892 = (12)(1199062

1+ 1199062

2+ 1199062

3+ 1199062

4) 1199050

=

0 119905119891

= 5 119891 = [1199092 (110)((119906

1+ 1199062) cos119909

5minus

(1199062+ 1199064) sin119909

5) 1199094 (110)((119906

1+ 1199063) sin119909

5+ (1199062+

1199064) cos119909

5) 1199096 (512)((119906

1+ 1199063) minus (119906

2+ 1199064))]119879 1199090=

[0 0 0 0 0 0]119879 120595 = [119909

1minus 4 1199092 1199093minus 4 1199094 1199095 1199096]119879

(4) (MSNIC) [20] 120601 = 1199093 1199050= 0 119905119891= 1 119891 = [119909

2 minus1199092+

119906 1199092

1+ 1199092

2+ 0005119906

2

]119879 119889 = [minus(119906 minus 20)(119906 + 20) 119909

2+

05 minus 8(119905 minus 05)2

]119879 1199090= [0 minus1 0]

119879

(5) (CSTCR) [20] 119892 = 1199092

1+ 1199092

2+ 01119906

2 1199050= 0 119905119891= 078

119891 = [minus(2 + 119906)(1199091+ 025) + (119909

2+ 05) exp(25119909

1(1199091+

2)) 05 minus 1199092minus (1199092+ 05) exp(25119909

1(1199091+ 2))]

119879 1199090=

[009 009]119879

(6) [34] 119892 = 21199091 1199050= 0 119905119891= 3 119891 = [119909

2 119906]119879 119889 = [minus(2 +

119906)(2 minus 119906) minus(6 + 1199091)]119879 1199090= [2 0]

119879(7) [36] 119892 = 119906

2 1199050= 0 119905

119891= 1 119891 = (12)119909

2 sin119909 + 1199061199090= 0 120595 = 119909 minus 05

(8) [26] 119892 = 1199092cos2119906 119905

0= 0 119905119891= 120587 119891 = sin(1199062) 119909

0=

1205872(9) [26] 119892 = (12)(119909

2

1+ 1199092

2+ 1199062

) 1199050= 0 119905

119891= 5 119891 =

[1199092 minus1199091+ (1 minus 119909

2

1)1199092+ 119906]119879 119889 = minus(119909

2+ 025) 119909

0=

[1 0]119879

(10) [26] 119892 = 1199092

1+ 1199092

2+ 0005119906

2

1199050= 0 119905

119891= 1 119891 =

[1199092 minus1199092+119906]119879


05) minus 05 minus 1199092)]119879

1199090= [0 minus1]

119879(11) [26] 119892 = (12)119906

2 1199050= 0 119905

119891= 2 119891 = [119909

2 119906]119879 1199090=

[1 1]119879 120595 = [119909

1 1199092]119879

(12) [26] 119892 = minus1199092 1199050= 0 119905

119891= 1 119891 = [119909

2 119906]119879 119889 =

minus(1 minus 119906)(1 + 119906) 1199090= [0 0]

119879 120595 = 1199092

(13) [26] 119892 = (12)(1199092

1+ 1199092

2+ 01119906

2

) 1199050= 0 119905

119891= 078

119891 = [minus2(1199091+ 025) + (119909

2+ 05) exp(25119909

1(1199091+ 2)) minus

(1199091+025)119906 05minus119909

2minus(1199092+05) exp(25119909

1(1199091+2))]119879

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(14) [26] 119892 = (12)1199062 1199050= 0 119905

119891= 10 119891 = [cos 119906 minus

1199092 sin 119906]119879 119889 = minus(120587 + 119906)(120587 minus 119906) 119909

0= [366 minus186]

119879120595 = [119909

1 1199092]119879

(15) [26]119892 = (12)(1199092

1+1199092

2) 1199050= 0 119905119891= 078119891 = [minus2(119909

1+

025)+(1199092+05) exp(25119909

1(1199091+2))minus(119909

1+025)119906 05minus

1199092minus(1199092+05) exp(25119909

1(1199091+2))]119879 119889 = minus(1minus119906)(1+119906)

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(16) [26] 120601 = 1199093 1199050= 0 119905

119891= 1 119891 = [119909

2 119906 (12)119906

2

]119879

119889 = 1199091minus 19 119909

0= [0 0 0]

119879 120595 = [1199091 1199092+ 1]119879

(17) [26] 120601 = minus1199093 1199050= 0 119905

119891= 5 119891 = [119909

2 minus2 + 119906119909

3


0= [10 minus2 10]

119879120595 = [119909

1 1199092]119879

(18) [26] 120601 = (1199091minus1)2

+1199092

2+1199092

3 119892 = (12)119906

2 1199050= 0 119905119891= 5

119891 = [1199093cos 119906 119909

3sin 119906 sin 119906]119879 119909

0= [0 0 0]

