Review Article A Review of Piecewise Linearization Methodsprocess, Li et al. [ ] developed a...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 101376 8 pageshttpdxdoiorg1011552013101376

Review ArticleA Review of Piecewise Linearization Methods

Ming-Hua Lin1 John Gunnar Carlsson2 Dongdong Ge3 Jianming Shi4 and Jung-Fa Tsai5

1 Department of Information Technology and Management Shih Chien University No 70 Dazhi Street Taipei 10462 Taiwan2 Program in Industrial and Systems Engineering University of Minnesota 111 Church Street SE Minneapolis MN 55455 USA3 School of Information Management and Engineering Shanghai University of Finance and Economics Shanghai 200433 China4 School of Management Tokyo University of Science 500 Shimokiyoku Kuki Saitama 346-8512 Japan5Department of Business Management National Taipei University of Technology Section 3 No 1 Chung-Hsiao E RoadTaipei 10608 Taiwan

Correspondence should be addressed to Jung-Fa Tsai jftsaintutedutw

Received 3 July 2013 Accepted 9 September 2013

Academic Editor Yi-Chung Hu

Copyright copy 2013 Ming-Hua Lin et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Various optimization problems in engineering and management are formulated as nonlinear programming problems Because ofthe nonconvexity nature of this kind of problems no efficient approach is available to derive the global optimum of the problemsHow to locate a global optimal solution of a nonlinear programming problem is an important issue in optimization theory Inthe last few decades piecewise linearization methods have been widely applied to convert a nonlinear programming problem intoa linear programming problem or a mixed-integer convex programming problem for obtaining an approximated global optimalsolution In the transformation process extra binary variables continuous variables and constraints are introduced to reformulatethe original problemThese extra variables and constraints mainly determine the solution efficiency of the converted problemThisstudy therefore provides a review of piecewise linearizationmethods and analyzes the computational efficiency of various piecewiselinearization methods

1 Introduction

Piecewise linear functions are frequently used in variousapplications to approximate nonlinear programs with non-convex functions in the objective or constraints by addingextra binary variables continuous variables and constraintsThey naturally appear as cost functions of supply chain prob-lems to model quantity discount functions for bulk procure-ment and fixed charges For example the transportation costinventory cost and production cost in a supply chain networkare often constructed as a sum of nonconvex piecewise linearfunctions due to economies of scale [1] Optimization prob-lems with piecewise linear costs arise in many applicationdomains including transportation telecommunications andproduction planning Specific applications include variantsof the minimum cost network flow problem with nonconvexpiecewise linear costs [2ndash7] the network loading problem [8ndash11] the facility location problem with staircase costs [12 13]the merge-in-transit problem [14] and the packing problem

[15ndash17] Other applications also include production plan-ning [18] optimization of electronic circuits [19] operationplanning of gas networks [20] process engineering [21 22]engineering design [23 24] appointment scheduling [25]and other network flow problems with nonconvex piecewiselinear objective functions [7]

Various methods of piecewisely linearizing a nonlinearfunction have been proposed in the literature [26ndash39] Twowell-known mixed-integer formulations for piecewise linearfunctions are the incremental cost [40] and the convexcombination [41] formulations Padberg [35] compared thelinear programming relaxations of the two mixed-integerprogramming models for piecewise linear functions in thesimplest case when no constraint exists He showed thatthe feasible set of the linear programming relaxation of theincremental cost formulation is integral that is the binaryvariables are integers at every vertex of the set He calledsuch formulations locally ideal On the other hand theconvex combination formulation is not locally ideal and it

2 Mathematical Problems in Engineering

strictly contains the feasible set of the linear programmingrelaxation of the incremental cost formulation Then Sherali[42] proposed a modified convex combination formulationthat is locally ideal Alternatively Beale and Tomlin [43]suggested a formulation for the piecewise linear functionsimilar to convex combination except that no binary variableis included in the model and the nonlinearities are enforcedalgorithmically directly in the branch-and-bound algorithmby branching on sets of variables which they called specialordered sets of type 2 (SOS2) It is also possible to formulatepiecewise linear functions similar to incremental cost butwithout binary variables and enforcing the nonlinearitiesdirectly in the branch-and-bound algorithm Two advantagesof eliminating binary variables are the substantial reductionin the size of themodel and the use of the polyhedral structureof the problem [44 45] Keha et al [46] studied formulationsof linear programs with piecewise linear objective functionswith andwithout additional binary variables and showed thatadding binary variables does not improve the bound of thelinear programming relaxationKeha et al [47] also presenteda branch-and-cut algorithm for solving linear programswith continuous separable piecewise-linear cost functionsInstead of introducing auxiliary binary variables and otherlinear constraints to represent SOS2 constraints used inthe traditional approach they enforced SOS2 constraints bybranching on them without auxiliary binary variables

Due to the broad applications of piecewise linear func-tions many studies have conducted related research on thistopic The main purpose of these studies is to find a betterway to represent a piecewise linear function or to tighten thelinear programming relaxation A superior representation ofpiecewise linear functions can effectively reduce the problemsize and enhance the computational efficiency However forexpressing a piecewise linear function of a single variable 119909

with119898+1 break points most of themethods in the textbooksand literature require adding extra 119898 binary variables and4m constraints which may cause a heavy computationalburden when 119898 is large Recently Li et al [48] developeda representation method for piecewise linear functions withfewer binary variables compared to the traditional methodsAlthough their method needs only lceillog

2119898rceil extra binary

variables to piecewisely linearize a nonlinear function with119898 + 1 break points the approximation process still requires8 + 8lceillog

2119898rceil extra constraints 2119898 nonnegative continuous

variables and 2lceillog2119898rceil free-signed continuous variables

Vielma et al [39] presented a note on Li et alrsquos paperand showed that two representations for piecewise linearfunctions introduced by Li et al [48] are both theoreticallyand computationally inferior to standard formulations forpiecewise linear functions Tsai and Lin [49] applied theVielma et al [39] techniques to express a piecewise linearfunction for solving a posynomial optimization problemCroxton et al [31] indicated that most models of expressingpiecewise linear functions are equivalent to each otherAdditionally it is well known that the numbers of extravariables and constraints required in the linearization processfor a nonlinear function obviously impact the computational

performance of the converted problemTherefore this paperfocuses on discussing and reviewing the recent advancesin piecewise linearization methods Section 2 reviews thepiecewise linearization methods Section 3 compares theformulations of various methods with the numbers of extrabinarycontinuous variables and constraints Section 4 dis-cusses error evaluation in piecewise linear approximationConclusions are made in Section 5

2 Formulations of PiecewiseLinearization Functions

Consider a general nonlinear function 119891(119909) of a singlevariable 119909 119891(119909) is a continuous function and 119909 is withinthe interval [119886

0 119886119898] Most commonly used textbooks of

nonlinear programming [26ndash28] approximate the nonlinearfunction by a piecewise linear function as follows

