Competitive Differential Evolution Applied to CEC 2013 Problems

Josef Tvrdık and Radka Polakova

Abstract—A variant of adaptive differential evolution (DE)with twelve competing strategies was applied to the set ofbenchmark problems proposed for the CEC 2013 competition.Competitive adaptation of DE strategies and parameter settingsavoids tuning of most control-parameter values because only thesize of population must be set by the user of the algorithm. Theexperimental results showed acceptable reliability of the searchfirst of all in the problems with unrotated objective functionsof smaller dimension. The possible cause of unreliability of thealgorithm in the rotated functions is analyzed.

I. INTRODUCTION

Differential evolution (DE) was proposed by Storn andPrice [1], [2] as a global optimizer for continuous single-objective optimization problems with a real-value objectivefunction. The search space (domain) Ω is specified bylower (𝑎𝑗) and upper (𝑏𝑗) limits of each component 𝑗,Ω =

∏𝐷𝑗=1[𝑎𝑗 , 𝑏𝑗 ], 𝑎𝑗 < 𝑏𝑗 , 𝑗 = 1, 2, . . . , 𝐷 , 𝐷 is the

dimension of the problem. The global minimum point 𝒙∗satisfying the condition 𝑓(𝒙∗) ≤ 𝑓(𝒙) for ∀𝒙 ∈ Ω is thesolution of the problem.

DE algorithm has become one of the most frequentlyevolutionary algorithms used for solving the continuousglobal optimization problems in recent years [3]. Recentresults of research in DE are comprehensively summarizedby Neri and Tirronen [4] and by Das and Suganthan [5].

The remaining part of the paper is organized as fol-lows: DE algorithm is described in Section II. Section IIIpresents the adaptive mechanism based on the competition ofstrategies and parameter settings. The variant of the adaptiveDE used in the experiments is described in Section IV.Experimental setting is specified in Section V, the resultsof experiments are presented in Section VI including thetime complexity of the algorithm. Comparison of the resultsin pairs of unrotated and rotated functions are shown anddiscussed in Section VII and conclusions are drawn inSection VIII.

II. DE ALGORITHM

Algorithm of DE works with a population of individuals(NP points in domain Ω) that are considered as candidates ofsolution. Parameter NP is called the size of the population.The population develops iteratively by using evolutionaryoperators of selection, mutation, and crossover. Each iteration

Josef Tvrdık is with the Department of Computer Sciences and Centreof Excellence IT4Innovations, Institute for Research and Applications ofFuzzy Modeling, University of Ostrava, 30. dubna 22, 701 03 Ostrava, CzechRepublic (phone: +420 59709 2183; email: josef.tvrdik@osu.cz).

Radka Polakova is with the Centre of Excellence IT4Innovations, Institutefor Research and Applications of Fuzzy Modeling, University of Ostrava,30. dubna 22, 701 03 Ostrava, Czech Republic (phone: +420 59709 1409;email: radka.polakova@osu.cz).

Algorithm 1 Differential evolution1: generate an initial population 𝑃 = (𝒙1,𝒙2, . . . ,𝒙𝑁𝑃 ),

𝒙𝑖 ∈ 𝛺 distributed uniformly2: evaluate 𝑓(𝒙𝑖), 𝑖 = 1, 2, . . . , 𝑁𝑃3: while stopping condition not reached do4: for 𝑖 = 1 to NP do5: generate a trial vector 𝒚6: evaluate 𝑓(𝒚)7: if 𝑓(𝒚) ≤ 𝑓(𝒙𝑖) then8: insert 𝒚 into new generation 𝑄9: else

10: insert 𝒙𝑖 into new generation 𝑄11: end if12: end for13: 𝑃 := 𝑄14: end while

corresponds to an evolutionary generation. Let us denotetwo subsequent generations by 𝑃 and 𝑄. Applications ofevolutionary operators in the old generation 𝑃 create a newgeneration 𝑄. After completing the new generation 𝑄, thegeneration 𝑄 becomes the old generation 𝑃 for next iteration.The basic structure of DE algorithm is shown in Algorithm 1.