119879


(19) [26] 119892 = 45(1199092

3+ 1199092

6) + 05(119906

2

1+ 1199062

2) 1199050= 0 119905

119891= 1

119891 = [91199094 91199095 91199096 9(1199061+ 1725119909

3) 91199062 minus(9119909

2)(1199061minus

270751199093+ 211990951199096)]119879 1199090= [0 22 0 0 minus1 0]

119879 120595 =

[1199091minus 10 119909

2minus 14 119909

3 1199094minus 25 119909

5 1199096]119879


2 1199093minus

4 1199094 1199095minus 1205874 119909

6]119879



References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1 15 2 25 3 35 40

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 80

200

400

600

800

1000

1200

1400

1600

IterFi

nal c

ondi

tion

(b)


2 4 6 8 10 12 14 16 18 20

50

100

150

200

250

300

350

Iter

Perfo

rman

ce in

dex

(a)

1 2 3 4 5 6 7 8 9 10 11 120

05

1

15

2

25

3

Iter

Fina

l con

ditio

n

(b)



1199011=

91198731199012

= 121198731199051= 31119873

1199052= 91119873

119892= 100 and119873

119894= 50 with








Output 1198731199051


119869

11 35 15333 00069 15286 1559821 35 15328 00097 15285 1584931 35 15329 00076 15285 1569941 35 15455 00607 15285 18449

Total 140 15357 00294 15285 18449

120593119891


952 times 10minus8

1 times 10minus12

452 times 10minus7


235 times 10minus7

1 times 10minus12

141 times 10minus6


491 times 10minus8

1 times 10minus12

221 times 10minus7


285 times 10minus7

1 times 10minus12

126 times 10minus6


Time

11 35 897838 272081 456614 155439421 35 1098147 460264 604192 284031031 35 1163591 300227 772673 179510341 35 1448290 393091 810425 2849514

Total 140 1142131 410891 456615 2849514

119864119869

11 35 00031 00045 0 0020421 35 00028 00064 0 0036931 35 00029 00050 0 0027141 35 00111 00397 0 02070

Total 140 00047 00192 0 02070

119870120595

11 35 00031 00045 475 times 10minus9

00204

21 35 00027 00064 550 times 10minus10

00369

31 35 00029 00050 154 times 10minus13

00271

41 35 00111 00397 193 times 10minus10

02070



Parameters 119869 119864119869

120593119891

Time 119870120595


1198731199051

1294 1296 0624 1079 12961198731199052

110572 393 0868 5939 3931198731199011

1945 1835 1271 4317 18351198731199012

2740 2478 1011 2302 2478119873119892

3541 3591 0491 0890 3591119873119894

0495 0309 0301 5849 1046

119875 value

1198731199051

0280 0279 0601 0 02791198731199052

0 0005 0489 0 00051198731199011

0125 0143 0286 0006 01431198731199012

0021 0034 0413 0 0034119873119892

0009 0008 0743 0472 0008119873119894

0739 0819 0877 0 0386



7 The Impact of SQP






119891119870120595

Time 119869 120593119891

119870120595


minus9


minus8




minus8




minus9


minus9




minus9


minus9




minus9



00252 57797 00151 139 times 10minus8

00472 66098Number 14 34761 672 times 10

minus6

00237 104585 34290 757 times 10minus6



111 times 10minus8


minus4

704 times 10minus9




minus10




minus8















Problem no1 2 3 4 5 6 7 8 9 10


120601119891

330 times 10minus9



11 12 13 14 15 16 17 18 19 20119869 48816 minus01273 00153 45727 979 times 10

minus4 23182 minus90882 00336 2513 20274120601119891

167 times 10minus5


minus4

594 times 10minus3


8 Conclusions


Appendix

Test Problems


(1) (VDP) [9] 119892 = (12)(1199092

1+ 1199092

2+ 1199062

) 1199050= 0 119905119891= 5 119891 =

[1199092 minus1199092+(1minus119909

2

1)1199092+119906]119879 1199090= [1 0]

119879120595 = 1199091minus1199092+1

(2) (CRP) [9] 119892 = (12)(1199092

1+1199092

2+01119906

2

) 1199050= 0 119905119891= 078

119891 = [1199091minus2(1199091+025)+(119909

2+05) exp(25119909

1(1199091+2))minus

(1199091+025)119906 05minus119909

2minus(1199092+05) exp(25119909

1(1199091+2))]119879

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(3) (FFRP) [9] 119892 = (12)(1199062