Firstly denote 119886119896(119896 = 0 2 119898) as the break points of

119891(119909) 1198860lt 1198861lt sdot sdot sdot lt 119886

119898 and Figure 1 indicates the piecewise

linearization of 119891(119909)119891(119909) can then be approximately linearized over the

interval [1198860 119886119898] as

119871 (119891 (119909)) =

119898

sum

119896=0

119891 (119886119896) 119905119896 (1)

where 119909 = sum119898

119896=0119886119896119905119896 sum119898119896=0

119905119896= 1 119905

119896ge 0 in which only two

adjacent 119905119896rsquos are allowed to be nonzero A nonlinear function

is then converted into the following expressions

Method 1 Consider

119871 (119891 (119909)) =

119898

sum

119896=1

119891 (119886119896) 119905119896

119909 =

119898

sum

119896=1

119886119896119905119896

1199050le 1199100

119905119896le 119910119896minus1

+ 119910119896 for 119896 = 1 2 119898 minus 1 119905

119898le 119910119898minus1

119898minus1

sum

119896=0

119910119896= 1

119898

sum

119896=0

119905119896= 1

(2)

where 119910119896isin 0 1 119905

119896ge 0 119896 = 0 1 119898 minus 1

The above expressions involve 119898 new binary variables1199100 1199101 119910

119898minus1 The number of newly added 0-1 variables for

piecewisely linearizing a function 119891(119909) equals the number ofbreaking intervals (ie119898) If119898 is large it may cause a heavycomputational burden

Li and Yu [33] proposed another global optimizationmethod for nonlinear programming problems where theobjective function and the constraints might be nonconvexA univariate function is initially expressed by a piecewise

Mathematical Problems in Engineering 3

a0

a1

a2

am

x

f(x)

L(f(x))

Figure 1 Piecewise linearization of 119891(119909)

linear function with a summation of absolute terms Denote119904119896(119896 = 0 1 119898minus 1) as the slopes of line segments between

119886119896and 119886119896+1

expressed as 119904119896= [119891(119886

119896+1) minus 119891(119886

119896)][119886119896+1

minus 119886119896]

119891(119909) can then be written as follows

119871 (119891 (119909)) = 119891 (1198860) + 1199040(119909 minus 119886

0)

+

119898minus1

sum

119896=1

119904119896minus 119904119896minus1

2

(1003816100381610038161003816119909 minus 119886119896

1003816100381610038161003816+ 119909 minus 119886

119896)

(3)

119891(119909) is convex in the interval [119886119896minus1

119886119896] if 119904119896minus 119904119896minus1

ge 0and otherwise 119891(119909) is a non-convex function which needs tobe linearized by adding extra binary variables By linearizingthe absolute terms Li and Yu [33] converted the nonlinearfunction into a piecewise linear function as shown below

Method 2 Consider

119871 (119891 (119909)) = 119891 (1198860) + 1199040(119909 minus 119886

0)

+ sum

119896119904119896gt119904119896minus1

(119904119896minus 119904119896minus1

)(119909 minus 119886119896+

119896minus1

sum

119897=0

119889119897)

+

1

2

sum

119896 119904119896lt119904119896minus1

(119904119896minus 119904119896minus1

) (119909 minus 2119911119896+ 2119886119896119906119896minus 119886119896)

119909 +

119898minus2

sum

119897=0

119889119897ge 119886119898minus1

0 le 119889119897le 119886119897+1

minus 119886119897

where 119904119896gt 119904119896minus1

119909 + 119909 (119906119896minus 1) le 119911

119896 119911

119896ge 0 where 119904

119896lt 119904119896minus1

(4)

where 119909 ge 0 119889119897ge 0 119911

119896ge 0 119906

119896isin 0 1 119909 are upper bounds

of 119909 and 119906119896are extra binary variables used to linearize a non-

convex function 119891(119909) for the interval [119886119896minus1

119886119896]

Comparing Method 2 with Method 1 Method 1 usesbinary variables to linearize 119891(119909) for whole 119909 interval Butthe binary variables used in Method 2 are only applied tolinearize the non-convex parts of 119891(119909) Method 2 thereforeuses fewer 0-1 variables than Method 1 However for 119891(119909)

with 119902 intervals of the non-convex parts Method 2 stillrequires 119902 binary variables to linearize 119891(119909)

Another general form of representing a piecewise linearfunction is proposed in the articles of Croxton et al [31] Li[32] Padberg [35] Topaloglu and Powell [36] and Li and Tsai[38] The expressions are formulated as shown below

Method 3 Consider

119886119896minus (119886119898

minus 1198860) (1 minus 120582

119896)

le 119909 le 119886119896+1

minus (119886119898minus 1198860) (1 minus 120582

119896) 119896 = 0 1 119898 minus 1(5)

where sum119898minus1

119896=0120582119896= 1 120582

119896isin 0 1 and

119891 (119886119896) + 119904119896(119909 minus 119886

119896) minus 119872(1 minus 120582

119896)

le 119891 (119909)

le 119891 (119886119896) + 119904119896(119909 minus 119886

119896)

+ 119872(1 minus 120582119896) 119896 = 0 1 119898 minus 1

(6)

where119872 is a large constant and 119904119896= (119891(119886

119896+1)minus119891(119886

119896))(119886119896+1

minus

119886119896)

The above expressions require extra 119898 binary variablesand 4119898 constraints where 119898 + 1 break points are used torepresent a piecewise linear function

Form the above discussions we can know that Methods1 2 and 3 require a number of extra binary variables andextra constraints linear in 119898 to express a piecewise linearfunction To approximate a nonlinear function by using apiecewise linear function the numbers of extra binary vari-able and constraints significantly influence the computationalefficiency If fewer binary variables and constraints are usedto represent a piecewise linear function then less CPU timeis needed to solve the transformed problem For decreasingthe extra binary variables involved in the approximationprocess Li et al [48] developed a representation methodfor piecewise linear functions with the number of binaryvariables logarithmic in 119898 Consider the same piecewiselinear function 119891(119909) discussed above where 119909 is within theinterval [119886

0 119886119898] and119898+ 1 break points exist within [119886

0 119886119898]

Let 120579 be an integer 0 le 120579 le 119898 minus 1 expressed as

120579 =

ℎ

sum

119895=1

2119895minus1

119906119895 ℎ = lceillog

2119898rceil 119906

119895isin 0 1 (7)

Let 119866(120579) sube 1 2 ℎ be a set composed of all indicessuch that sum

119895isin119866(120579)2119895minus1

= 120579 For instance 119866(0) = 120601 119866(3) =

1 2Denote 119866(120579) to be the number of elements in 119866(120579) For

instance 119866(0) = 0 119866(3) = 2To approximate a univariate nonlinear function by using

a piecewise linear function the following expressions arededuced by the Li et al [48] method


Method 4 Consider

119898minus1

sum

120579=0

119903120579119886120579le 119909 le

119898minus1

sum

120579=0

119903120579119886120579+1

119891 (119909) =

119898minus1

sum

120579=0

(119891 (119886120579) minus 119904120579(119886120579minus 1198860)) 119903120579

+

119898minus1

sum

120579=0

119904120579119908120579 119904120579=

119891 (119886120579+1

) minus 119891 (119886120579)