A new trial point 𝒚 (line 5 in Algorithm 1) is generatedby using mutation and crossover. There are various typesof mutation and crossover. The most popular mutation typeproposed in [2] and called DE/rand/1 generates the mutantpoint 𝒖 by adding the weighted difference of two points,

𝒖 = 𝒓1 + 𝐹 (𝒓2 − 𝒓3) , 𝐹 > 0 , (1)

where 𝒓1, 𝒓2, and 𝒓3 are three mutually distinct pointsrandomly taken from population 𝑃 , not coinciding with thecurrent point 𝒙𝑖, and 𝐹 is an input parameter.

Kaelo and Ali [6] proposed a new kind of mutationcalled it random localization. It is slightly different fromDE/rand/1 (1):

𝒖 = ��1 + 𝐹 (��2 − ��3) , (2)

where ��1 is the tournament best among 𝒓1, 𝒓2, and 𝒓3, i.e.��1 = argmin𝑖∈{1,2,3} 𝑓(𝒓𝑖), where 𝒓1, 𝒓2, and 𝒓3 are threemutually distinct points randomly taken from population 𝑃 ,not coinciding with the current point 𝒙𝑖, like in the case ofthe DE/rand/1. The points ��2 and ��3 are the remaining pointsof the three randomly selected points. This type of mutationis denoted DE/randrl/1 hereafter. Based on the experimentalresults in [6], it was found that this DE/randrl/1 mutationcan increase the speed of the search by almost 30 % withoutdecreasing the reliability of the search when compared to theDE/rand/1 mutation.

2013 IEEE Congress on Evolutionary Computation June 20-23, Cancún, México

978-1-4799-0454-9/13/$31.00 ©2013 IEEE 1651

The elements 𝑦𝑗 , 𝑗 = 1, 2, . . . , 𝐷, of the trial point 𝒚are built up by the crossover of the current point 𝒙𝑖 and themutant point 𝒖. The most frequently used kind of crossoveris called binomial. It uses the following rule of combinationof parents’ elements

𝑦𝑗 =

{𝑢𝑗 if 𝑈𝑗 ≤ CR or 𝑗 = 𝑙𝑥𝑖𝑗 if 𝑈𝑗 > CR and 𝑗 ∕= 𝑙 ,

(3)

where 𝑙 is a randomly chosen integer from {1, 2, . . . , 𝐷},𝑈1, 𝑈2, . . . , 𝑈𝐷 are independent random variables uniformlydistributed in [0, 1), and CR ∈ [0, 1] is an input parameterinfluencing the number of elements to be exchanged bycrossover. The rule given by Eq. (3) ensures that at leastone element of vector 𝒙𝑖 is changed, even if CR = 0. TheDE variants applying binomial crossover according to (3) aredenoted DE/ ▪ / ▪ /bin in literature.

The exponential crossover usually denoted DE/ ▪ / ▪ /expwas also proposed by Price and Storn [2]. The exponentialcrossover in DE is similar to the two-point crossover ingenetic algorithms. For exponential crossover, the startingposition of crossover (𝑘 = 1) is randomly chosen from{1, . . . , 𝐷}, and 𝐿 consecutive elements (counted in circularmanner) are taken from the mutant vector 𝒖. Probabilityof replacing the 𝑘-th element in the sequence 1, 2, . . . , 𝐿,𝐿 ≤ 𝐷 decreases exponentially with increasing 𝑘.