1+ 1199062

2+ 1199062

3+ 1199062

4) 1199050

=

0 119905119891

= 5 119891 = [1199092 (110)((119906

1+ 1199062) cos119909

5minus

(1199062+ 1199064) sin119909

5) 1199094 (110)((119906

1+ 1199063) sin119909

5+ (1199062+

1199064) cos119909

5) 1199096 (512)((119906

1+ 1199063) minus (119906

2+ 1199064))]119879 1199090=

[0 0 0 0 0 0]119879 120595 = [119909

1minus 4 1199092 1199093minus 4 1199094 1199095 1199096]119879

(4) (MSNIC) [20] 120601 = 1199093 1199050= 0 119905119891= 1 119891 = [119909

2 minus1199092+

119906 1199092

1+ 1199092

2+ 0005119906

2

]119879 119889 = [minus(119906 minus 20)(119906 + 20) 119909

2+

05 minus 8(119905 minus 05)2

]119879 1199090= [0 minus1 0]

119879

(5) (CSTCR) [20] 119892 = 1199092

1+ 1199092

2+ 01119906

2 1199050= 0 119905119891= 078

119891 = [minus(2 + 119906)(1199091+ 025) + (119909

2+ 05) exp(25119909

1(1199091+

2)) 05 minus 1199092minus (1199092+ 05) exp(25119909

1(1199091+ 2))]

119879 1199090=

[009 009]119879

(6) [34] 119892 = 21199091 1199050= 0 119905119891= 3 119891 = [119909

2 119906]119879 119889 = [minus(2 +

119906)(2 minus 119906) minus(6 + 1199091)]119879 1199090= [2 0]

119879(7) [36] 119892 = 119906

2 1199050= 0 119905

119891= 1 119891 = (12)119909

2 sin119909 + 1199061199090= 0 120595 = 119909 minus 05

(8) [26] 119892 = 1199092cos2119906 119905

0= 0 119905119891= 120587 119891 = sin(1199062) 119909

0=

1205872(9) [26] 119892 = (12)(119909

2

1+ 1199092

2+ 1199062

) 1199050= 0 119905

119891= 5 119891 =

[1199092 minus1199091+ (1 minus 119909

2

1)1199092+ 119906]119879 119889 = minus(119909

2+ 025) 119909

0=

[1 0]119879

(10) [26] 119892 = 1199092

1+ 1199092

2+ 0005119906

2

1199050= 0 119905

119891= 1 119891 =

[1199092 minus1199092+119906]119879


05) minus 05 minus 1199092)]119879

1199090= [0 minus1]

119879(11) [26] 119892 = (12)119906

2 1199050= 0 119905

119891= 2 119891 = [119909

2 119906]119879 1199090=

[1 1]119879 120595 = [119909

1 1199092]119879

(12) [26] 119892 = minus1199092 1199050= 0 119905

119891= 1 119891 = [119909

2 119906]119879 119889 =

minus(1 minus 119906)(1 + 119906) 1199090= [0 0]

119879 120595 = 1199092

(13) [26] 119892 = (12)(1199092

1+ 1199092

2+ 01119906

2

) 1199050= 0 119905

119891= 078

119891 = [minus2(1199091+ 025) + (119909

2+ 05) exp(25119909

1(1199091+ 2)) minus

(1199091+025)119906 05minus119909

2minus(1199092+05) exp(25119909

1(1199091+2))]119879

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(14) [26] 119892 = (12)1199062 1199050= 0 119905

119891= 10 119891 = [cos 119906 minus

1199092 sin 119906]119879 119889 = minus(120587 + 119906)(120587 minus 119906) 119909

0= [366 minus186]

119879120595 = [119909

1 1199092]119879

(15) [26]119892 = (12)(1199092

1+1199092

2) 1199050= 0 119905119891= 078119891 = [minus2(119909

1+

025)+(1199092+05) exp(25119909

1(1199091+2))minus(119909

1+025)119906 05minus

1199092minus(1199092+05) exp(25119909

1(1199091+2))]119879 119889 = minus(1minus119906)(1+119906)

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(16) [26] 120601 = 1199093 1199050= 0 119905

119891= 1 119891 = [119909

2 119906 (12)119906

2

]119879

119889 = 1199091minus 19 119909

0= [0 0 0]

119879 120595 = [1199091 1199092+ 1]119879

(17) [26] 120601 = minus1199093 1199050= 0 119905

119891= 5 119891 = [119909

2 minus2 + 119906119909

3


0= [10 minus2 10]

119879120595 = [119909

1 1199092]119879

(18) [26] 120601 = (1199091minus1)2

+1199092

2+1199092

3 119892 = (12)119906

2 1199050= 0 119905119891= 5

119891 = [1199093cos 119906 119909

3sin 119906 sin 119906]119879 119909

0= [0 0 0]