119886120579+1

minus 119886120579

119898minus1

sum

120579=0

119903120579= 1 119903

120579ge 0

119898minus1

sum

120579=0

119903120579

119866 (120579) +

ℎ

sum

119895=1

119911119895= 0

minus1199061015840

119895le 119911119895le 1199061015840

119895 119895 = 1 2 ℎ

119898minus1

sum

120579=0

119903120579119888120579119895

minus (1 minus 1199061015840

119895) le 119911119895le

119898minus1

sum

120579=0

119903120579119888120579119895

+ (1 minus 1199061015840

119895)

119895 = 1 2 ℎ

119898minus1

sum

120579=0

119908120579= 119909 minus 119886

0

119898minus1

sum

120579=0

119908120579

119866 (120579) +

ℎ

sum

119895=1

120575119895= 0

minus (119886119898

minus 1198860) 1199061015840

119895le 120575119895le (119886119898

minus 1198860) 1199061015840

119895 119895 = 1 2 ℎ

119898minus1

sum

120579=0

119908120579119888120579119895

minus (119886119898

minus 1198860) (1 minus 119906

1015840

119895)

le 120575119895le

119898minus1

sum

120579=0

119908120579119888120579119895

+ (119886119898

minus 1198860) (1 minus 119906

1015840

119895) 119895 = 1 2 ℎ

ℎ

sum

119895=1

2119895minus1

1199061015840

119895le 119898

(8)

where 1199061015840119895isin 0 1 119888

120579119895 119911119895 and 120575

119895are free continuous variables

119903120579and 119908

120579are nonnegative continuous and all the variables

are the same as defined before

The expressions of Method 4 for representing a piecewiselinear function 119891(119909) with 119898 + 1 break points use lceillog

2119898rceil

binary variables 8 + 8lceillog2119898rceil constraints 2119898 non-negative


Comparing with Methods 1 2 and 3 Method 4 indeedreduces the number of binary variables used such that thecomputational efficiency is improved Although Li et al [48]developed a superior way of expressing a piecewise linearfunction by using fewer binary variables Vielma et al [39]

investigated that this representation for piecewise linear func-tions is theoretically and computationally inferior to standardformulations for piecewise linear functions Vielma andNemhauser [50] recently developed a novel piecewise linearexpression requiring fewer variables and constraints than thecurrent piecewise linearization techniques to approximatethe univariate nonlinear functions Their method needsa logarithmic number of binary variables and constraintsto express a piecewise linear function The formulation isdescribed as shown below

Let 119875 = 0 1 2 119898 and 119901 isin 119875 An injective function119861 1 2 119898 rarr 0 1

120579 120579 = lceillog2119898rceil where the vectors

119861(119901) and 119861(119901 + 1) differ in at most one component for all119901 isin 1 2 119898 minus 1

Let 119861(119901) = (1199061 1199062 119906

120579) for all 119906

119896isin 0 1 119896 =

1 2 120579 and 119861(0) = 119861(1) Some notations are introducedbelow

119878+(119896) a set composed of all 119901 where 119906

119896= 1 of 119861(119901) and

119861(119901 + 1) for 119901 = 1 2 119898 minus 1 or 119906119896

= 1 of 119861(119901) for 119901 isin

0119898 that is 119878+(119896) = 119901 | forall 119861(119901) and119861(119901 + 1) 119906119896= 1 119901 =

1 2 119898 minus 1 cup 119901 | forall 119861(119901) 119906119896= 1 119901 isin 0119898

119878minus(119896) a set composed of all 119901 where 119906

119896= 0 of 119861(119901) and

119861(119901 + 1) for 119901 = 1 2 119898 minus 1 or 119906119896

= 0 of 119861(119901) for 119901 isin

0119898 that is 119878minus(119896) = 119901 | forall 119861(119901) and119861(119901 + 1) 119906119896= 0 119901 =


The linear approximation of a univariate 119891(119909) 1198860

le

119909 le 119886119898 by the technique of Vielma and Nemhauser [50] is

formulated as follows

Method 5 Denote 119871(119891(119909)) as the piecewise linear function of119891(119909) where 119886

0lt 1198861lt 1198862lt sdot sdot sdot lt 119886

119898be the119898+1 break points

of 119871(119891(119909)) 119871(119891(119909)) can be expressed as

119871 (119891 (119909)) =

119898

sum

119901=0

119891 (119886119901) 120582119901

119909 =

119898

sum

119901=0

119886119901120582119901

119898

sum

119901=0

120582119901= 1

sum

119901isin119878+(119896)

120582119901le 119906119896

sum

119901isin119878minus(119896)

120582119901le 1 minus 119906

119896 forall120582

119901isin R+ forall119906119896isin 0 1

(9)

Method 5 uses lceillog2119898rceil binary variables119898+1 continuous

variables and 3+2lceillog2119898rceil constraints to express a piecewise

linearization function with119898 line segments

3 Formulation Comparisons

The comparison results of the above five methods in termsof the numbers of binary variables continuous variables andconstraints are listed in Table 1 The number of extra binaryvariables of Methods 1 and 3 is linear in the number of linesegments Methods 4 and 5 have the logarithmic number of


extra binary variables with119898 line segments and the numberof extra binary variables of Method 2 is equal to the numberof concave piecewise line segments In the deterministicglobal optimization for a minimization problem inversepower and exponential transformations generate nonconvexexpressions that require to be linearly approximated in thereformulated problem That means Methods 4 and 5 aresuperior to Methods 1 2 and 3 in terms of the numbers ofextra binary variables and constraints as shown in Table 1Moreover Method 5 has fewer extra continuous variablesand constraints than Method 4 in linearizing a nonlinearfunction

Till et al [51] reviewed the literature on the complexityof mixed-integer linear programming (MILP) problems andsummarized that the computational complexity varies from119874(119889sdot119899

2) to119874(2

119889sdot1198993) where 119899 is the number of constraints and

119889 is the number of binaries Therefore reducing constraintsand binary variables makes a greater impact than reducingcontinuous variables on computational efficiency of solvingMILP problems For finding a global solution of a nonlinearprogramming problem by a piecewise linearization methodif the linearization method generates a large number ofadditional constraints and binaries the computational effi-ciency will decrease and cause heavy computational burdensAccording to the above discussions Method 5 is morecomputationally efficient than the other fourmethods Exper-iment results from the literature [39 48 49] also support thestatement

Beale and Tomlin [43] suggested a formulation forpiecewise linear functions by using continuous variablesin special ordered sets of type 2 (SOS2) Although nobinary variables are included in the SOS2 formulation thenonlinearities are enforced algorithmically and directly inthe branch-and-bound algorithm by branching on sets ofvariables Since the traditional SOS2 branching schemes havetoo many dichotomies the piecewise linearization techniquein Method 5 induces an independent branching scheme oflogarithm depth and provides a significant computationaladvantage [50] The computational results in Vielma andNemhauser [50] show that Method 5 outperforms the SOS2model without binary variables

The factors affecting the computational efficiency in solv-ing nonlinear programming problems include the tightnessof the constructed convex underestimator the efficiency ofthe piecewise linearization technique and the number of thetransformed variables An appropriate variable transforma-tion constructs a tighter convex underestimator and makesfewer break points required in the linearization process tosatisfy the same optimality tolerance and feasibility toleranceVielma andNemhauser [50] indicated that the formulation ofMethod 5 is sharp and locally ideal andhas favorable tightnesspropertiesThey presented experimental results showing thatMethod 5 significantly outperforms othermethods especiallywhen the number of break points becomes large Vielma et al[39] explained that the formulation of Method 4 is not sharpand is theoretically and computationally inferior to standardMILP formulations (convex combinationmodel logarithmicconvex combination model) for piecewise linear functions

