An Improved LSHADE-RSP Algorithm with the Cauchy Perturbation: iLSHADE-RSP
AAn Improved LSHADE-RSP Algorithm with the Cauchy Perturbation: iLSHADE-RSP
Tae Jong Choi a , Chang Wook Ahn b a Department of Artifical Intelligence and Software, Kyungil University50, Gamasil-gil, Hayang-eup, Gyeongsan-si, Gyeongsangbuk-do, Republic of Korea b Artificial Intelligence Graduate School, Gwangju Institute of Science and Technology (GIST)123, Cheomdangwagi-ro, Buk-gu, Gwangju, Republic of Korea
Abstract
A new method for improving the optimization performance of a state-of-the-art di ff erential evolution (DE) variant is proposed inthis paper. The technique can increase the exploration by adopting the long-tailed property of the Cauchy distribution, which helpsthe algorithm to generate a trial vector with great diversity. Compared to the previous approaches, the proposed approach perturbs atarget vector instead of a mutant vector based on a jumping rate. We applied the proposed approach to LSHADE-RSP ranked secondplace in the CEC 2018 competition on single objective real-valued optimization. A set of 30 di ff erent and di ffi cult optimizationproblems is used to evaluate the optimization performance of the improved LSHADE-RSP. Our experimental results verify thatthe improved LSHADE-RSP significantly outperformed not only its predecessor LSHADE-RSP but also several cutting-edge DEvariants in terms of convergence speed and solution accuracy. Keywords:
Artificial Intelligence, Evolutionary Algorithm, Di ff erential Evolution, Mathematical Optimization
1. Introduction
A population-based metaheuristic optimization methodcalled evolutionary algorithms (EAs) is designed based on Dar-win’s theory of natural selection. EAs generate a set of initialcandidate solutions and update them iteratively with artificiallydesigned evolutionary operators. As compared to traditionalsearch algorithms, EAs are global, robust, and can be appliedto any problem.It is important to establish a balance between exploration andexploitation to improve the optimization performance of EAs.Matej et al. [1] stated that “Exploration is the process of visit-ing entirely new regions of a search space, whilst exploitationis the process of visiting those regions of a search space withinthe neighborhood of previously visited points.” If an EA hastoo strong exploration, it might not be beneficial from exist-ing candidate solutions [2]. On the other hand, if an EA hastoo strong exploitation, the probability of finding an optimalsolution might be decreased [2]. Many researchers have inves-tigated a number of approaches for balancing the two corner-stones [1].Di ff erential evolution (DE) proposed by Storn and Price[3, 4] is one of the most successful EAs to deal with mathemati-cal optimization. DE distributes its candidate solutions over thesearch boundaries of an optimization problem and updates themiteratively with vector di ff erence based evolutionary operators.DE has two main advantages over other EAs: 1) it has a sim-ple structure and a few control parameters, and 2) the e ff ective-ness of DE has been demonstrated on various real-world prob-lems [5, 6]. Among numerous DE variants, L-SHADE variants[7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18] frequently performvery well on various optimization problems. LSHADE-RSP [17] was recently proposed and ranked second place in the CEC2018 competition on single objective real-valued optimization.Although LSHADE-RSP has shown excellent performance,it has the following problem: LSHADE-RSP uses a rank-basedselective pressure scheme, which increases the greediness andboosts the convergence speed. However, it may cause prema-ture convergence in which all the candidate solutions fall intothe local optimum of an optimization problem and cannot es-cape from there [19, 20, 21, 22]. Although LSHADE-RSP usesa setting for increasing the number of p best individuals in ane ff ort to mitigate the problem, it may not be su ffi cient. In otherwords, LSHADE-RSP may fail to achieve exploration and ex-ploitation.In this paper, we proposed a new method for improvingthe optimization performance of LSHADE-RSP. The techniqueperturbs a target vector with the Cauchy distribution based on ajumping rate, which helps the algorithm to generate a trial vec-tor with great diversity. Therefore, the technique can increasethe probability of finding an optimal solution by adopting thelong-tailed property of the Cauchy distribution. In the literatureof DE, some researchers have demonstrated the e ff ectiveness ofusing the Cauchy distribution in the phase of the recombination[23, 24, 25, 26, 27]. The novelty of the proposed approach liesin the perturbation of a target vector instead of a mutant vector.We named the combination of LSHADE-RSP and the proposedapproach as iLSHADE-RSP.We carried out experiments to evaluate the optimization per-formance of the proposed algorithm on the CEC 2017 testsuite [28]. Our experimental results verify that the proposedalgorithm significantly outperformed not only its predecessorLSHADE-RSP but also several state-of-the-art DE variants in Preprint submitted to Knowledge-Based Systems June 5, 2020 a r X i v : . [ c s . N E ] J un erms of convergence speed and solution accuracy.The rest of this paper is organized as follows: We introducethe background of this paper in Section 2. In Section 3, wereview the relevant literature to know, especially for L-SHADEvariants. In Section 4, the details of the proposed algorithmis explained. We describe the experimental setup in Section 5.We present the experimental results and discussion in Section6. Finally, we conclude this paper in Section 7.