Let us consider the probability 𝑝𝑚 of replacing theelement of 𝒙𝑖 by the element of the mutant vector 𝒖. Themean value of replaced elements is then 𝑝𝑚×𝐷. The relationbetween the 𝑝𝑚 and control parameter CR was studied indetail by Zaharie [7]. For binomial crossover, the relationbetween 𝑝𝑚 and control parameter CR is linear,

𝑝𝑚 = CR(1− 1/𝐷) + 1/𝐷, (4)

while for exponential crossover the relationship is stronglynon-linear and the difference from linearity increases withincreasing dimension of the problem. Zaharie’s result for ex-ponential crossover can be rewritten in the form of polynomequation

CR𝐷 − 𝐷 𝑝𝑚 CR + 𝐷 𝑝𝑚 − 1 = 0. (5)

The crossover parameter CR satisfies the conditions: CR = 0for 𝑝𝑚 = 1/𝐷 and CR = 1 for 𝑝𝑚 = 1. It is apparentthat the equation (5) has the only one real root in the openinterval of (0, 1) for 𝑝𝑚 ∈ (1/𝐷, 1). Thus, for given 𝑝𝑚we are able to find unique corresponding value of CR from(5). This is applied in the implementation of competitive DEvariant used in this study.

Differential evolution has a few control parametersonly, namely the size of population NP, mutation strategy,crossover type, and couple of parameters 𝐹 and CR. How-ever, the efficiency of differential evolution is very sensitiveespecially regarding the setting of 𝐹 and CR values. Themost suitable control-parameter values for a specific problemmay be found by trial-and-error tuning, which requires a lotof time. There are some recommendations for setting of theseparameters but they are not applicable for all the optimizationproblems. Due to these facts, several new adaptive variants ofDE have been recently proposed, some of them [8], [9], [10],[11] are considered as the state-of-the-art adaptive variantsof DE.

III. COMPETITIVE DIFFERENTIAL EVOLUTION

Competitive setting of the DE strategies and the controlparameters was originally proposed in [12]. In this self-adaptive approach, we choose randomly among 𝐻 differentDE strategies or settings of control parameters (𝐹 , CR) withprobabilities 𝑞ℎ, ℎ = 1, 2, . . . , 𝐻 .

These probabilities change according to the success rateof the settings in preceding steps of the search process. Theℎth setting is considered successful if it generates a trial point𝒚 better than its counter-partner 𝒙𝑖 in the old generation 𝑃 ,i.e. 𝑓(𝒚) ≤ 𝑓(𝒙𝑖), see line 7 of Algorithm 1. Probability 𝑞ℎis evaluated as the relative frequency

𝑞ℎ =𝑛ℎ + 𝑛0∑𝐻

𝑗=1(𝑛𝑗 + 𝑛0), (6)

where 𝑛ℎ is the current count of the ℎth setting’s successes,and 𝑛0 > 0 is a constant. The input parameter 𝑛0 > 1prevents a dramatic change in 𝑞ℎ by one random successfuluse of the ℎth parameter setting. To avoid degeneration ofthe search process, the current values of 𝑞ℎ are reset to theirstarting values 𝑞ℎ = 1/𝐻 if any probability 𝑞ℎ decreasesbellow the given limit 𝛿 > 0 during the search process. Thevariants of DE using this competitive mechanism are denotedCDE hereafter.

Several combinations of DE strategies and control pa-rameter settings were compared in different benchmarktests [13], [14], [15]. The CDE variant denoted b6e6rl usingtwelve different DE strategies or parameter settings wasfound as one of the most efficient among all the testedCDE variants. This variant was also compared with thestate-of-the-art adaptive DE algorithms [8], [9], [10], [11]and two versions of DE algorithm with composite trialvector generation strategies and control parameters [16]. Theb6e6rl variant of CDE appeared the most reliable and thesecond fastest algorithm among those seven algorithms inthe benchmark set of six shifted functions at three levels ofdimension (𝐷 = 10, 30, and 100) [17].