4 Error Evaluation

For evaluating the error of piecewise linear approximationTsai and Lin [49 52] and Lin and Tsai [53] utilized theexpression |119891(119909) minus 119871(119891(119909))| to estimate the error indicatedin Figure 2 If 119891(119909) is the objective function 119892

119894(119909) lt 0

is the 119894th constraint and 119909lowast is the solution derived from

the transformed program then the linearization does notrequire to be refined until |119891(119909

lowast) minus 119871(119891(119909

lowast))| le 120576

1and

Max119894(119892119894(119909lowast)) le 1205762 where |119891(119909

lowast) minus 119871(119891(119909

lowast))| is the evaluated

error in objective 1205761is the optimality tolerance 119892

119894(119909lowast) is the

error in the 119894th constraint and 1205762is the feasibility tolerance

The accuracy of the linear approximation significantlydepends on the selection of break points and more breakpoints can increase the accuracy of the linear approximationSince adding numerous break points leads to a significantincrease in the computational burden the break point selec-tion strategies can be applied to improve the computationalefficiency in solving optimization problems by the determin-istic approaches Existing break point selection strategies areclassified into three categories as follows [54]

(i) add a new break point at themidpoint of each intervalof existing break points

(ii) add a new break point at the point with largestapproximation error of each interval

(iii) add a new break point at the previously obtainedsolution point

According to the deterministic optimization methodsfor solving nonconvex nonlinear problems [29 33 38 3948 49 53ndash56] the inverse or logarithmic transformation isrequired to be approximated by the piecewise linearizationfunction For example the function 119910 = ln119909 or 119910 = 119909

minus1 isrequired to be piecewisely linearized by using an appropriatebreakpoint selection strategy if a new break point is addedat the midpoint of each interval of existing break points orat the point with largest approximation error the numberof line segments becomes double in each iteration If anew breakpoint is added at the previously obtained solutionpoint only one breakpoint is added in each iteration Howto improve the computational efficiency by a better breakpoint selection strategy still needs more investigations orexperiments to get concrete results

5 Conclusions

This study provides an overview on some of the mostcommonly used piecewise linearization methods in deter-ministic optimization From the formulation point of viewthe numbers of extra binaries continuous variables andconstraints are decreasing in the latest development methodsespecially for the number of extra binaries which may causeheavy computational burdens Additionally a good piecewiselinearization method must consider the tightness propertiessuch as sharp and locally ideal Since effective break pointsselection strategy is important to enhance the computationalefficiency in linear approximation more work should bedone to study the optimal positioning of the break points


Table 1 Comparison results of five methods in expressing a piecewise linearization function with119898 line segments (ie119898+ 1 break points)

Items Method 1 Method 2 Method 3 Method 4 Method 5No of binary variables 119898 119902 (no of concave piecewise segments) 119898 lceillog

2119898)rceil lceillog

2119898rceil

No of continuous variables 119898 + 1 119898 + 1 0 2119898 + 2 lceillog2119898rceil 119898 + 1

No of constraints 119898 + 5 119898 + 1 4119898 8 + 8 lceillog2119898rceil 3 + 2 lceillog

2119898rceil

a0

a1

x

Error

f(xlowast

)

xlowast

L(f(xlowast

))

f(x)

f(x)

(a)

a1

a2

x

Error

xlowasta

0

f(xlowast

)

L(f(xlowast

))

f(x)

(b)

Figure 2 Error evaluation of the linear approximation

Although a logarithmic piecewise linearization method withgood tightness properties has been proposed it is stilltoo time consuming for finding an approximately globaloptimum of a large scale nonconvex problem Developing anefficient polynomial time algorithm for solving nonconvexproblems by piecewise linearization techniques is still achallenging question Obviously this contribution gives onlya few preliminary insights and might point toward issuesdeserving additional research

Acknowledgment

The research is supported by Taiwan NSC Grants NSC 101-2410-H-158-002-MY2 and NSC 102-2410-H-027-012-MY3

References

[1] J F Tsai ldquoAn optimization approach for supply chain manage-ment models with quantity discount policyrdquo European Journalof Operational Research vol 177 no 2 pp 982ndash994 2007

[2] E H Aghezzaf and L A Wolsey ldquoModelling piecewise linearconcave costs in a tree partitioning problemrdquo Discrete AppliedMathematics vol 50 no 2 pp 101ndash109 1994

[3] A Balakrishnan and S Graves ldquoA composite algorithm for aconcave-cost network flow problemrdquo Networks vol 19 no 2pp 175ndash202 1989

[4] K L CroxtonModeling and solving network flow problems withpiecewise linear costs with applications in supply chain manage-ment [PhD thesis] Operations Research CenterMassachusettsInstitute of Technology Cambridge Mass USA 1999

[5] L M A Chan A Muriel Z J Shen and D Simchi-LevildquoOn the effectiveness of zero-inventory-ordering policies for theeconomic lot-sizing model with a class of piecewise linear coststructuresrdquo Operations Research vol 50 no 6 pp 1058ndash10672002

[6] L M A Chan A Muriel Z J M Shen et al ldquoEffec-tive zero-inventory-ordering policies for the single-warehousemultiretailer problem with piecewise linear cost structuresrdquoManagement Science vol 48 no 11 pp 1446ndash1460 2002

[7] K L Croxton B Gendron and T L Magnanti ldquoVariabledisaggregation in network flow problems with piecewise linearcostsrdquo Operations Research vol 55 no 1 pp 146ndash157 2007

[8] D Bienstock and O Gunluk ldquoCapacitated network designpolyhedral structure and computationrdquo INFORMS Journal onComputing vol 8 no 3 pp 243ndash259 1996

[9] V Gabrel A Knippel and M Minoux ldquoExact solution ofmulticommodity network optimization problems with generalstep cost functionsrdquo Operations Research Letters vol 25 no 1pp 15ndash23 1999

[10] O Gunluk ldquoA branch-and-cut algorithm for capacitated net-work design problemsrdquo Mathematical Programming A vol 86no 1 pp 17ndash39 1999

[11] T L Magnanti P Mirchandani and R Vachani ldquoModeling andsolving the two-facility capacitated network loading problemrdquoOperations Research vol 43 no 1 pp 142ndash157 1995

[12] K Holmberg ldquoSolving the staircase cost facility location prob-lem with decomposition and piecewise linearizationrdquo EuropeanJournal of Operational Research vol 75 no 1 pp 41ndash61 1994

[13] KHolmberg and J Ling ldquoA Lagrangean heuristic for the facilitylocation problem with staircase costsrdquo European Journal ofOperational Research vol 97 no 1 pp 63ndash74 1997


[14] K L Croxton B Gendron and T L Magnanti ldquoModelsand methods for merge-in-transit operationsrdquo TransportationScience vol 37 no 1 pp 1ndash22 2003

[15] H L Li C T Chang and J F Tsai ldquoApproximately global opti-mization for assortment problems using piecewise linearizationtechniquesrdquo European Journal of Operational Research vol 140no 3 pp 584ndash589 2002