2. Background ff erential Evolution Since it was introduced, DE [3, 4] has received much atten-tion because of simplicity and applicability. At the beginningof an optimization process, DE generates a set of NP initialcandidate solutions as follows. P g = ( x , g , x , g , · · · , x NP , g ) (1)where P g denotes a population at generation g . Each candidatesolution denoted by x i , g = ( x i , g , x i , g , · · · , x Di , g ) is a D -dimensionalvector. DE updates the candidate solutions iteratively with vec-tor di ff erence based evolutionary operators, such as mutation,crossover, and selection, to search for the global optimum of anoptimization problem. The mutation and crossover operatorscreate a set of NP o ff spring, and the selection operator createsa population for the next generation by comparing the fitnessvalue of a candidate solution and that of its corresponding o ff -spring. DE returns the current best solution when it reaches themaximum number of generations G max or function evaluations NFE max . At the beginning of an optimization process, DE distributesits candidate solutions over the search boundaries of an opti-mization problem with the initialization operator. Each candi-date solution is initialized as follows. x ji , = x jmin + rand ji · ( x jmax − x jmin ) (2)where x min = ( x min , x min , · · · , x Dmin ) and x max = ( x max , x max , · · · , x Dmax ) denote the lower and upper searchboundaries of an optimization problem, respectively. Also, rand ji denotes a uniformly distributed random number between[0 , A mutant vector v i , g is created in the mutation operator. Thesix frequently used classical mutation strategies are listed asfollows. • DE / rand / v i , g = x r , g + F · ( x r , g − x r , g ) • DE / rand / v i , g = x r , g + F · ( x r , g − x r , g ) + F · ( x r , g − x r , g ) • DE / best / v i , g = x best , g + F · ( x r , g − x r , g ) • DE / best / v i , g = x best , g + F · ( x r , g − x r , g ) + F · ( x r , g − x r , g ) • DE / current-to-best / v i , g = x i , g + F · ( x best , g − x i , g ) + F · ( x r , g − x r , g ) • DE / current-to-rand / v i , g = x i , g + K · ( x r , g − x i , g ) + F · ( x r , g − x r , g )where r , r , r , r , r denote mutually di ff erent random indiceswithin { , , · · · , NP } , which are also di ff erent from i . More-over, x best , g denotes the current best candidate solution. Fur-thermore, F denotes a scaling factor, and K denotes a uniformlydistributed random number between [0 , A trial vector u i , g is created in the crossover operator. Thebinomial crossover frequently used creates a trial vector as fol-lows. u ji , g = v ji , g if rand ji < CR or j = j rand x ji , g otherwise (3)where j rand denotes a random index within { , , · · · , D } . Also, CR denotes a crossover rate. The exponential crossover createsa trial vector as follows. u ji , g = v ji , g if j = (cid:104) n (cid:105) D , (cid:104) n + (cid:105) D , · · · , (cid:104) n + L − (cid:105) D x ji , g otherwise (4)where n denotes a random index within { , , · · · , D } , and L denotes the number of elements, which can be calculated asfollows. L = { L = L + } WHILE (( rand ji < CR ) AND ( L < D ))Also, (cid:104)·(cid:105) D denotes the modulo of D . DE creates a population for the next generation with the se-lection operator. The selection operator compares the fitnessvalue of a candidate solution ( x i , g ) and that of its correspond-ing o ff spring ( u i , g ) and picks the better one in terms of solutionaccuracy as follows. x i , g + = (cid:40) u i , g if f ( u i , g ) ≤ f ( x i , g ) x i , g otherwise (5)where f ( x ) denotes an optimization problem to be minimized. The Cauchy distribution is a family of continuous probabil-ity distributions, which is stable and has a probability densityfunction (PDF), which can be expressed analytically. As com-pared to the Gaussian distribution, the Cauchy distribution has2 igure 1: The four di ff erent PDFs of the Cauchy distribution a higher peak and a longer tail. The Cauchy distribution hastwo parameters: the location parameter x and the scale param-eter γ . The Cauchy distribution has a short and wide PDF if thescale parameter is high, while a tall and narrow PDF if the scaleparameter is low. The PDF of the Cauchy distribution with x and γ can be defined as follows. f ( x ; x , γ ) = πγ [1 + ( x − x γ ) ] = π (cid:20) γ ( x − x ) + γ (cid:21) (6)Additionally, the cumulative distribution function of theCauchy distribution with x and γ can be defined as follows. F ( x ; x , γ ) = π arctan (cid:18) x − x γ (cid:19) +
12 (7)Fig. 1 shows the four di ff erent PDFs of the Cauchy distribution.
3. Literature Review
Since it was introduced, many researchers have developednew methods for DE, such as • Adaptive parameters with single mutation strategy DE al-gorithms [29, 30, 31, 32, 33, 34, 35, 36, 37], • Adaptive parameters with multiple mutation strategy DEalgorithms [38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48], • Hybrid DE algorithms [49, 50, 51, 52, 27, 26, 25, 24, 53,23], • DE algorithms with sampling explicit probabilistic models[54, 55, 56].For more detailed information, please refer to the following pa-pers [5, 6, 57, 58].Among numerous DE variants, L-SHADE variants [7, 8,9, 10, 11, 12, 13, 14, 15, 16, 17, 18] frequently perform very well on various optimization problems. JADE [7] pro-posed by Zhang and Sanderson is considered to be the ori-gin of L-SHADE variants. JADE uses a new mutation strat-egy DE / current-to- p best / / current-to-best /
1, and the external archive suppliesthe progress of an evolutionary process. These modificationsimprove the exploration of the algorithm. JADE also uses alearning process-based adaptive parameter control to adjust thecontrol parameters F and CR automatically. In the experi-ments, JADE outperformed several algorithms, including jDE[29], SaDE [38], and PSO [59].Tanabe and Fukunaga proposed SHADE [8], an enhance-ment to JADE. The main di ff erence between SHADE and JADEis that SHADE utilizes historical memories, which store the av-erage of successfully evolved individuals’ control parameters F and CR to adjust the control parameters F and CR auto-matically. In the experiments, SHADE outperformed severalalgorithms, including CoDE [40], EPSDE [39], and dynNP-DE [30]. Tanabe and Fukunaga later proposed L-SHADE [9],an enhancement to SHADE. The main di ff erence between L-SHADE and SHADE is that L-SHADE utilizes linear popula-tion size reduction (LPSR), which gradually reduces the pop-ulation size as a linear function to establish a balance betweenexploration and exploitation. In the experiments, L-SHADEoutperformed several algorithms, including NBIPOP- A CMA-ES [60] and iCMAES-ILS [61].Brest et al. proposed an improved version of L-SHADEcalled iL-SHADE [10]. iL-SHADE updates historical mem-ories µ F and µ CR by calculating the average of old and newvalues. iL-SHADE also gradually reduces the p value ofDE / current-to- p best / / current-to- p best-w /
1, which assignsa lower scaling factor F w at the early stage of an evolutionaryprocess and a higher scaling factor F w at the late stage of anevolutionary process. jSO was ranked second place in the CEC2017 competitions on single objective real-valued optimization.Awad et al. proposed LSHADE-EpSin [12] based on L-SHADE, which utilizes a new ensemble sinusoidal approachfor tuning the scaling factor F in an adaptive manner. Thenew ensemble sinusoidal approach is the combination of twosinusoidal waves whose objective is to establish a balance be-tween exploration and exploitation. LSHADE-EpSin also uti-lizes a random walk at the late stage of an evolutionary pro-cess. Awad et al. proposed L- conv SHADE [13] based on L-SHADE, which utilizes a new crossover operator based on co-variance matrix adaptation with Euclidean neighborhood forrotation invariance. Awad et al. later proposed LSHADE- cn EpSin [14], which is the combination of LSHADE-EpSin andL- conv
SHADE with two major modifications: 1) a new ensem-ble sinusoidal approach with a learning process-based adaptiveparameter control and 2) a new crossover operator based oncovariance matrix adaptation with Euclidean neighborhood forrotation invariance. Finally, Awad et al. proposed EsDE r -NR315] based on LSHADE-EpSin, which gradually reduces thepopulation size with a niching-based approach.Mohamed et al. proposed LSHADE-SPACMA [16], an en-hancement to L-SHADE, which is a hybrid algorithm betweenLSHADE-SPA [16] and a modified version of CMA-ES [62].LSHADE-SPA uses a new semi-parameter control for tuningthe scaling factor F in an adaptive manner. The modified ver-sion of CMA-ES undergoes the phase of the crossover, whichimproves the exploration of the algorithm.Yet et al. propose mL-SHADE [18], an enhancement to L-SHADE, in which three major modifications are made: 1) aterminal value for the control parameter CR is excluded, 2) apolynomial mutation strategy is included, and 3) a perturbationfor historical memories is included. These modifications im-prove the exploration of the algorithm.