119879


(19) [26] 119892 = 45(1199092

3+ 1199092

6) + 05(119906

2

1+ 1199062

2) 1199050= 0 119905

119891= 1

119891 = [91199094 91199095 91199096 9(1199061+ 1725119909

3) 91199062 minus(9119909

2)(1199061minus

270751199093+ 211990951199096)]119879 1199090= [0 22 0 0 minus1 0]

119879 120595 =

[1199091minus 10 119909

2minus 14 119909

3 1199094minus 25 119909

5 1199096]119879


2 1199093minus

4 1199094 1199095minus 1205874 119909

6]119879



References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Output 1198731199051


119869

11 35 15333 00069 15286 1559821 35 15328 00097 15285 1584931 35 15329 00076 15285 1569941 35 15455 00607 15285 18449

Total 140 15357 00294 15285 18449

120593119891


952 times 10minus8

1 times 10minus12

452 times 10minus7


235 times 10minus7

1 times 10minus12

141 times 10minus6


491 times 10minus8

1 times 10minus12

221 times 10minus7


285 times 10minus7

1 times 10minus12

126 times 10minus6


Time

11 35 897838 272081 456614 155439421 35 1098147 460264 604192 284031031 35 1163591 300227 772673 179510341 35 1448290 393091 810425 2849514

Total 140 1142131 410891 456615 2849514

119864119869

11 35 00031 00045 0 0020421 35 00028 00064 0 0036931 35 00029 00050 0 0027141 35 00111 00397 0 02070

Total 140 00047 00192 0 02070

119870120595

11 35 00031 00045 475 times 10minus9

00204

21 35 00027 00064 550 times 10minus10

00369

31 35 00029 00050 154 times 10minus13

00271

41 35 00111 00397 193 times 10minus10

02070



Parameters 119869 119864119869

120593119891

Time 119870120595


1198731199051

1294 1296 0624 1079 12961198731199052

110572 393 0868 5939 3931198731199011

1945 1835 1271 4317 18351198731199012

2740 2478 1011 2302 2478119873119892

3541 3591 0491 0890 3591119873119894

0495 0309 0301 5849 1046

119875 value

1198731199051

0280 0279 0601 0 02791198731199052

0 0005 0489 0 00051198731199011

0125 0143 0286 0006 01431198731199012

0021 0034 0413 0 0034119873119892

0009 0008 0743 0472 0008119873119894

0739 0819 0877 0 0386



7 The Impact of SQP






119891119870120595

Time 119869 120593119891

119870120595


minus9


minus8




minus8




minus9


minus9




minus9


minus9




minus9



00252 57797 00151 139 times 10minus8

00472 66098Number 14 34761 672 times 10

minus6

00237 104585 34290 757 times 10minus6



111 times 10minus8


minus4

704 times 10minus9




minus10




minus8















Problem no1 2 3 4 5 6 7 8 9 10


120601119891

330 times 10minus9



11 12 13 14 15 16 17 18 19 20119869 48816 minus01273 00153 45727 979 times 10

minus4 23182 minus90882 00336 2513 20274120601119891

167 times 10minus5


minus4

594 times 10minus3


8 Conclusions


Appendix

Test Problems


(1) (VDP) [9] 119892 = (12)(1199092

1+ 1199092

2+ 1199062

) 1199050= 0 119905119891= 5 119891 =

[1199092 minus1199092+(1minus119909

2

1)1199092+119906]119879 1199090= [1 0]

119879120595 = 1199091minus1199092+1

(2) (CRP) [9] 119892 = (12)(1199092

1+1199092

2+01119906

2

) 1199050= 0 119905119891= 078

119891 = [1199091minus2(1199091+025)+(119909

2+05) exp(25119909

1(1199091+2))minus

(1199091+025)119906 05minus119909

2minus(1199092+05) exp(25119909

1(1199091+2))]119879

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(3) (FFRP) [9] 119892 = (12)(1199062

1+ 1199062

2+ 1199062

3+ 1199062

4) 1199050

=

0 119905119891

= 5 119891 = [1199092 (110)((119906

1+ 1199062) cos119909

5minus

(1199062+ 1199064) sin119909

5) 1199094 (110)((119906

1+ 1199063) sin119909

5+ (1199062+

1199064) cos119909

5) 1199096 (512)((119906

1+ 1199063) minus (119906

2+ 1199064))]119879 1199090=

[0 0 0 0 0 0]119879 120595 = [119909

1minus 4 1199092 1199093minus 4 1199094 1199095 1199096]119879

(4) (MSNIC) [20] 120601 = 1199093 1199050= 0 119905119891= 1 119891 = [119909

2 minus1199092+

119906 1199092

1+ 1199092

2+ 0005119906

2

]119879 119889 = [minus(119906 minus 20)(119906 + 20) 119909

2+

05 minus 8(119905 minus 05)2

]119879 1199090= [0 minus1 0]