[16] J F Tsai andH L Li ldquoA global optimizationmethod for packingproblemsrdquoEngineeringOptimization vol 38 no 6 pp 687ndash7002006

[17] J F Tsai P C Wang and M H Lin ldquoAn efficient determin-istic optimization approach for rectangular packing problemsrdquoOptimization 2013

[18] R Fourer DM Gay and BW KernighanAMPLmdashAModelingLanguage for Mathematical Programming The Scientific PressSan Francisco Calif USA 1993

[19] T Graf P van Hentenryck C Pradelles-Lasserre and LZimmer ldquoSimulation of hybrid circuits in constraint logicprogrammingrdquo Computers amp Mathematics with Applicationsvol 20 no 9-10 pp 45ndash56 1990

[20] A Martin M Moller and S Moritz ldquoMixed integer models forthe stationary case of gas network optimizationrdquoMathematicalProgramming vol 105 no 2-3 pp 563ndash582 2006

[21] M L Bergamini P Aguirre and I Grossmann ldquoLogic-basedouter approximation for globally optimal synthesis of processnetworksrdquo Computers and Chemical Engineering vol 29 no 9pp 1914ndash1933 2005

[22] M L Bergamini I Grossmann N Scenna and P AguirreldquoAn improved piecewise outer-approximation algorithm for theglobal optimization of MINLP models involving concave andbilinear termsrdquo Computers and Chemical Engineering vol 32no 3 pp 477ndash493 2008

[23] J F Tsai ldquoGlobal optimization of nonlinear fractional program-ming problems in engineering designrdquo Engineering Optimiza-tion vol 37 no 4 pp 399ndash409 2005

[24] M H Lin J F Tsai and P C Wang ldquoSolving engineeringoptimization problems by a deterministic global approachrdquoApplied Mathematics and Information Sciences vol 6 no 7 pp21Sndash27S 2012

[25] D Ge G Wan Z Wang and J Zhang ldquoA note on appointmentscheduling with piece-wise linear cost functionsrdquo Workingpaper 2012

[26] M S Bazaraa H D Sherali and C M Shetty NonlinearProgrammingmdashTheory andAlgorithms JohnWileyampSonsNewYork NY USA 2nd edition 1993

[27] F S Hillier and G J Lieberman Introduction to OperationsResearch McGraw-Hill New York NY USA 6th edition 1995

[28] H A Taha Operations Research Macmillan New York NYUSA 7th edition 2003

[29] C A Floudas Deterministic Global OptimizationmdashTheoryMethods and Applications Kluwer Academic Publishers Dor-drecht The Netherlands 1999

[30] S Vajda Mathematical Programming Addison-Wesley NewYork NY USA 1964

[31] K L Croxton B Gendron and T L Magnanti ldquoA comparisonof mixed-integer programming models for nonconvex piece-wise linear cost minimization problemsrdquo Management Sciencevol 49 no 9 pp 1268ndash1273 2003

[32] H L Li ldquoAn efficient method for solving linear goal program-ming problemsrdquo Journal of Optimization Theory and Applica-tions vol 90 no 2 pp 465ndash469 1996

[33] H L Li and C S Yu ldquoGlobal optimization method for non-convex separable programming problemsrdquo European Journal ofOperational Research vol 117 no 2 pp 275ndash292 1999

[34] H L Li and H C Lu ldquoGlobal optimization for generalizedgeometric programswithmixed free-sign variablesrdquoOperationsResearch vol 57 no 3 pp 701ndash713 2009

[35] M Padberg ldquoApproximating separable nonlinear functions viamixed zero-one programsrdquo Operations Research Letters vol 27no 1 pp 1ndash5 2000

[36] H Topaloglu and W B Powell ldquoAn algorithm for approximat-ing piecewise linear concave functions from sample gradientsrdquoOperations Research Letters vol 31 no 1 pp 66ndash76 2003

[37] S Kontogiorgis ldquoPractical piecewise-linear approximation formonotropic optimizationrdquo INFORMS Journal on Computingvol 12 no 4 pp 324ndash340 2000

[38] H L Li and J F Tsai ldquoTreating free variables in generalizedgeometric global optimization programsrdquo Journal of GlobalOptimization vol 33 no 1 pp 1ndash13 2005

[39] J P Vielma S Ahmed and G Nemhauser ldquoA note on lsquoasuperior representation method for piecewise linear functionsrsquordquoINFORMS Journal on Computing vol 22 no 3 pp 493ndash4972010

[40] H M Markowitz and A S Manne ldquoOn the solution of discreteprogramming problemsrdquo Econometrica vol 25 no 1 pp 84ndash110 1957

[41] G B Dantzig ldquoOn the significance of solving linear program-ming problems with some integer variablesrdquo Econometrica vol28 no 1 pp 30ndash44 1960

[42] H D Sherali ldquoOn mixed-integer zero-one representations forseparable lower-semicontinuous piecewise linear functionsrdquoOperations Research Letters vol 28 no 4 pp 155ndash160 2001

[43] E L M Beale and J A Tomlin ldquoSpecial facilities in a generalmathematical programming system for nonconvex problemsusing ordered sets of variablesrdquo in Proceedings of the 5thInternational Conference on Operations Research J LawrenceEd pp 447ndash454 Tavistock Publications London UK 1970

[44] I R de Farias Jr E L Johnson and G L Nemhauser ldquoBranch-and-cut for combinatorial optimization problems without aux-iliary binary variablesrdquoThe Knowledge Engineering Review vol16 no 1 pp 25ndash39 2001

[45] T Nowatzki M Ferris K Sankaralingam and C Estan Opti-mization and Mathematical Modeling in Computer ArchitectureMorgan amp Claypool Publishers San Rafael Calif USA 2013

[46] A B Keha I R de Farias and G L Nemhauser ldquoModelsfor representing piecewise linear cost functionsrdquo OperationsResearch Letters vol 32 no 1 pp 44ndash48 2004

[47] A B Keha I R de Farias andG L Nemhauser ldquoA branch-and-cut algorithmwithout binary variables for nonconvex piecewiselinear optimizationrdquoOperations Research vol 54 no 5 pp 847ndash858 2006

[48] H L Li H C Lu C H Huang and N Z Hu ldquoA superior rep-resentation method for piecewise linear functionsrdquo INFORMSJournal on Computing vol 21 no 2 pp 314ndash321 2009

[49] J F Tsai andM H Lin ldquoAn efficient global approach for posyn-omial geometric programming problemsrdquo INFORMS Journalon Computing vol 23 no 3 pp 483ndash492 2011

[50] J P Vielma and G L Nemhauser ldquoModeling disjunctiveconstraints with a logarithmic number of binary variables andconstraintsrdquo Mathematical Programming vol 128 no 1-2 pp49ndash72 2011


[51] J Till S Engell S Panek and O Stursberg ldquoApplied hybridsystem optimization an empirical investigation of complexityrdquoControl Engineering Practice vol 12 no 10 pp 1291ndash1303 2004

[52] J F Tsai and M H Lin ldquoGlobal optimization of signomialmixed-integer nonlinear programming problems with freevariablesrdquo Journal of Global Optimization vol 42 no 1 pp 39ndash49 2008