4. Proposed Algorithm
This section describes an improved LSHADE-RSP callediLSHADE-RSP, which employs a modified recombination op-erator, which calculates a perturbation of a target vector withthe Cauchy distribution.
The proposed algorithm uses a new mutation strategyDE / current-to- p best / r [17]. The strategy is designed based ona rank-based selective pressure scheme [63], which proportion-ally selects two donor vectors x r , g and x r , g with respect to thefitness value. The higher the ranking of a candidate solutionhas, the more opportunity it will be selected. The strategy canbe defined as follows. • DE / current-to- p best / r: v i , g = x i , g + F w · ( x pbest , g − x i , g ) + F · ( x pr , g − ˜x pr , g )where F w = . · F if 0 ≤ NFE < . · NFE max . · F if 0 . · NFE max ≤ NFE < . · NFE max . · F otherwisewhere x pbest , g denotes one of the top 100 p % individuals with p ∈ (0 , x pr , g denotes a random donor vector from apopulation based on rank-based probabilities, and ˜x pr , g denotesa random donor vector from a population based on rank-basedprobabilities or from an external archive. The probability of the i th individual being selected can be calculated as follows. pr i = Rank i (cid:80) NP g j = ( Rank j ) (8)where Rank i = k · ( NP g − i ) + k denotes a rank greediness factor. Additionally,LSHADE-RSP uses a setting for increasing the number of p bestindividuals, which can be calculated as follows. p = . · (cid:16) + NFENFE max (cid:17) (10)
The proposed algorithm uses linear population size reduc-tion (LPSR) [9] to establish a balance between exploration andexploitation. The idea behind LPSR is to use a higher popula-tion size at the beginning of an optimization process and grad-ually reduce it as a linear function. At the end of each gener-ation, LPSR calculates the population size for the next genera-tion NP g + as follows. NP g + = round (cid:104) NP init − NESNES max · (cid:16) NP init − NP f in (cid:17)(cid:105) (11)where NP init and NP f in denote the initial and final populationsizes, respectively. If the next population size NP g + is smallerthan the current one NP g , the worst NP g − NP g + candidatesolutions with respect to the fitness value are discarded. For theinitial and final population sizes, LSHADE-RSP uses NP init = round (cid:16) sqrt ( D ) ∗ log ( D ) ∗ (cid:17) and NP f in = The proposed algorithm uses a learning process-based adap-tive parameter control [11] to adjust the control parameters F and CR automatically. Each candidate solution has its controlparameters F i , g and CR i , g . At each generation, the control pa-rameters F i , g and CR i , g are calculated as follows. F i , g = rndc i ( M F , r , .
1) (12) CR i , g = rndn i ( M CR , r , .
1) (13)where rndc i and rndn i denote the Cauchy and Gaussian distri-butions, respectively. Also, M F , r and M CR , r denote randomlyselected values from historical memories µ F and µ CR , respec-tively. The scaling factor is recalculated if F i , g ≤ F i , g >
1. The crossover rate is first truncated to [0 , CR i , g = . CR i < . NFE < . · NFE max . CR i < . NFE < . · NFE max CR i , g otherwiseThe historical memories µ F and µ CR store the successfullyevolved candidate solutions’ control parameters. The capacityof the historical memories is H . At the beginning of an op-timization process, all the entries, except the last one, of thememory µ F are initialized to 0.3. Similarly, all the entries, ex-cept the last one, of the memory µ CR are initialized to 0.8. Thelast entry of the memories µ F and µ CR always keep 0.9 duringthe optimization process. After the selection operator, the suc-cessfully evolved candidate solutions’ control parameters arestored in S F and S CR . Then, one of the entries of the memoriesis updated as follows.4 F , k = (cid:40) mean WL ( S F ) if S F (cid:44) ∅ M F , k otherwise (14) M CR , k = (cid:40) mean WL ( S CR ) if S CR (cid:44) ∅ M CR , k otherwise (15)where meanw WL denotes the weighted Lehmer mean, whichtakes into consideration the improvement in fitness values be-tween candidate solutions and their corresponding o ff spring. LSHADE-RSP [17] uses a rank-based selective pressurescheme, which tends to select higher ranking candidate solu-tions as donor vectors. Therefore, the scheme can increase thegreediness, which can boost the convergence speed. However,the scheme may decrease the solution accuracy because of pre-mature convergence in which all the candidate solutions fallinto the local optimum of an optimization problem and cannotescape from there [19, 20, 21, 22]. Although LSHADE-RSPuses a setting for increasing the number of p best individuals, itmay not be su ffi cient to compensate for the increased greedi-ness.To improve the exploration property of EAs, many re-searchers have developed new methods. Among them, usinga long-tailed stable distribution, such as the Cauchy or L´evydistribution, in the phase of the recombination is one of thepopular ones. By using a long-tailed stable distribution, EAscan generate candidate solutions over large distances, whichcan improve the exploration property. In the literature of DE,some researchers have demonstrated the e ff ectiveness of using along-tailed stable distribution in the phase of the recombination[26, 27, 25]. This motivated us to devise a modified recombi-nation operator for LSHADE-RSP.The idea behind the modified recombination operator is sim-ple. When creating a trial vector, the operator first perturbs atarget vector with the Cauchy distribution. After that, the op-erator creates a trial vector by recombining the perturbed targetvector and its corresponding mutant vector. Therefore, a muchdi ff erent trial vector can be created by adopting the long-tailproperty of the Cauchy distribution. The novelty of the pro-posed approach lies in the perturbation of a target vector in-stead of a mutant vector. Fig. 2 illustrates the behavior of themodified recombination operator with DE / rand / / bin.As we mentioned earlier, the modified recombination opera-tor is an extension of the recombination operator of LSHADE-RSP. The original operator can be defined as follows. u ji , g = x ji , g + F w · ( x jpbest , g − x ji , g ) if rand ji < CR or j = j rand + F · ( x jpr , g − ˜ x jpr , g ) x ji , g otherwise (16)where x jpbest , g denotes the j th component of one of the top100 p % individuals with p ∈ (0 , x jpr , g denotes the j thcomponent of a random donor vector from a population based (a) Original DE / rand / / bin(b) Modified DE / rand / / binFigure 2: This figure illustrates the behavior of the modified recombina-tion operator with DE / rand / / bin. For simplicity of explanation, we choseDE / rand / / bin instead of DE / current-to- p best / r. As we can see from the figure,the modified DE / rand / / bin can explore larger feasible regions than the originalDE / rand / / bin. This is because the modified DE / rand / / bin perturbs a targetvector, which increases the number of possible locations for its correspondingtrial vector significantly. on rank-based probabilities, and ˜ x jpr , g denotes the j th compo-nent of a random donor vector from a population based on rank-based probabilities or from an external archive. The modifiedoperator can be defined as follows. u ji , g = x ji , g + F w · ( x jpbest , g − x ji , g ) if rand ji < CR or j = j rand + F · ( x jpr , g − ˜ x jpr , g ) rndc ji ( x ji , g , .
1) otherwise (17)where rndc ji denotes the Cauchy distribution.The proposed algorithm alternately applies one of the two re-combination operators according to the jumping rate p j . Whencreating a trial vector, the proposed algorithm applies the orig-inal operator if a random number is higher than or equal to therate. Otherwise, the proposed algorithm applies the modifiedoperator. Algorithm 1 shows the pseudo-code of the proposedalgorithm.5 lgorithm 1: iLSHADE-RSP Input :
Objective function f ( x ), lower bound x min , upperbound x max , maximum number of functionevaluations NFE max , and jumping rate p j Output:
Final best objective value f ( x best , G max ) /* Initialization */ Set function evaluation
NFE ← Set generation g ← Initialize population P g = ( x , g , · · · , x NP , g ) randomly; Set archive A ← ∅ ; Set all elements in µ F to 0.3; Set all elements in µ CR to 0.8; /* Iteration */ while None of termination criteria is satisfied do /* Recombination operator */ Set S F ← ∅ , S CR ← ∅ ; for i = i < NP ; i = i + do Assign F i , g , CR i , g using Algorithm 2; if rand i ≤ p j then u i , g ← the modified operator using Eq. (21); else u i , g ← the original operator using Eq. (16); end end /* Selection operator */ for i = i < NP ; i = i + do if f ( u i , g ) ≤ f ( x i , g ) then x i , g + ← u i , g ; x i , g → A ; F i , g → S F , CR i , g → S CR ; else x i , g + ← x i , g ; end end Shrink P g + by discarding worst solutions; Shrink A by discarding random solutions; Update µ F , µ CR ; p ← . · (cid:16) + NFENFE max (cid:17) ; g ← g + end5. Experimental Setup All the following experiments were performed on Windows10 Pro 64 bit of a PC with AMD Ryzen Threadripper 2990WX@ 3.0GHz. The proposed and comparison algorithms were de-veloped in the C ++ programming language with Visual Studio2019 64 bit. We used the following test algorithms for the comparativeanalysis. • iLSAHDE-RSP: the proposed algorithm. Algorithm 2:
Parameter Assignment Select r i from [1 , H ] randomly; if r i = H then M F , r i ← . M CR , r i ← . end F i , g ← rndc i ( M F , r i , . if g < . · NFE max and F i , g > . then F i , g ← . end if M CR , r i < then CR i , g ← else CR i , g ← rndn i ( M CR , r i , . end if g < . · NFE max then CR i , g ← max ( CR i , g , . else if g < . · NFE max then CR i , g ← max ( CR i , g , . end end • LSHADE-RSP [17]: ranked the second place in the CEC2018 competition on single objective optimization. • jSO [11]: ranked the second place in the CEC 2017 com-petition on single objective optimization. • L-SHADE [9]: ranked the first place in the CEC 2014competition on single objective optimization. • SHADE [8]: ranked the fourth place in the CEC 2013competition on single objective optimization. • JADE [7]: the origin of L-SHADE variants. • EDEV [46]: a multi-population-based DE variant. • MPEDE [45]: a multi-population-based DE variant. • CoDE [40]: a composite DE variant. • EPSDE [39]: an ensemble DE variant. • SaDE [38]: a self-adaptive DE variant. • dynNP-DE [30]: a self-adaptive DE variant.The algorithms are six L-SHADE variants, two multi-population-based DE variants, and four well-known classicalDE variants. The proposed algorithm introduces the jump-ing rate p j . As shown in Section 6, the proposed algorithmworks best with p j ∈ [0 . , . p j = . .3. Test Functions To compare the proposed and comparison algorithms exper-imentally, we carried out experiments on the CEC 2017 testsuite [28] in 10, 30, 50 and 100 dimensions. The CEC 2017test suite has 30 di ff erent and di ffi cult optimization problems,such as three unimodal test functions ( F - F ), seven simplemultimodal test functions ( F - F ), ten expanded multimodaltest functions ( F - F ), and ten hybrid composition test func-tions ( F - F ). A function is said to be unimodal if it has nolocal optima, while a function is said to be multimodal if it hasmultiple local optima.According to the experimental setups of the test suite, themaximum number of function evaluations NFE max was set to10 , · D . Moreover, the search boundaries of the test suitewere set to [ − , D . Furthermore, all the experimental re-sults were obtained by 51 runs independently. For more detailedinformation, please refer to the following papers [28]. The function error value (FEV) is utilized to assess the test al-gorithm’s accuracy. The FEV is the di ff erence between the finalbest objective value of a test algorithm and the global optimumof an optimization problem, which can be defined as follows.FEV = f ( x best , G max ) − f ( x ∗ ) (18)where f ( x ) denotes an objective function. Also, x best , G max and x ∗ denote the final best objective value and the global optimum,respectively. We utilized the Wilcoxon rank-sum test and the Friedmantest with Hochberg’s post hoc for the comparative analysis. Theformer is used to test the statistical significance of two test algo-rithms, while the latter is used to test the statistical significanceof multiple test algorithms [64].