IV. DESCRIPTION OF THE VARIANT OF CDE USED FORCEC 2013 PROBLEMS

The b6e6rl variant of CDE was chosen for the solutionof CEC 2013 problems. This variant uses two DE strategies,namely DE/randrl/1/bin and DE/randrl/1/exp, each with sixdifferent settings of (𝐹,CR). The strategies and settings arelisted in Table I. The values of 𝑝1, 𝑝2, and 𝑝3 are set upequidistantly in the open interval of (1/𝐷, 1). The valueof 𝑝2 is in the middle of (1/𝐷, 1), 𝑝1 is in the middle of(1/𝐷, 𝑝2), and 𝑝3 in the middle of (𝑝2, 1). The values of𝑝𝑖, 𝑖 = 1, 2, 3, are set up automatically with respect to thedimension of the problem at the start of the algorithm as wellas the corresponding values of CR𝑖 evaluated as a root of thepolynom (5). The setting of 𝑝1, 𝑝2, and 𝑝3 for exponentialcrossover together with the dependence of CR on 𝑝𝑚 forboth types of crossover for 𝐷 = 30 is depicted in Figure 1.

The values of mutation probability and the correspondingvalues of CR applied to the problems of 𝐷 = 10, 30, and 50are shown in Table II.

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TABLE I. DE STRATEGIES AND PARAMETER SETTINGS COMPETING

IN b6e6rl VARIANT OF CDE

ℎ mutation crossover 𝐹 CR 𝑝𝑚

1 0.5 02 0.5 0.53 randrl/1 binomial 0.5 14 0.8 05 0.8 0.56 0.8 1

7 0.5 CR1 𝑝1

8 0.5 CR2 𝑝2

9 randrl/1 exponential 0.5 CR3 𝑝3

10 0.8 CR1 𝑝1

11 0.8 CR2 𝑝2

12 0.8 CR3 𝑝3

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Probability of mutation

CR

p2

p1

p3

Exponential

Binomial

Fig. 1. Dependence of CR on 𝑝𝑚 and values of 𝑝1, 𝑝2, 𝑝3 for 𝐷 = 30

TABLE II. VALUES OF MUTATION PROBABILITY AND THE

CORRESPONDING VALUES OF CR FOR EXPONENTIAL CROSSOVER

𝐷 = 10 𝐷 = 30 𝐷 = 50𝑖 𝑝𝑖 CR𝑖 𝑝𝑖 CR𝑖 𝑝𝑖 CR𝑖

1 0.3250 0.7011 0.2750 0.8815 0.2650 0.92622 0.5500 0.8571 0.5167 0.9488 0.5100 0.96883 0.7750 0.9418 0.7583 0.9801 0.7550 0.9880

Mutation according to (2) can cause that a new trial point𝒚 moves out of the domain Ω, which means the violationof the boundary constraints. In such a case, the values of𝑦𝑗 ∕∈ [𝑎𝑗 , 𝑏𝑗 ] can be repaired by turn over into Ω by usingtransformation 𝑦𝑗 ← 2×𝑎𝑗 − 𝑦𝑗 or 𝑦𝑗 ← 2× 𝑏𝑗 − 𝑦𝑗 for theviolated components. This technique is frequently used andit is also implemented in b6e6rl variant of CDE algorithmfor the CEC 2013 problems.

The parameters controlling the competition of DE strate-gies (and control-parameter settings) were set up to therecommended values succeeded in previous applications ofthe algorithm [17], i.e. 𝑛0 = 2 and 𝛿 = 1/(𝐻 ∗ 5) = 0.0167.Different values of 𝛿 were tested in [18] but no different valueimproving the performance of the algorithm significantly wasfound.

V. EXPERIMENTAL SETTING

All computations were carried out on a standard PC withWindows 7 Professional, Intel(R) Core(TM)2 Quad CPUQ6600, 2.40GH, 2.39GHz, 2 GB RAM.

The b6e6rl algorithm, which is a variant of competitiveDE described above, is implemented in Matlab 2010a andthis environment was used for experiments. Experimentalsetting follows the requirements given in the report [19].28 minimization problems are defined in the report, thesource code of functions in C (cec13 func.cpp from January28, 2013) was downloaded from the web page [19] andcompiled by Lcc-win32 C 2.4.1 via mex command (mexcec13 func.cpp -DWINDOWS).