[53] M H Lin and J F Tsai ldquoRange reduction techniques forimproving computational efficiency in global optimization ofsignomial geometric programming problemsrdquo European Jour-nal of Operational Research vol 216 no 1 pp 17ndash25 2012

[54] A Lundell Transformation techniques for signomial functionsin global optimization [PhD thesis] Abo Akademi University2009

[55] A Lundell and T Westerlund ldquoOn the relationship betweenpower and exponential transformations for positive signomialfunctionsrdquo Chemical Engineering Transactions vol 17 pp 1287ndash1292 2009

[56] A Lundell J Westerlund and T Westerlund ldquoSome transfor-mation techniques with applications in global optimizationrdquoJournal of Global Optimization vol 43 no 2-3 pp 391ndash4052009

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


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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


strictly contains the feasible set of the linear programmingrelaxation of the incremental cost formulation Then Sherali[42] proposed a modified convex combination formulationthat is locally ideal Alternatively Beale and Tomlin [43]suggested a formulation for the piecewise linear functionsimilar to convex combination except that no binary variableis included in the model and the nonlinearities are enforcedalgorithmically directly in the branch-and-bound algorithmby branching on sets of variables which they called specialordered sets of type 2 (SOS2) It is also possible to formulatepiecewise linear functions similar to incremental cost butwithout binary variables and enforcing the nonlinearitiesdirectly in the branch-and-bound algorithm Two advantagesof eliminating binary variables are the substantial reductionin the size of themodel and the use of the polyhedral structureof the problem [44 45] Keha et al [46] studied formulationsof linear programs with piecewise linear objective functionswith andwithout additional binary variables and showed thatadding binary variables does not improve the bound of thelinear programming relaxationKeha et al [47] also presenteda branch-and-cut algorithm for solving linear programswith continuous separable piecewise-linear cost functionsInstead of introducing auxiliary binary variables and otherlinear constraints to represent SOS2 constraints used inthe traditional approach they enforced SOS2 constraints bybranching on them without auxiliary binary variables

Due to the broad applications of piecewise linear func-tions many studies have conducted related research on thistopic The main purpose of these studies is to find a betterway to represent a piecewise linear function or to tighten thelinear programming relaxation A superior representation ofpiecewise linear functions can effectively reduce the problemsize and enhance the computational efficiency However forexpressing a piecewise linear function of a single variable 119909

with119898+1 break points most of themethods in the textbooksand literature require adding extra 119898 binary variables and4m constraints which may cause a heavy computationalburden when 119898 is large Recently Li et al [48] developeda representation method for piecewise linear functions withfewer binary variables compared to the traditional methodsAlthough their method needs only lceillog

2119898rceil extra binary

variables to piecewisely linearize a nonlinear function with119898 + 1 break points the approximation process still requires8 + 8lceillog

2119898rceil extra constraints 2119898 nonnegative continuous


Vielma et al [39] presented a note on Li et alrsquos paperand showed that two representations for piecewise linearfunctions introduced by Li et al [48] are both theoreticallyand computationally inferior to standard formulations forpiecewise linear functions Tsai and Lin [49] applied theVielma et al [39] techniques to express a piecewise linearfunction for solving a posynomial optimization problemCroxton et al [31] indicated that most models of expressingpiecewise linear functions are equivalent to each otherAdditionally it is well known that the numbers of extravariables and constraints required in the linearization processfor a nonlinear function obviously impact the computational

performance of the converted problemTherefore this paperfocuses on discussing and reviewing the recent advancesin piecewise linearization methods Section 2 reviews thepiecewise linearization methods Section 3 compares theformulations of various methods with the numbers of extrabinarycontinuous variables and constraints Section 4 dis-cusses error evaluation in piecewise linear approximationConclusions are made in Section 5

2 Formulations of PiecewiseLinearization Functions

Consider a general nonlinear function 119891(119909) of a singlevariable 119909 119891(119909) is a continuous function and 119909 is withinthe interval [119886

0 119886119898] Most commonly used textbooks of

nonlinear programming [26ndash28] approximate the nonlinearfunction by a piecewise linear function as follows

Firstly denote 119886119896(119896 = 0 2 119898) as the break points of

119891(119909) 1198860lt 1198861lt sdot sdot sdot lt 119886

119898 and Figure 1 indicates the piecewise

linearization of 119891(119909)119891(119909) can then be approximately linearized over the

interval [1198860 119886119898] as

119871 (119891 (119909)) =

119898

sum

119896=0

119891 (119886119896) 119905119896 (1)

where 119909 = sum119898

119896=0119886119896119905119896 sum119898119896=0

119905119896= 1 119905

119896ge 0 in which only two

adjacent 119905119896rsquos are allowed to be nonzero A nonlinear function

is then converted into the following expressions

Method 1 Consider

119871 (119891 (119909)) =

119898

sum

119896=1

119891 (119886119896) 119905119896

119909 =

119898

sum

119896=1

119886119896119905119896

1199050le 1199100

119905119896le 119910119896minus1

+ 119910119896 for 119896 = 1 2 119898 minus 1 119905

119898le 119910119898minus1

119898minus1

sum

119896=0

119910119896= 1

119898

sum

119896=0

119905119896= 1

(2)

where 119910119896isin 0 1 119905

119896ge 0 119896 = 0 1 119898 minus 1

The above expressions involve 119898 new binary variables1199100 1199101 119910

119898minus1 The number of newly added 0-1 variables for

piecewisely linearizing a function 119891(119909) equals the number ofbreaking intervals (ie119898) If119898 is large it may cause a heavycomputational burden

Li and Yu [33] proposed another global optimizationmethod for nonlinear programming problems where theobjective function and the constraints might be nonconvexA univariate function is initially expressed by a piecewise


a0

a1

a2

am

x

f(x)

L(f(x))



119886119896and 119886119896+1

expressed as 119904119896= [119891(119886

119896+1) minus 119891(119886

119896)][119886119896+1

minus 119886119896]


119871 (119891 (119909)) = 119891 (1198860) + 1199040(119909 minus 119886

0)

+

119898minus1

sum

119896=1

119904119896minus 119904119896minus1

2

(1003816100381610038161003816119909 minus 119886119896

1003816100381610038161003816+ 119909 minus 119886

119896)

(3)


119886119896] if 119904119896minus 119904119896minus1


Method 2 Consider

119871 (119891 (119909)) = 119891 (1198860) + 1199040(119909 minus 119886

0)

+ sum

119896119904119896gt119904119896minus1

(119904119896minus 119904119896minus1

)(119909 minus 119886119896+

119896minus1

sum

119897=0

119889119897)

+

1

2

sum

119896 119904119896lt119904119896minus1

(119904119896minus 119904119896minus1

) (119909 minus 2119911119896+ 2119886119896119906119896minus 119886119896)

119909 +

119898minus2

sum

119897=0

119889119897ge 119886119898minus1

0 le 119889119897le 119886119897+1

minus 119886119897

where 119904119896gt 119904119896minus1

119909 + 119909 (119906119896minus 1) le 119911

119896 119911

119896ge 0 where 119904

119896lt 119904119896minus1

(4)

where 119909 ge 0 119889119897ge 0 119911

119896ge 0 119906




119886119896]