6. Experimental Results and Discussion
In this section, we present the experimental results and dis-cussion on the CEC 2017 test suite in 10, 30, 50, and 100 di-mensions.
We present the comparative analysis of the test algorithmsin this subsection. Tables 1, 3, 5, and 7 present the means andstandard deviations of the FEVs of the test algorithms in 10, 30,50, and 100 dimension, respectively. In the tables, the symbols“ + ”, “ = ”, and “-” denote that the corresponding algorithm hasstatistically better, similar or worse performance compared tothe proposed algorithm, respectively. Moreover, Tables 2, 4, 6,and 8 present the results of the Friedman test with Hochberg’spost hoc in 10, 30, 50, and 100 dimension, respectively. Fur-thermore, Figs. 3, 4, 5, 6, and 7 provide the convergence graphsof the test algorithms in 100 dimension. Table 1 presents the means and standard deviations of theFEVs of the test algorithms in 10 dimension, obtained by 51independent runs. As can be seen from the table, the proposedalgorithm performs better performance than all of the other testalgorithms. Specifically, iLSHADE-RSP found significantlybetter solutions with lower FEVs than SHADE, JADE, EDEV,MPEDE, CoDE, EPSDE, and SaDE on more than 50 percent ofthe test functions. In particular, MPEDE and CoDE were sig-nificantly outperformed by iLSHADE-RSP on approximately80 percent of the test functions. As compared to its predeces-sor LSHADE-RSP, the proposed algorithm considerably out-performed on 2 test functions and underperformed it on 0 testfunctions. In addition, Table 2 presents the Friedman test withHochberg’s post hoc, which supports the comparative analy-sis in Table 1 where iLSHADE-RSP ranked the first amongthe test algorithms, and the outperformance over JADE, EDEV,MPEDE, CoDE, EPSDE, and SaDE was statistically signifi-cant.The means and standard deviations of the FEVs of the pro-posed and comparison algorithms in 30 dimension are shownin Table 3, collected by 51 independent runs. As can be seenfrom the table, the proposed algorithm performs better per-formance than all of the other test algorithms. Specifically,iLSHADE-RSP found significantly better solutions with lowerFEVs than SHADE, JADE, EDEV, MPEDE, CoDE, EPSDE,SaDE, and dynNP-DE on more than 80 percent of the test func-tions. In particular, JADE, EDEV, MPEDE, CoDE, EPSDE,and SaDE were not able to outperform iLSHADE-RSP on anyof the test functions. As compared to its predecessor LSHADE-RSP, the proposed algorithm considerably outperformed on 6test functions and underperformed it on 1 test functions. Ad-ditionally, Table 4 presents the Friedman test with Hochberg’spost hoc, which supports the comparative analysis in Table 3where iLSHADE-RSP ranked the first among the test algo-rithms, and the outperformance over SHADE, JADE, EDEV,MPEDE, CoDE, EPSDE, SaDE, and dynNP-DE was statisti-cally significant.Table 5 presents the means and standard deviations of theFEVs of the test algorithms in 50 dimension, obtained by 51independent runs. As can be seen from the table, the proposedalgorithm performs better performance than all of the other testalgorithms. Specifically, iLSHADE-RSP found significantlybetter solutions with lower FEVs than all of the other test al-gorithms except LSHADE-RSP on more than 50 percent of thetest functions. In particular, SHADE, JADE, EDEV, MPEDE,CoDE, EPSDE, SaDE, and dynNP-DE were significantly out-performed by iLSHADE-RSP on approximately 90 percent ofthe test functions. As compared to its predecessor LSHADE-RSP, the proposed algorithm considerably outperformed on 8test functions and underperformed it on 4 test functions. Inaddition, Table 6 presents the Friedman test with Hochberg’spost hoc, which supports the comparative analysis in Table 5where iLSHADE-RSP ranked the first among the test algo-rithms, and the outperformance over SHADE, JADE, EDEV,MPEDE, CoDE, EPSDE, SaDE, and dynNP-DE was statisti-cally significant.The means and standard deviations of the FEVs of the pro-7 able 1: Means and standard deviations of FEVs of test algorithms on CEC 2017 test suite in 10 dimension iLSHADE-RSP LSHADE-RSP jSO L-SHADE SHADE JADE EDEVMEAN (STD DEV) MEAN (STD DEV) MEAN (STD DEV) MEAN (STD DEV) MEAN (STD DEV) MEAN (STD DEV) MEAN (STD DEV)F1 0.00E +