119879

(5) (CSTCR) [20] 119892 = 1199092

1+ 1199092

2+ 01119906

2 1199050= 0 119905119891= 078

119891 = [minus(2 + 119906)(1199091+ 025) + (119909

2+ 05) exp(25119909

1(1199091+

2)) 05 minus 1199092minus (1199092+ 05) exp(25119909

1(1199091+ 2))]

119879 1199090=

[009 009]119879

(6) [34] 119892 = 21199091 1199050= 0 119905119891= 3 119891 = [119909

2 119906]119879 119889 = [minus(2 +

119906)(2 minus 119906) minus(6 + 1199091)]119879 1199090= [2 0]

119879(7) [36] 119892 = 119906

2 1199050= 0 119905

119891= 1 119891 = (12)119909

2 sin119909 + 1199061199090= 0 120595 = 119909 minus 05

(8) [26] 119892 = 1199092cos2119906 119905

0= 0 119905119891= 120587 119891 = sin(1199062) 119909

0=

1205872(9) [26] 119892 = (12)(119909

2

1+ 1199092

2+ 1199062

) 1199050= 0 119905

119891= 5 119891 =

[1199092 minus1199091+ (1 minus 119909

2

1)1199092+ 119906]119879 119889 = minus(119909

2+ 025) 119909

0=

[1 0]119879

(10) [26] 119892 = 1199092

1+ 1199092

2+ 0005119906

2

1199050= 0 119905

119891= 1 119891 =

[1199092 minus1199092+119906]119879


05) minus 05 minus 1199092)]119879

1199090= [0 minus1]

119879(11) [26] 119892 = (12)119906

2 1199050= 0 119905

119891= 2 119891 = [119909

2 119906]119879 1199090=

[1 1]119879 120595 = [119909

1 1199092]119879

(12) [26] 119892 = minus1199092 1199050= 0 119905

119891= 1 119891 = [119909

2 119906]119879 119889 =

minus(1 minus 119906)(1 + 119906) 1199090= [0 0]

119879 120595 = 1199092

(13) [26] 119892 = (12)(1199092

1+ 1199092

2+ 01119906

2

) 1199050= 0 119905

119891= 078

119891 = [minus2(1199091+ 025) + (119909

2+ 05) exp(25119909

1(1199091+ 2)) minus

(1199091+025)119906 05minus119909

2minus(1199092+05) exp(25119909

1(1199091+2))]119879

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(14) [26] 119892 = (12)1199062 1199050= 0 119905

119891= 10 119891 = [cos 119906 minus

1199092 sin 119906]119879 119889 = minus(120587 + 119906)(120587 minus 119906) 119909

0= [366 minus186]

119879120595 = [119909

1 1199092]119879

(15) [26]119892 = (12)(1199092

1+1199092

2) 1199050= 0 119905119891= 078119891 = [minus2(119909

1+

025)+(1199092+05) exp(25119909

1(1199091+2))minus(119909

1+025)119906 05minus

1199092minus(1199092+05) exp(25119909

1(1199091+2))]119879 119889 = minus(1minus119906)(1+119906)

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(16) [26] 120601 = 1199093 1199050= 0 119905

119891= 1 119891 = [119909

2 119906 (12)119906

2

]119879

119889 = 1199091minus 19 119909

0= [0 0 0]

119879 120595 = [1199091 1199092+ 1]119879

(17) [26] 120601 = minus1199093 1199050= 0 119905

119891= 5 119891 = [119909

2 minus2 + 119906119909

3


0= [10 minus2 10]

119879120595 = [119909

1 1199092]119879

(18) [26] 120601 = (1199091minus1)2

+1199092

2+1199092

3 119892 = (12)119906

2 1199050= 0 119905119891= 5

119891 = [1199093cos 119906 119909

3sin 119906 sin 119906]119879 119909

0= [0 0 0]

119879


(19) [26] 119892 = 45(1199092

3+ 1199092

6) + 05(119906

2

1+ 1199062

2) 1199050= 0 119905

119891= 1

119891 = [91199094 91199095 91199096 9(1199061+ 1725119909

3) 91199062 minus(9119909

2)(1199061minus

270751199093+ 211990951199096)]119879 1199090= [0 22 0 0 minus1 0]

119879 120595 =

[1199091minus 10 119909

2minus 14 119909

3 1199094minus 25 119909

5 1199096]119879


2 1199093minus

4 1199094 1199095minus 1205874 119909

6]119879



References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119891119870120595