Search range for all the test functions is [−100, 100]𝐷 .The tests were carried out at three levels of dimension,51 repeated runs per the test function and the dimensionof the problem. The random generator was set up byrand(’state’,sum(100*clock)) statement at the start of eachrun. The run was stopped if the prescribed MaxFES wasreached, i.e. when MaxFES = 1×105 for 𝐷 = 10, MaxFES= 3× 105 for 𝐷 = 30, and MaxFES= 5× 105 for 𝐷 = 50.

Due to adaptive features of the b6e6rl algorithm, theonly control parameter needed to be set is the populationsize. It was found in previous experiments with adaptive DEvariants [17] that for good performance it is sufficient muchsmaller population size than NP ≃ 10 × 𝐷 recommendedfor many evolutionary algorithms including standard DE [2].The adaptive DE variants performed well with the populationsize from 20 to 5 × 𝐷 in the problems of low dimension

TABLE III. VALUES OF FUNCTION ERRORS FOR 𝐷 = 10 (NP = 50)

F Best Worst Median Mean Std1 0 0 0 0 02 0 9.113E-03 0 1.823E-04 1.276E-033 0 6.31503 3.898E-04 3.939E-01 1.49584 0 6.124E-08 0 2.124E-09 1.071E-085 0 0 0 0 06 0 9.81242 0 4.23281 4.908137 1.031E-05 3.687E-02 1.013E-03 2.956E-03 6.414E-038 20.2252 20.5553 20.3775 20.3834 7.901E-029 0 5.77369 1.15017 1.78046 1.87017

10 0 9.699E-02 2.948E-02 3.511E-02 2.539E-0211 0 0 0 0 012 2.40123 12.5531 7.97506 8.06705 2.2483513 3.20013 16.8923 9.22708 9.57340 3.4094814 0 1.874E-01 6.245E-02 7.103E-02 6.372E-0215 336.986 1053.27 702.706 699.176 172.14516 2.191E-01 1.38809 9.829E-01 9.475E-01 2.488E-0117 10.1224 10.1224 10.1224 10.1224 1.256E-1418 14.0523 32.9305 25.6201 24.9139 4.7457519 1.831E-01 4.185E-01 3.374E-01 3.383E-01 4.127E-0220 1.63623 2.87660 2.32367 2.31043 2.984E-0121 100 400.194 400.194 382.532 62.360222 0 102.845 10.5983 19.8074 25.343823 283.993 1142.44 701.662 705.280 201.62324 200 210.476 206.671 204.262 3.5157525 110.236 209.098 200 200.125 13.146526 102.835 200.017 200.017 154.989 46.408227 300 486.348 300 324.674 62.498728 100 300 300 292.157 39.2078

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and with NP ≤ 100 in the problems of higher dimension.Thus, the population size NP was set up in dependence onthe dimension of problems but we took into account notonly dimension of the problem but also experience frompast computations and the results of short preliminary tuningexperiments. The experiments for the CEC 2013 competitionwere carried out with the population size of the followingvalues: NP = 50 for 𝐷 = 10 and NP = 100 for the problemsof 𝐷 = 30 and 𝐷 = 50.

TABLE IV. VALUES OF FUNCTION ERRORS FOR 𝐷 = 30 (NP = 100)

F Best Worst Median Mean Std1 0 0 0 0 02 6217.58 211988 58961.6 69955.9 44104.73 5.330E-02 59230 21.0825 4364.6 135854 4.461E-04 1.632E-01 6.470E-03 1.807E-02 2.883E-025 0 0 0 0 06 5.710E-02 26.4074 7.800E-01 5.24973 9.902447 2.95037 46.3012 24.6193 24.4497 8.961098 20.8089 21.0349 20.935 20.9382 4.729E-029 26.2038 30.9152 28.6155 28.63075 1.15321