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00 (0.00E + = +
00 (0.00E + = +
00 (0.00E + = F2 0.00E +
00 (0.00E +
00) 0.00E +
00 (0.00E + = +
00 (0.00E + = +
00 (0.00E + = +
00 (0.00E + = +
00 (0.00E + = = F3 0.00E +
00 (0.00E +
00) 0.00E +
00 (0.00E + = +
00 (0.00E + = +
00 (0.00E + = +
00 (0.00E + = +
00 (0.00E + = +
00 (0.00E + = F4 0.00E +
00 (0.00E +
00) 0.00E +
00 (0.00E + = +
00 (0.00E + = +
00 (0.00E + = +
00 (0.00E + = +
00 (0.00E + = +
00 (0.00E + = F5 1.29E +
00 (8.03E-01) 1.29E +
00 (9.39E-01) = +
00 (8.74E-01) - 2.46E +
00 (9.21E-01) - 2.64E +
00 (7.38E-01) - 3.48E +
00 (8.39E-01) - 4.68E +
00 (9.38E-01) -F6 2.91E-14 (5.02E-14) 1.56E-14 (3.96E-14) = +
00 (0.00E + + +
00 (0.00E + + + +
01 (6.28E-01) 1.18E +
01 (4.92E-01) = +
01 (6.40E-01) = +
01 (7.14E-01) = +
01 (7.39E-01) - 1.37E +
01 (8.66E-01) - 1.54E +
01 (1.28E +
00) -F8 1.56E +
00 (8.02E-01) 1.37E +
00 (9.32E-01) = +
00 (7.82E-01) - 2.61E +
00 (8.56E-01) - 2.55E +
00 (8.80E-01) - 3.59E +
00 (9.42E-01) - 5.37E +
00 (1.19E +
00) -F9 0.00E +
00 (0.00E +
00) 0.00E +
00 (0.00E + = +
00 (0.00E + = +
00 (0.00E + = +
00 (0.00E + = +
00 (0.00E + = +
00 (0.00E + = F10 4.01E +
01 (7.47E +
01) 2.18E +
01 (4.56E + = +
01 (5.92E + = +
01 (4.19E + = +
01 (5.46E +
01) - 7.73E +
01 (5.30E +
01) - 2.12E +
02 (7.48E +
01) -F11 0.00E +
00 (0.00E +
00) 0.00E +
00 (0.00E + = +
00 (0.00E + = = +
00 (6.89E-01) - 2.11E +
00 (6.85E-01) -F12 3.55E-01 (2.09E-01) 3.71E-01 (1.63E-01) = +
00 (1.68E + = +
01 (5.22E +
01) - 9.18E +
01 (7.31E +
01) - 9.22E +
01 (7.82E +
01) - 1.06E +
02 (8.74E +
01) -F13 3.19E +
00 (2.39E +
00) 3.25E +
00 (2.39E + = +
00 (2.46E + = +
00 (2.14E + = +
00 (2.44E + = +
00 (3.06E + = +
00 (2.06E +
00) -F14 1.95E-02 (1.39E-01) 1.56E-01 (3.65E-01) = = +
00 (5.09E-01) -F15 2.14E-01 (2.25E-01) 2.00E-01 (2.26E-01) = = = = = + = +
00 (7.72E-01) - 2.93E +
00 (1.42E +
00) -F17 6.29E-01 (4.22E-01) 6.49E-01 (4.42E-01) = + + + = +
00 (7.46E-01) -F18 1.78E-01 (1.96E-01) 2.06E-01 (2.18E-01) = = = + = +
00 (4.83E +
00) -F19 1.24E-02 (9.75E-03) 1.03E-02 (1.05E-02) = = = = + +
00 (0.00E + + + + + F21 1.16E +
02 (3.77E +
01) 1.16E +
02 (3.76E + = +
02 (4.98E + = +
02 (5.07E +
01) - 1.27E +
02 (4.29E +
01) - 1.51E +
02 (4.88E +
01) - 1.13E +
02 (3.47E + = F22 1.00E +
02 (0.00E +
00) 1.00E +
02 (0.00E + = +
01 (7.76E + = +
02 (0.00E + = +
01 (1.80E + = +
01 (1.70E + = +
01 (4.20E + + F23 3.01E +
02 (1.64E +
00) 2.95E +
02 (4.22E + = +
02 (1.74E + = +
02 (1.56E +
00) - 3.03E +
02 (1.59E +
00) - 3.05E +
02 (1.47E +
00) - 3.06E +
02 (1.31E +
00) -F24 2.49E +
02 (1.16E +
02) 2.53E +
02 (1.09E + = +
02 (1.03E + = +
02 (5.17E +
01) - 2.79E +
02 (9.13E + = +
02 (8.16E +
01) - 2.24E +
02 (1.18E + = F25 4.07E +
02 (1.80E +
01) 4.00E +
02 (8.82E + = +
02 (1.94E + = +
02 (2.13E + = +
02 (2.27E +
01) - 4.18E +
02 (2.28E +
01) - 4.07E +
02 (1.82E + = F26 3.00E +
02 (0.00E +
00) 3.00E +
02 (0.00E + = +
02 (0.00E + = +
02 (0.00E + = +
02 (0.00E + = +
02 (4.20E + = +
02 (0.00E + = F27 3.86E +
02 (2.67E +
00) 3.90E +
02 (4.28E-01) - 3.90E +
02 (3.85E-01) - 3.90E +
02 (4.01E-01) - 3.90E +
02 (1.40E +
00) - 3.89E +
02 (5.66E-01) - 3.89E +
02 (9.00E-01) -F28 3.08E +
02 (3.92E +
01) 3.14E +
02 (6.03E + = +
02 (8.53E + = +
02 (1.02E + = +
02 (1.38E +
02) - 3.63E +
02 (1.22E + = +
02 (0.00E + = F29 2.34E +
02 (3.56E +
00) 2.34E +
02 (2.97E + = +
02 (3.19E + = +