Method 3 Consider

119886119896minus (119886119898

minus 1198860) (1 minus 120582

119896)

le 119909 le 119886119896+1


119896) 119896 = 0 1 119898 minus 1(5)


119896=0120582119896= 1 120582

119896isin 0 1 and

119891 (119886119896) + 119904119896(119909 minus 119886

119896) minus 119872(1 minus 120582

119896)

le 119891 (119909)

le 119891 (119886119896) + 119904119896(119909 minus 119886

119896)

+ 119872(1 minus 120582119896) 119896 = 0 1 119898 minus 1

(6)


119896+1)minus119891(119886

119896))(119886119896+1

minus

119886119896)




0 119886119898]


120579 =

ℎ

sum

119895=1

2119895minus1

119906119895 ℎ = lceillog

2119898rceil 119906

119895isin 0 1 (7)


119895isin119866(120579)2119895minus1

= 120579 For instance 119866(0) = 120601 119866(3) =





Method 4 Consider

119898minus1

sum

120579=0

119903120579119886120579le 119909 le

119898minus1

sum

120579=0

119903120579119886120579+1

119891 (119909) =

119898minus1

sum

120579=0

(119891 (119886120579) minus 119904120579(119886120579minus 1198860)) 119903120579

+

119898minus1

sum

120579=0

119904120579119908120579 119904120579=

119891 (119886120579+1

) minus 119891 (119886120579)

119886120579+1

minus 119886120579

119898minus1

sum

120579=0

119903120579= 1 119903

120579ge 0

119898minus1

sum

120579=0

119903120579

119866 (120579) +

ℎ

sum

119895=1

119911119895= 0

minus1199061015840

119895le 119911119895le 1199061015840

119895 119895 = 1 2 ℎ

119898minus1

sum

120579=0

119903120579119888120579119895

minus (1 minus 1199061015840

119895) le 119911119895le

119898minus1

sum

120579=0

119903120579119888120579119895

+ (1 minus 1199061015840

119895)

119895 = 1 2 ℎ

119898minus1

sum

120579=0

119908120579= 119909 minus 119886

0

119898minus1

sum

120579=0

119908120579

119866 (120579) +

ℎ

sum

119895=1

120575119895= 0

minus (119886119898

minus 1198860) 1199061015840

119895le 120575119895le (119886119898

minus 1198860) 1199061015840

119895 119895 = 1 2 ℎ

119898minus1

sum

120579=0

119908120579119888120579119895

minus (119886119898

minus 1198860) (1 minus 119906

1015840

119895)

le 120575119895le

119898minus1

sum

120579=0

119908120579119888120579119895

+ (119886119898

minus 1198860) (1 minus 119906

1015840

119895) 119895 = 1 2 ℎ

ℎ

sum

119895=1

2119895minus1

1199061015840

119895le 119898

(8)

where 1199061015840119895isin 0 1 119888

120579119895 119911119895 and 120575


119903120579and 119908




2119898rceil








Let 119861(119901) = (1199061 1199062 119906

120579) for all 119906

119896isin 0 1 119896 =



119896= 1 of 119861(119901) and

119861(119901 + 1) for 119901 = 1 2 119898 minus 1 or 119906119896

= 1 of 119861(119901) for 119901 isin




119896= 0 of 119861(119901) and

119861(119901 + 1) for 119901 = 1 2 119898 minus 1 or 119906119896

= 0 of 119861(119901) for 119901 isin




le







119871 (119891 (119909)) =

119898

sum

119901=0

119891 (119886119901) 120582119901

119909 =

119898

sum

119901=0

119886119901120582119901

119898

sum

119901=0

120582119901= 1

sum

119901isin119878+(119896)

120582119901le 119906119896

sum

119901isin119878minus(119896)

120582119901le 1 minus 119906

119896 forall120582


(9)









2) to119874(2





4 Error Evaluation


119894(119909) lt 0



lowast) minus 119871(119891(119909

lowast))| le 120576

1and


lowast) minus 119871(119891(119909











5 Conclusions






2119898rceil



2119898rceil

a0

a1

x

Error

f(xlowast

)

xlowast

L(f(xlowast

))

f(x)

f(x)

(a)

a1

a2

x

Error

xlowasta

0

f(xlowast

)

L(f(xlowast

))

f(x)

(b)



Acknowledgment


References


































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










a0

a1

a2

am

x

f(x)

L(f(x))



119886119896and 119886119896+1

expressed as 119904119896= [119891(119886

119896+1) minus 119891(119886

119896)][119886119896+1

minus 119886119896]


119871 (119891 (119909)) = 119891 (1198860) + 1199040(119909 minus 119886

0)

+

119898minus1

sum

119896=1

119904119896minus 119904119896minus1

2

(1003816100381610038161003816119909 minus 119886119896

1003816100381610038161003816+ 119909 minus 119886

119896)

(3)


119886119896] if 119904119896minus 119904119896minus1


Method 2 Consider

119871 (119891 (119909)) = 119891 (1198860) + 1199040(119909 minus 119886

0)

+ sum

119896119904119896gt119904119896minus1

(119904119896minus 119904119896minus1

)(119909 minus 119886119896+

119896minus1

sum

119897=0

119889119897)

+

1

2

sum

119896 119904119896lt119904119896minus1

(119904119896minus 119904119896minus1

) (119909 minus 2119911119896+ 2119886119896119906119896minus 119886119896)

119909 +

119898minus2

sum

119897=0

119889119897ge 119886119898minus1

0 le 119889119897le 119886119897+1

minus 119886119897

where 119904119896gt 119904119896minus1

119909 + 119909 (119906119896minus 1) le 119911

119896 119911

119896ge 0 where 119904

119896lt 119904119896minus1

(4)

where 119909 ge 0 119889119897ge 0 119911

119896ge 0 119906




119886119896]




Method 3 Consider

119886119896minus (119886119898

minus 1198860) (1 minus 120582

119896)

le 119909 le 119886119896+1


119896) 119896 = 0 1 119898 minus 1(5)


119896=0120582119896= 1 120582

119896isin 0 1 and

119891 (119886119896) + 119904119896(119909 minus 119886

119896) minus 119872(1 minus 120582

119896)

le 119891 (119909)

le 119891 (119886119896) + 119904119896(119909 minus 119886

119896)

+ 119872(1 minus 120582119896) 119896 = 0 1 119898 minus 1

(6)


119896+1)minus119891(119886

119896))(119886119896+1

minus

119886119896)




0 119886119898]


120579 =

ℎ

sum

119895=1

2119895minus1

119906119895 ℎ = lceillog

2119898rceil 119906

119895isin 0 1 (7)


119895isin119866(120579)2119895minus1

= 120579 For instance 119866(0) = 120601 119866(3) =





Method 4 Consider

119898minus1

sum

120579=0

119903120579119886120579le 119909 le

119898minus1

sum

120579=0

119903120579119886120579+1

119891 (119909) =

119898minus1

sum

120579=0

(119891 (119886120579) minus 119904120579(119886120579minus 1198860)) 119903120579

+

119898minus1

sum

120579=0

119904120579119908120579 119904120579=

119891 (119886120579+1

) minus 119891 (119886120579)