10 0 5.416E-02 1.724E-02 1.912E-02 1.336E-0211 0 0 0 0 012 54.3466 115.409 82.7053 83.5686 12.970613 82.7194 144.878 114.126 113.617 13.972814 0 8.328E-02 2.082E-02 2.368E-02 2.499E-0215 3901.65 5309.89 4718.87 4661.29 339.04316 4.594E-01 2.46614 2.03667 1.97934 3.853E-0117 30.4337 30.4338 30.4337 30.4337 4.15E-0518 135.923 200.559 174.493 172.412 13.458619 1.51332 2.07279 1.85544 1.84490 1.460E-0120 10.9401 12.3172 11.8901 11.8396 3.242E-0121 200 443.544 300 296.773 85.500622 59.5431 149.375 123.513 122.948 16.297923 4034.83 5658.4 5029.9 5009.54 406.29224 220.124 274.431 248.822 251.539 13.832825 241.246 295.937 281.683 275.313 17.603426 200.001 367.578 200.007 209.754 39.376827 722.464 1122.72 1026.9 1007.08 74.417428 300 300 300 300 0

VI. RESULTS

The report [19] stated the obligatory content and structureof the experimental results, which is presented below.

A. Function errors

The values of function errors are shown in Table III for𝐷 = 10, in Table IV for 𝐷 = 30, and Table V for 𝐷 = 50.The tables are presented in the structure required by [19], thevalues of function error smaller than 1×10−8 were replacedby zero.

The results in Table III show the relatively good reliabilityof the search on the most of functions of lower hardnessexcept Function 15, in the case of Function 8 and Function17 the search is trapped systematically at outside the rangeof the global minimum. Bad performance appeared in thecomposition functions (from F21 to F28) but these functionsare expected hard for any algorithm. The performance of thealgorithm is a bit worse in the problems of higher dimension,see Tables IV and V. Systematic termination of the searchoutside the range of the global minimum is found again in the

case of Function 8 and Function 17, additionally in Function20, and for 𝐷 = 50 in Function 6.

The algorithm usually performs worse in the problems,where the objective function is rotated. This matter is dis-cussed in more details in Section VII.

TABLE V. VALUES OF FUNCTION ERRORS FOR 𝐷 = 50 (NP = 100)

F Best Worst Median Mean Std1 0 0 0 0 02 147987 897424 297699 323455 1560293 30800 1.365E+08 1.643E+06 8.608E+06 2.327E+074 1.027E-02 1.45604 9.869E-02 2.323E-01 3.116E-015 0 0 0 0 06 43.4473 43.4473 43.4473 43.4473 1.435E-147 35.1307 112.053 83.9725 82.6468 15.56398 21.0093 21.2105 21.1342 21.1271 4.652E-029 49.6790 61.1880 56.8302 56.6850 2.56926

10 7.396E-03 6.893E-02 2.959E-02 3.539E-02 1.851E-0211 0 0 0 0 012 117.405 236.969 190.472 189.168 24.036013 175.652 316.420 249.612 252.762 30.784414 4.177E-08 7.495E-02 3.748E-02 3.650E-02 2.058E-0215 7780.11 10040.4 9277.65 9265.51 457.02416 7.081E-01 3.27929 2.45341 2.29560 6.409E-0117 50.7858 50.7858 50.7858 50.7858 6.459E-1418 116.497 377.309 334.156 327.966 38.426819 2.93741 3.86017 3.44528 3.43122 1.876E-0120 20.0777 22.0084 21.5208 21.4610 3.863E-0121 200 1122.19 200 460.027 410.30422 15.0256 130.404 30.1790 36.0068 24.446323 8531.88 10976.7 9785.16 9776.30 533.19524 264.961 354.264 336.088 332.903 15.381925 295.594 385.544 370.798 364.253 20.874226 200.018 449.556 429.387 328.202 120.83227 1103.56 1880.86 1747.56 1729.24 107.47328 400 400 400 400 0

B. Algorithm complexity

Algorithm complexity of the algorithm was measuredaccording to the guidelines in [19], the results are presentedin Table VI, the values of time are in seconds. T0 is thetime needed for computing the program given in [19] carringout 1 × 106 times of the prescribed sequence of arithmeticoperations, T1 is the time of 2×105 evaluations of Function14 and T2 is the average time of five runs of the algorithmwith MaxFES = 2× 105 on Function 14.