02 (2.54E + = +
02 (6.32E +
00) - 2.44E +
02 (5.07E +
00) - 2.49E +
02 (5.39E +
00) -F30 3.84E +
02 (3.29E +
01) 3.95E +
02 (0.00E +
00) - 2.49E +
04 (1.75E +
05) - 1.64E +
04 (1.14E +
05) - 4.20E +
02 (2.74E +
01) - 3.26E +
04 (1.60E +
05) - 1.16E +
03 (1.43E +
03) - +/=/ - 0 / / / / / / / /
15 1 / /
19 2 / / +
02 (4.59E +
02) - 3.18E-05 (2.96E-05) - 0.00E +
00 (0.00E + = +
02 (2.22E +
03) - 0.00E +
00 (0.00E + = F2 1.88E-01 (1.34E +
00) - 2.33E-06 (1.68E-06) - 4.77E-08 (1.17E-07) - 1.12E-04 (5.93E-04) - 1.39E-14 (9.55E-14) = F3 3.87E-06 (2.51E-05) - 4.27E-02 (7.38E-02) - 0.00E +
00 (0.00E + = +
00 (0.00E + = F4 4.19E-02 (2.81E-01) - 3.43E-04 (1.97E-04) - 0.00E +
00 (0.00E + = +
00 (1.53E +
00) - 8.32E +
00 (1.72E +
00) - 4.25E +
00 (1.17E +
00) - 4.12E +
00 (1.01E +
00) - 6.41E +
00 (3.56E +
00) -F6 5.56E-05 (2.25E-05) - 8.05E-14 (5.25E-14) - 0.00E +
00 (0.00E + + = +
00 (0.00E + + F7 1.99E +
01 (1.63E +
00) - 2.10E +
01 (3.05E +
00) - 1.54E +
01 (1.45E +
00) - 1.62E +
01 (1.30E +
00) - 2.18E +
01 (5.34E +
00) -F8 8.60E +
00 (2.26E +
00) - 9.12E +
00 (1.89E +
00) - 4.66E +
00 (1.30E +
00) - 4.63E +
00 (1.12E +
00) - 7.19E +
00 (3.61E +
00) -F9 7.52E-11 (8.70E-11) - 0.00E +
00 (0.00E + = +
00 (0.00E + = +
00 (0.00E + = F10 4.06E +
02 (9.26E +
01) - 4.02E +
02 (1.18E +
02) - 1.79E +
02 (8.55E +
01) - 1.70E +
02 (8.52E +
01) - 2.90E +
02 (2.61E +
02) -F11 3.41E +
00 (6.36E-01) - 5.61E-01 (6.85E-01) - 1.80E +
00 (1.06E +
00) - 2.19E +
00 (1.35E +
00) - 3.25E-01 (4.67E-01) -F12 3.64E +
04 (1.20E +
05) - 2.03E +
03 (1.09E +
03) - 2.22E +
02 (2.07E +
02) - 8.25E +
03 (2.49E +
04) - 1.15E +
00 (3.00E + = F13 1.03E +
02 (9.72E +
01) - 8.48E +
00 (3.12E +
00) - 5.32E +
00 (2.76E +
00) - 4.16E +
00 (3.43E + = +
00 (2.33E + = F14 1.08E +
01 (2.33E +
00) - 6.52E-05 (2.81E-04) - 3.03E-01 (3.94E-01) - 4.96E-01 (5.37E-01) - 4.68E-01 (5.75E-01) -F15 3.75E +
00 (6.97E-01) - 5.22E-01 (3.15E-01) - 2.92E-01 (4.64E-01) = = = F16 5.91E +
00 (2.34E +
00) - 6.40E-01 (3.42E-01) = +
00 (1.08E +
00) - 3.38E-01 (2.33E-01) + F17 1.28E +
01 (2.76E +
00) - 2.68E-01 (2.02E-01) + + = + F18 9.83E +
01 (1.30E +
02) - 4.56E-01 (2.36E-01) - 7.12E +
00 (9.67E +
00) - 3.81E-01 (5.38E-01) - 5.97E-02 (1.30E-01) + F19 2.19E +
00 (4.34E-01) - 5.18E-02 (2.58E-02) - 3.48E-03 (5.60E-03) = +
01 (6.17E +
00) - 7.61E-03 (9.57E-03) + F20 2.21E +
00 (9.06E-01) - 3.06E-02 (9.37E-02) + + + + F21 1.24E +
02 (3.00E +
01) - 1.46E +
02 (5.54E +
01) - 1.61E +
02 (5.34E +
01) - 1.33E +
02 (4.63E +
01) - 1.21E +
02 (4.29E + = F22 9.83E +
01 (1.19E + = +
01 (4.07E +
01) - 9.07E +
01 (2.86E + = +
01 (1.52E + = +
01 (4.29E + = F23 3.09E +
02 (1.72E +
00) - 3.09E +
02 (2.25E +
00) - 3.06E +
02 (1.48E +
00) - 3.06E +
02 (1.18E +
00) - 3.06E +
02 (2.06E +
00) -F24 2.65E +
02 (8.54E +
01) - 2.83E +
02 (1.03E +
02) - 3.10E +
02 (6.99E +
01) - 2.23E +
02 (1.10E + = +
02 (1.17E + = F25 4.03E +
02 (1.46E + = +
02 (9.02E + = +
02 (2.29E +
01) - 4.08E +
02 (1.87E + = +
02 (0.00E + = F26 3.00E +
02 (0.00E + = +
02 (0.00E + = +
02 (0.00E + = +
02 (0.00E + = +
02 (0.00E + = F27 3.89E +
02 (5.02E-01) - 3.88E +
02 (1.03E +
00) - 3.90E +
02 (1.75E +
00) - 3.90E +
02 (7.79E-01) - 3.89E +
02 (9.23E-01) -F28 3.00E +
02 (0.00E + = +
02 (0.00E + = +
02 (8.95E + = +
02 (5.85E + = +
02 (0.00E + = F29 2.55E +
02 (6.25E +
00) - 2.50E +
02 (6.93E +
00) - 2.46E +
02 (3.72E +
00) - 2.42E +
02 (9.24E +
00) - 2.31E +
02 (2.76E + + F30 4.34E +
03 (4.90E +
03) - 1.03E +
03 (5.21E +
02) - 3.27E +
04 (1.60E +
05) - 6.82E +
02 (6.40E +
02) - 4.01E +
02 (8.43E +
00) - +/=/ - 0 / /
26 2 / /
23 3 / /
18 1 / /
20 7 / / The symbols “ +/=/ -” indicate that the corresponding algorithm performed significantly better ( + ), not significantly better or worse ( = ), or significantly worse ( − )compared to iLSHADE-RSP using the Wilcoxon rank-sum test with α = .