Time 119869 120593119891

119870120595


minus9


minus8




minus8




minus9


minus9




minus9


minus9




minus9



00252 57797 00151 139 times 10minus8

00472 66098Number 14 34761 672 times 10

minus6

00237 104585 34290 757 times 10minus6



111 times 10minus8


minus4

704 times 10minus9




minus10




minus8















Problem no1 2 3 4 5 6 7 8 9 10


120601119891

330 times 10minus9



11 12 13 14 15 16 17 18 19 20119869 48816 minus01273 00153 45727 979 times 10

minus4 23182 minus90882 00336 2513 20274120601119891

167 times 10minus5


minus4

594 times 10minus3


8 Conclusions


Appendix

Test Problems


(1) (VDP) [9] 119892 = (12)(1199092

1+ 1199092

2+ 1199062

) 1199050= 0 119905119891= 5 119891 =

[1199092 minus1199092+(1minus119909

2

1)1199092+119906]119879 1199090= [1 0]

119879120595 = 1199091minus1199092+1

(2) (CRP) [9] 119892 = (12)(1199092

1+1199092

2+01119906

2

) 1199050= 0 119905119891= 078

119891 = [1199091minus2(1199091+025)+(119909

2+05) exp(25119909

1(1199091+2))minus

(1199091+025)119906 05minus119909

2minus(1199092+05) exp(25119909

1(1199091+2))]119879

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(3) (FFRP) [9] 119892 = (12)(1199062

1+ 1199062

2+ 1199062

3+ 1199062

4) 1199050

=

0 119905119891

= 5 119891 = [1199092 (110)((119906

1+ 1199062) cos119909

5minus

(1199062+ 1199064) sin119909

5) 1199094 (110)((119906

1+ 1199063) sin119909

5+ (1199062+

1199064) cos119909

5) 1199096 (512)((119906

1+ 1199063) minus (119906

2+ 1199064))]119879 1199090=

[0 0 0 0 0 0]119879 120595 = [119909

1minus 4 1199092 1199093minus 4 1199094 1199095 1199096]119879

(4) (MSNIC) [20] 120601 = 1199093 1199050= 0 119905119891= 1 119891 = [119909

2 minus1199092+

119906 1199092

1+ 1199092

2+ 0005119906

2

]119879 119889 = [minus(119906 minus 20)(119906 + 20) 119909

2+

05 minus 8(119905 minus 05)2

]119879 1199090= [0 minus1 0]

119879

(5) (CSTCR) [20] 119892 = 1199092

1+ 1199092

2+ 01119906

2 1199050= 0 119905119891= 078

119891 = [minus(2 + 119906)(1199091+ 025) + (119909

2+ 05) exp(25119909

1(1199091+

2)) 05 minus 1199092minus (1199092+ 05) exp(25119909

1(1199091+ 2))]

119879 1199090=

[009 009]119879

(6) [34] 119892 = 21199091 1199050= 0 119905119891= 3 119891 = [119909

2 119906]119879 119889 = [minus(2 +

119906)(2 minus 119906) minus(6 + 1199091)]119879 1199090= [2 0]

119879(7) [36] 119892 = 119906

2 1199050= 0 119905

119891= 1 119891 = (12)119909

2 sin119909 + 1199061199090= 0 120595 = 119909 minus 05

(8) [26] 119892 = 1199092cos2119906 119905

0= 0 119905119891= 120587 119891 = sin(1199062) 119909

0=

1205872(9) [26] 119892 = (12)(119909

2

1+ 1199092

2+ 1199062

) 1199050= 0 119905

119891= 5 119891 =

[1199092 minus1199091+ (1 minus 119909

2

1)1199092+ 119906]119879 119889 = minus(119909

2+ 025) 119909

0=

[1 0]119879

(10) [26] 119892 = 1199092

1+ 1199092

2+ 0005119906

2

1199050= 0 119905

119891= 1 119891 =

[1199092 minus1199092+119906]119879


05) minus 05 minus 1199092)]119879

1199090= [0 minus1]

119879(11) [26] 119892 = (12)119906

2 1199050= 0 119905

119891= 2 119891 = [119909

2 119906]119879 1199090=

[1 1]119879 120595 = [119909

1 1199092]119879

(12) [26] 119892 = minus1199092 1199050= 0 119905

119891= 1 119891 = [119909

2 119906]119879 119889 =

minus(1 minus 119906)(1 + 119906) 1199090= [0 0]

119879 120595 = 1199092

(13) [26] 119892 = (12)(1199092

1+ 1199092

2+ 01119906

2

) 1199050= 0 119905

119891= 078

119891 = [minus2(1199091+ 025) + (119909

2+ 05) exp(25119909

1(1199091+ 2)) minus

(1199091+025)119906 05minus119909

2minus(1199092+05) exp(25119909

1(1199091+2))]119879

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(14) [26] 119892 = (12)1199062 1199050= 0 119905