TABLE VI. COMPUTATIONAL COMPLEXITY

T0 T1 (s) T2 (s) (T2 − T1)/T0

𝐷 = 10 2.83 18.66 71.69𝐷 = 30 0.221 5.03 21.44 74.32𝐷 = 50 6.78 25.14 83.15

C. Parameters

This section is required in [19] in the prescribed structure.

∙ Population size is the only parameter to be adjustedto the solved problem. The values of the parameters

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controlling the competition remains on their defaultsetting, that is 𝑛0 = 2 and 𝛿 = 1/(𝐻 ∗ 5) = 0.0167.

∙ No dynamic range of parameter value is applied.

∙ Population size NP was set up in dependence on thedimension of problems but we took into account notonly dimension of the problem but also experiencefrom past computations and the results of shortpreliminary tuning experiments.

∙ Computation time needed to the tuning of the pop-ulation size can be estimated by the 6 × MaxFESfunction evaluations per problem. A quick analysisof the results of tuning took about two hours.

∙ Values of the population size were set up to NP = 50for 𝐷 = 10 and NP = 100 for the problems of𝐷 = 30 and 𝐷 = 50.

VII. COMPARISON OF UNROTATED AND ROTATEDFUNCTIONS

Among the benchmark functions [19] there are pairs offunctions that differs only by the rotation, namely Functions11 and 12, Functions 14 and 15, and Functions 22 and23. The convergence plots of these pairs of functions fordimension 𝐷 = 10, 𝐷 = 30, and 𝐷 = 50 are drawn inFigures 2, 3, and 4, respectively. In each plot, the minimumerror value in the current generation is shown on verticalaxis, median, worst and best values of 51 runs are depictedfor each reported FES.

0 2 4 6 8 10

x 104

0

1

2

3

4

5F11 Rastrigin

FES

Err

or

minmedianmax

0 2 4 6 8 10

x 104

0

10

20

30

40

50F12 Rastrigin rotated

FES

Err

or

minmedianmax

0 2 4 6 8 10

x 104

0

50

100

150

200

250

300F14 Schwefel

FES

Err

or

minmedianmax

0 2 4 6 8 10

x 104

0

500

1000

1500

2000F15 Schwefel rotated

FES

Err

or

minmedianmax

0 2 4 6 8 10

x 104

0

200

400

600

800F22 CF not rotated

FES

Err

or

minmedianmax

0 2 4 6 8 10

x 104

0

500

1000

1500

2000

2500F23 CF rotated

FES

Err

or

minmedianmax

Fig. 2. Convergence plots for pairs of unrotated and rotated functions,𝐷 = 10

0 1 2 3

x 105

0

20

40

60

80

100F11 Rastrigin

FES

Err

or

minmedianmax

0 1 2 3

x 105

50

100

150

200

250F12 Rastrigin rotated

FES

Err

or

minmedianmax

0 1 2 3

x 105

0

1000

2000

3000

4000F14 Schwefel

FES

Err

or

minmedianmax

0 1 2 3

x 105

3000

4000

5000

6000

7000

8000F15 Schwefel rotated

FES

Err

or

minmedianmax

0 1 2 3

x 105

0

1000

2000

3000

4000F22 CF not rotated

FES

Err

or

minmedianmax

0 1 2 3

x 105

4000

5000

6000

7000

8000

9000F23 CF rotated

FES

Err

or

minmedianmax

Fig. 3. Convergence plots for pairs of unrotated and rotated functions,𝐷 = 30

We can see that the algorithm finds an acceptable solutionvery quickly for the unrotated functions and the differencesbetween the worst and the best runs are small, whilst inthe case of the rotated functions the convergence is slowwith a larger differences between the worst and the bestruns. The explanation of bad convergence in the problemswith the rotated functions also gives Table VII, where thepercentage of successful trial points is presented for threepairs of function and 𝐷 = 30. Notice, that a trial point 𝒚 issuccessful if 𝑓(𝒚) ≤ 𝑓(𝒙𝑖), i.e. the fitness of the populationin next generation is increased due to inserting the new trialpoint 𝒚.