119886120579+1

minus 119886120579

119898minus1

sum

120579=0

119903120579= 1 119903

120579ge 0

119898minus1

sum

120579=0

119903120579

119866 (120579) +

ℎ

sum

119895=1

119911119895= 0

minus1199061015840

119895le 119911119895le 1199061015840

119895 119895 = 1 2 ℎ

119898minus1

sum

120579=0

119903120579119888120579119895

minus (1 minus 1199061015840

119895) le 119911119895le

119898minus1

sum

120579=0

119903120579119888120579119895

+ (1 minus 1199061015840

119895)

119895 = 1 2 ℎ

119898minus1

sum

120579=0

119908120579= 119909 minus 119886

0

119898minus1

sum

120579=0

119908120579

119866 (120579) +

ℎ

sum

119895=1

120575119895= 0

minus (119886119898

minus 1198860) 1199061015840

119895le 120575119895le (119886119898

minus 1198860) 1199061015840

119895 119895 = 1 2 ℎ

119898minus1

sum

120579=0

119908120579119888120579119895

minus (119886119898

minus 1198860) (1 minus 119906

1015840

119895)

le 120575119895le

119898minus1

sum

120579=0

119908120579119888120579119895

+ (119886119898

minus 1198860) (1 minus 119906

1015840

119895) 119895 = 1 2 ℎ

ℎ

sum

119895=1

2119895minus1

1199061015840

119895le 119898

(8)

where 1199061015840119895isin 0 1 119888

120579119895 119911119895 and 120575


119903120579and 119908




2119898rceil








Let 119861(119901) = (1199061 1199062 119906

120579) for all 119906

119896isin 0 1 119896 =



119896= 1 of 119861(119901) and

119861(119901 + 1) for 119901 = 1 2 119898 minus 1 or 119906119896

= 1 of 119861(119901) for 119901 isin




119896= 0 of 119861(119901) and

119861(119901 + 1) for 119901 = 1 2 119898 minus 1 or 119906119896

= 0 of 119861(119901) for 119901 isin




le







119871 (119891 (119909)) =

119898

sum

119901=0

119891 (119886119901) 120582119901

119909 =

119898

sum

119901=0

119886119901120582119901

119898

sum

119901=0

120582119901= 1

sum

119901isin119878+(119896)

120582119901le 119906119896

sum

119901isin119878minus(119896)

120582119901le 1 minus 119906

119896 forall120582


(9)









2) to119874(2





4 Error Evaluation


119894(119909) lt 0



lowast) minus 119871(119891(119909

lowast))| le 120576

1and


lowast) minus 119871(119891(119909











5 Conclusions






2119898rceil



2119898rceil

a0

a1

x

Error

f(xlowast

)

xlowast

L(f(xlowast

))

f(x)

f(x)

(a)

a1

a2

x

Error

xlowasta

0

f(xlowast

)

L(f(xlowast

))

f(x)

(b)



Acknowledgment


References


































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Method 4 Consider

119898minus1

sum

120579=0

119903120579119886120579le 119909 le

119898minus1

sum

120579=0

119903120579119886120579+1

119891 (119909) =

119898minus1

sum

120579=0

(119891 (119886120579) minus 119904120579(119886120579minus 1198860)) 119903120579

+

119898minus1

sum

120579=0

119904120579119908120579 119904120579=

119891 (119886120579+1

) minus 119891 (119886120579)

119886120579+1

minus 119886120579

119898minus1

sum

120579=0

119903120579= 1 119903

120579ge 0

119898minus1

sum

120579=0

119903120579

119866 (120579) +

ℎ

sum

119895=1

119911119895= 0

minus1199061015840

119895le 119911119895le 1199061015840

119895 119895 = 1 2 ℎ

119898minus1

sum

120579=0

119903120579119888120579119895

minus (1 minus 1199061015840

119895) le 119911119895le

119898minus1

sum

120579=0

119903120579119888120579119895

+ (1 minus 1199061015840

119895)

119895 = 1 2 ℎ

119898minus1

sum

120579=0

119908120579= 119909 minus 119886

0

119898minus1

sum

120579=0

119908120579

119866 (120579) +

ℎ

sum

119895=1

120575119895= 0

minus (119886119898

minus 1198860) 1199061015840

119895le 120575119895le (119886119898

minus 1198860) 1199061015840

119895 119895 = 1 2 ℎ

119898minus1

sum

120579=0

119908120579119888120579119895

minus (119886119898

minus 1198860) (1 minus 119906

1015840

119895)

le 120575119895le

119898minus1

sum

120579=0

119908120579119888120579119895

+ (119886119898

minus 1198860) (1 minus 119906

1015840

119895) 119895 = 1 2 ℎ

ℎ

sum

119895=1

2119895minus1

1199061015840

119895le 119898

(8)

where 1199061015840119895isin 0 1 119888

120579119895 119911119895 and 120575


119903120579and 119908




2119898rceil








Let 119861(119901) = (1199061 1199062 119906

120579) for all 119906

119896isin 0 1 119896 =



119896= 1 of 119861(119901) and

119861(119901 + 1) for 119901 = 1 2 119898 minus 1 or 119906119896

= 1 of 119861(119901) for 119901 isin




119896= 0 of 119861(119901) and

119861(119901 + 1) for 119901 = 1 2 119898 minus 1 or 119906119896

= 0 of 119861(119901) for 119901 isin




le







119871 (119891 (119909)) =

119898

sum

119901=0

119891 (119886119901) 120582119901

119909 =

119898

sum

119901=0

119886119901120582119901

119898

sum

119901=0

120582119901= 1

sum

119901isin119878+(119896)

120582119901le 119906119896

sum

119901isin119878minus(119896)

120582119901le 1 minus 119906

119896 forall120582


(9)









2) to119874(2





4 Error Evaluation


119894(119909) lt 0



lowast) minus 119871(119891(119909

lowast))| le 120576

1and


lowast) minus 119871(119891(119909











5 Conclusions






2119898rceil



2119898rceil

a0

a1

x

Error

f(xlowast

)

xlowast

L(f(xlowast

))

f(x)

f(x)

(a)

a1

a2

x

Error

xlowasta

0

f(xlowast

)

L(f(xlowast

))

f(x)

(b)



Acknowledgment


References


































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












2) to119874(2





4 Error Evaluation


119894(119909) lt 0



lowast) minus 119871(119891(119909

lowast))| le 120576

1and


lowast) minus 119871(119891(119909











5 Conclusions






2119898rceil



2119898rceil

a0

a1

x

Error

f(xlowast

)

xlowast

L(f(xlowast

))

f(x)

f(x)

(a)

a1

a2

x

Error

xlowasta

0

f(xlowast

)

L(f(xlowast

))

f(x)

(b)



Acknowledgment


References


































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













2119898rceil



2119898rceil

a0

a1

x

Error

f(xlowast

)

xlowast

L(f(xlowast

))

f(x)

f(x)

(a)

a1

a2

x

Error

xlowasta

0

f(xlowast

)

L(f(xlowast

))

f(x)

(b)



Acknowledgment


References


































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Review Article A Review of Piecewise Linearization Methodsprocess, Li et al. [ ] developed a...

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