119891= 10 119891 = [cos 119906 minus

1199092 sin 119906]119879 119889 = minus(120587 + 119906)(120587 minus 119906) 119909

0= [366 minus186]

119879120595 = [119909

1 1199092]119879

(15) [26]119892 = (12)(1199092

1+1199092

2) 1199050= 0 119905119891= 078119891 = [minus2(119909

1+

025)+(1199092+05) exp(25119909

1(1199091+2))minus(119909

1+025)119906 05minus

1199092minus(1199092+05) exp(25119909

1(1199091+2))]119879 119889 = minus(1minus119906)(1+119906)

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(16) [26] 120601 = 1199093 1199050= 0 119905

119891= 1 119891 = [119909

2 119906 (12)119906

2

]119879

119889 = 1199091minus 19 119909

0= [0 0 0]

119879 120595 = [1199091 1199092+ 1]119879

(17) [26] 120601 = minus1199093 1199050= 0 119905

119891= 5 119891 = [119909

2 minus2 + 119906119909

3


0= [10 minus2 10]

119879120595 = [119909

1 1199092]119879

(18) [26] 120601 = (1199091minus1)2

+1199092

2+1199092

3 119892 = (12)119906

2 1199050= 0 119905119891= 5

119891 = [1199093cos 119906 119909

3sin 119906 sin 119906]119879 119909

0= [0 0 0]

119879


(19) [26] 119892 = 45(1199092

3+ 1199092

6) + 05(119906

2

1+ 1199062

2) 1199050= 0 119905

119891= 1

119891 = [91199094 91199095 91199096 9(1199061+ 1725119909

3) 91199062 minus(9119909

2)(1199061minus

270751199093+ 211990951199096)]119879 1199090= [0 22 0 0 minus1 0]

119879 120595 =

[1199091minus 10 119909

2minus 14 119909

3 1199094minus 25 119909

5 1199096]119879


2 1199093minus

4 1199094 1199095minus 1205874 119909

6]119879



References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Problem no1 2 3 4 5 6 7 8 9 10


120601119891

330 times 10minus9



11 12 13 14 15 16 17 18 19 20119869 48816 minus01273 00153 45727 979 times 10

minus4 23182 minus90882 00336 2513 20274120601119891

167 times 10minus5


minus4

594 times 10minus3


8 Conclusions


Appendix

Test Problems


(1) (VDP) [9] 119892 = (12)(1199092

1+ 1199092

2+ 1199062

) 1199050= 0 119905119891= 5 119891 =

[1199092 minus1199092+(1minus119909

2

1)1199092+119906]119879 1199090= [1 0]

119879120595 = 1199091minus1199092+1

(2) (CRP) [9] 119892 = (12)(1199092

1+1199092

2+01119906

2

) 1199050= 0 119905119891= 078

119891 = [1199091minus2(1199091+025)+(119909

2+05) exp(25119909

1(1199091+2))minus

(1199091+025)119906 05minus119909

2minus(1199092+05) exp(25119909

1(1199091+2))]119879

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(3) (FFRP) [9] 119892 = (12)(1199062

1+ 1199062

2+ 1199062

3+ 1199062

4) 1199050

=

0 119905119891

= 5 119891 = [1199092 (110)((119906

1+ 1199062) cos119909

5minus

(1199062+ 1199064) sin119909

5) 1199094 (110)((119906

1+ 1199063) sin119909

5+ (1199062+

1199064) cos119909

5) 1199096 (512)((119906

1+ 1199063) minus (119906

2+ 1199064))]119879 1199090=

[0 0 0 0 0 0]119879 120595 = [119909

1minus 4 1199092 1199093minus 4 1199094 1199095 1199096]119879

(4) (MSNIC) [20] 120601 = 1199093 1199050= 0 119905119891= 1 119891 = [119909

2 minus1199092+

119906 1199092

1+ 1199092

2+ 0005119906

2

]119879 119889 = [minus(119906 minus 20)(119906 + 20) 119909

2+

05 minus 8(119905 minus 05)2

]119879 1199090= [0 minus1 0]

119879

(5) (CSTCR) [20] 119892 = 1199092

1+ 1199092

2+ 01119906

2 1199050= 0 119905119891= 078

119891 = [minus(2 + 119906)(1199091+ 025) + (119909

2+ 05) exp(25119909

1(1199091+

2)) 05 minus 1199092minus (1199092+ 05) exp(25119909

1(1199091+ 2))]