TABLE VII. PERCENTAGE OF SUCCESSFUL TRIAL POINTS AT EARLY

STAGE (BEFORE 0.1× MAXFES) AND IN THE WHOLE PROCESS, 𝐷 = 30

Before 0.1× MaxFES TotalFunction mean std mean std

11 19.4 % 0.488 14.4 % 0.18412 8.8 % 0.301 2.1 % 0.06114 14.2 % 0.267 15.8 % 0.19115 2.8 % 0.153 0.9 % 0.02222 13.5 % 0.285 5.4 % 2.16523 3.0 % 0.177 1.1 % 0.642

It is apparent from the values in Table VII that theproportion of successful generating the trial points is muchsmaller in the case of rotated functions, both in the earlystage of the search and in the whole process, where thesuccess falls down even to values about 1 %. It indicates thatthe algorithm wastes computational time without an evident

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Fig. 4. Convergence plots for pairs of unrotated and rotated functions,𝐷 = 50

progress in the fitness of the population in the problems withrotated objective functions.

The distribution of successful generating trials pointsamong twelve competing settings was also compared for thepairs of unrotated and rotated functions. The distributionsfor the pair of Function 14 and 15 are shown in Figure 5as box-plots depicted from 51 runs. The order of settingson horizontal line is the same as in Table II. It is seen thatthe most frequent strategies (settings) are bin1 and bin4, i.e.those with binomial crossover and mutation applied only inone dimension of the search space. However, while the sumof their frequencies is less than 70 % and other strategies withfrequencies mostly about 5 % or more can help efficiently inthe search in the case of the unrotated function, in the case ofthe rotated function the sum of the bin1 and bin4 frequenciesis about 85 % and the frequencies of the other strategies falldown to 1 %. Thus, their role becomes almost negligible.The distributions for the rest pairs were very similar to thedistributions presented in Figure 5 and they are not shownhere in order to save space.

The results of the comparison of unrotated and rotatedfunctions indicate that the possible cause of algorithm failurein the problems with rotated functions is the absence ofanother strategy suitable for rotated problems in the poolof competing strategies.

VIII. CONCLUSIONS

The b6e6rl variant of competitive DE was proposedseveral years ago and it was used successfully in the solution

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Fig. 5. Distribution of successful generating trials points among twelvecompeting settings

of various optimization problems [17], [20]. This DE variantwas applied to the problems of CEC 2013 competitionwithout any changes. The algorithm is adaptive, so the onlyparameter to be set is the size of population. The CEC2013 problems were expected to be hard and the size ofthe population was used greater than in former applications.

The algorithm performed well especially in the problemsof lower dimension or in the problems with unrotated objec-tive functions. However, systematical failure was observedin the minimization of several functions at all three levelsof dimension, mostly in the problems with rotated objectivefunctions. Since the submission of the first version of thepaper several attempts how to modify the algorithm in orderto achieve a higher efficiency of the algorithm have been pro-posed and tested on CEC 2013 problems. The modificationswere focused on a refreshment of the population, when somekind of stagnation happens in the search process. However,any modification used in the experiments did not increasethe efficiency of the algorithm significantly.

After the analysis presented in Section VII, we canconclude that the possible cause of the algorithm failureon rotated functions is the lack of such a strategy in thepool of competing strategies that is able to cope well withthe minimization of rotated functions. To find a new DEstrategy pool which makes the algorithm more efficient inthe optimization problems with rotated objective functions isthe topic for next research.

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ACKNOWLEDGMENT

This work was supported by the European Regional De-velopment Fund in the IT4Innovations Centre of Excellenceproject (CZ.1.05/1.1.00/02.0070) and by SGS17/PRF/2013project of University of Ostrava.

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