119879 1199090=

[009 009]119879

(6) [34] 119892 = 21199091 1199050= 0 119905119891= 3 119891 = [119909

2 119906]119879 119889 = [minus(2 +

119906)(2 minus 119906) minus(6 + 1199091)]119879 1199090= [2 0]

119879(7) [36] 119892 = 119906

2 1199050= 0 119905

119891= 1 119891 = (12)119909

2 sin119909 + 1199061199090= 0 120595 = 119909 minus 05

(8) [26] 119892 = 1199092cos2119906 119905

0= 0 119905119891= 120587 119891 = sin(1199062) 119909

0=

1205872(9) [26] 119892 = (12)(119909

2

1+ 1199092

2+ 1199062

) 1199050= 0 119905

119891= 5 119891 =

[1199092 minus1199091+ (1 minus 119909

2

1)1199092+ 119906]119879 119889 = minus(119909

2+ 025) 119909

0=

[1 0]119879

(10) [26] 119892 = 1199092

1+ 1199092

2+ 0005119906

2

1199050= 0 119905

119891= 1 119891 =

[1199092 minus1199092+119906]119879


05) minus 05 minus 1199092)]119879

1199090= [0 minus1]

119879(11) [26] 119892 = (12)119906

2 1199050= 0 119905

119891= 2 119891 = [119909

2 119906]119879 1199090=

[1 1]119879 120595 = [119909

1 1199092]119879

(12) [26] 119892 = minus1199092 1199050= 0 119905

119891= 1 119891 = [119909

2 119906]119879 119889 =

minus(1 minus 119906)(1 + 119906) 1199090= [0 0]

119879 120595 = 1199092

(13) [26] 119892 = (12)(1199092

1+ 1199092

2+ 01119906

2

) 1199050= 0 119905

119891= 078

119891 = [minus2(1199091+ 025) + (119909

2+ 05) exp(25119909

1(1199091+ 2)) minus

(1199091+025)119906 05minus119909

2minus(1199092+05) exp(25119909

1(1199091+2))]119879

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(14) [26] 119892 = (12)1199062 1199050= 0 119905

119891= 10 119891 = [cos 119906 minus

1199092 sin 119906]119879 119889 = minus(120587 + 119906)(120587 minus 119906) 119909

0= [366 minus186]

119879120595 = [119909

1 1199092]119879

(15) [26]119892 = (12)(1199092

1+1199092

2) 1199050= 0 119905119891= 078119891 = [minus2(119909

1+

025)+(1199092+05) exp(25119909

1(1199091+2))minus(119909

1+025)119906 05minus

1199092minus(1199092+05) exp(25119909

1(1199091+2))]119879 119889 = minus(1minus119906)(1+119906)

1199090= [005 0]

119879 120595 = [1199091 1199092]119879

(16) [26] 120601 = 1199093 1199050= 0 119905

119891= 1 119891 = [119909

2 119906 (12)119906

2

]119879

119889 = 1199091minus 19 119909

0= [0 0 0]

119879 120595 = [1199091 1199092+ 1]119879

(17) [26] 120601 = minus1199093 1199050= 0 119905

119891= 5 119891 = [119909

2 minus2 + 119906119909

3


0= [10 minus2 10]

119879120595 = [119909

1 1199092]119879

(18) [26] 120601 = (1199091minus1)2

+1199092

2+1199092

3 119892 = (12)119906

2 1199050= 0 119905119891= 5

119891 = [1199093cos 119906 119909

3sin 119906 sin 119906]119879 119909

0= [0 0 0]

119879


(19) [26] 119892 = 45(1199092

3+ 1199092

6) + 05(119906

2

1+ 1199062

2) 1199050= 0 119905

119891= 1

119891 = [91199094 91199095 91199096 9(1199061+ 1725119909

3) 91199062 minus(9119909

2)(1199061minus

270751199093+ 211990951199096)]119879 1199090= [0 22 0 0 minus1 0]

119879 120595 =

[1199091minus 10 119909

2minus 14 119909

3 1199094minus 25 119909

5 1199096]119879


2 1199093minus

4 1199094 1199095minus 1205874 119909

6]119879



References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(19) [26] 119892 = 45(1199092

3+ 1199092

6) + 05(119906

2

1+ 1199062

2) 1199050= 0 119905

119891= 1

119891 = [91199094 91199095 91199096 9(1199061+ 1725119909

3) 91199062 minus(9119909

2)(1199061minus

270751199093+ 211990951199096)]119879 1199090= [0 22 0 0 minus1 0]

119879 120595 =

[1199091minus 10 119909

2minus 14 119909

3 1199094minus 25 119909

5 1199096]119879


2 1199093minus

4 1199094 1199095minus 1205874 119909

6]119879



References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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