05 significance level.Table 2: Friedman test with Hochberg’s post hoc for test algorithms on CEC 2017 test suite in 10 dimension
Algorithm Average ranking z-value p-value Adj. p-value (Hochberg) Sig. Test statistics1 iLSHADE-RSP 4.152 LSHADE-RSP 4.42 -2.86.E-01 7.75.E-01 7.75.E-01 No N 303 jSO 5.30 -1.24.E +
00 2.17.E-01 8.67.E-01 No Chi-Square 76.904 L-SHADE 5.23 -1.16.E +
00 2.45.E-01 7.34.E-01 No df 115 SHADE 6.12 -2.11.E +
00 3.46.E-02 1.73.E-01 No p-value 5.85.E-126 JADE 7.05 -3.12.E +
00 1.84.E-03 1.10.E-02 Yes Sig. Yes7 EDEV 7.35 -3.44.E +
00 5.87.E-04 4.11.E-03 Yes8 MPEDE 9.97 -6.25.E +
00 4.15.E-10 4.57.E-09 Yes9 CoDE 7.87 -3.99.E +
00 6.54.E-05 5.89.E-04 Yes10 EPSDE 7.62 -3.72.E +
00 1.96.E-04 1.57.E-03 Yes11 SaDE 7.90 -4.03.E +
00 5.62.E-05 5.62.E-04 Yes12 dynNP-DE 5.03 -9.49.E-01 3.43.E-01 6.85.E-01 No posed and comparison algorithms in 100 dimension are shownin Table 7, collected by 51 independent runs. As can be seenfrom the table, the proposed algorithm performs better per-formance than all of the other test algorithms. Specifically,iLSHADE-RSP found significantly better solutions with lower FEVs than all of the other test algorithms except LSHADE-RSP on more than 50 percent of the test functions. In particu-lar, MPEDE, CoDE, EPSDE, SaDE, and dynNP-DE were sig-nificantly outperformed by iLSHADE-RSP on approximately90 percent of the test functions. As compared to its predeces-8 able 3: Means and standard deviations of FEVs of test algorithms on CEC 2017 test suite in 30 dimension iLSHADE-RSP LSHADE-RSP jSO L-SHADE SHADE JADE EDEVMEAN (STD DEV) MEAN (STD DEV) MEAN (STD DEV) MEAN (STD DEV) MEAN (STD DEV) MEAN (STD DEV) MEAN (STD DEV)F1 1.67E-15 (4.62E-15) 8.35E-16 (3.37E-15) = = = = F2 0.00E +
00 (0.00E +
00) 0.00E +
00 (0.00E + = +
00 (0.00E + = = +
13 (6.16E +
13) -F3 2.56E-14 (2.85E-14) 1.89E-14 (2.70E-14) = + + +
03 (1.54E +
04) - 4.98E +
03 (1.28E + = F4 2.27E +
01 (7.02E-01) 5.86E +
01 (5.74E-14) - 5.86E +
01 (5.74E-14) - 5.87E +
01 (7.70E-01) - 3.62E +
01 (2.98E + = +
01 (2.39E +
01) - 5.16E +
01 (2.09E +
01) -F5 7.89E +
00 (2.38E +
00) 7.11E +
00 (2.09E + = +
00 (1.91E + = +
00 (1.60E + = +
01 (3.29E +
00) - 2.63E +
01 (4.00E +
00) - 3.41E +
01 (4.74E +
00) -F6 5.77E-08 (1.34E-07) 6.04E-09 (2.71E-08) = = = = +
01 (3.42E +
00) 3.95E +
01 (2.50E + + +
01 (2.10E + + +
01 (1.42E + + +
01 (3.56E +
00) - 5.46E +
01 (4.02E +
00) - 6.13E +
01 (4.25E +
00) -F8 7.93E +
00 (2.39E +
00) 7.38E +
00 (2.28E + = +
00 (2.36E + = +
00 (1.59E + = +
01 (2.86E +
00) - 2.64E +
01 (3.83E +
00) - 3.17E +
01 (5.62E +
00) -F9 0.00E +
00 (0.00E +
00) 0.00E +
00 (0.00E + = +
00 (0.00E + = +
00 (0.00E + = = = F10 1.90E +
03 (3.46E +
02) 1.92E +
03 (3.31E + = +
03 (3.36E + + +
03 (1.51E + + +
03 (2.79E + + +
03 (2.15E + = +
03 (3.23E +
02) -F11 3.41E +
00 (5.57E +
00) 2.62E +
00 (2.23E + = +
00 (8.72E + = +
01 (2.81E +
01) - 2.50E +
01 (2.72E +
01) - 3.68E +
01 (2.65E +
01) - 3.90E +
01 (2.90E +
01) -F12 1.20E +
02 (7.85E +
01) 9.51E +
01 (7.16E + = +
02 (1.14E +
02) - 1.06E +
03 (3.76E +
02) - 1.42E +
03 (7.73E +
02) - 1.15E +
03 (3.87E +
02) - 1.45E +
03 (1.21E +
03) -F13 1.80E +
01 (4.92E +
00) 1.73E +
01 (5.41E + = +
01 (4.93E + + +
01 (4.75E + = +
01 (2.37E +
01) - 4.79E +
01 (5.92E +
01) - 6.05E +
01 (7.21E +
01) -F14 2.17E +
01 (1.06E +
00) 2.15E +
01 (1.23E + = +
01 (1.21E +
00) - 2.16E +
01 (1.24E + = +
01 (6.52E +
00) - 7.30E +
03 (1.27E +
04) - 3.46E +
01 (1.44E +
01) -F15 1.08E +
00 (7.50E-01) 1.13E +
00 (7.73E-01) = = +
00 (1.46E +
00) - 2.15E +
01 (2.29E +
01) - 7.71E +
02 (1.82E +
03) - 2.46E +
01 (2.06E +
01) -F16 1.66E +
01 (6.67E +
00) 2.87E +
01 (4.35E +
01) - 7.32E +
01 (7.71E +
01) - 6.25E +
01 (7.43E +
01) - 3.09E +
02 (1.34E +
02) - 3.83E +
02 (1.45E +
02) - 4.73E +
02 (1.10E +
02) -F17 3.89E +
01 (7.18E +
00) 3.78E +
01 (7.04E + = +
01 (9.46E + + +
01 (6.94E + + +
01 (1.80E +
01) - 7.70E +
01 (3.12E +
01) - 1.01E +
02 (2.56E +
01) -F18 2.08E +
01 (2.88E-01) 2.08E +
01 (2.89E-01) = +
01 (4.08E-01) + +
01 (1.07E +
00) - 9.82E +
01 (7.62E +
01) - 1.58E +
04 (4.77E +
04) - 2.12E +
04 (6.72E +
04) -F19 3.31E +
00 (6.06E-01) 3.49E +
00 (1.08E + = +
00 (1.40E +
00) - 5.38E +
00 (1.40E +
00) - 1.26E +
01 (1.07E +
01) - 1.27E +
03 (3.62E +
03) - 1.67E +
01 (9.42E +
00) -F20 3.24E +
01 (7.02E +
00) 3.38E +
01 (9.50E + = +
01 (8.54E + = +
01 (8.81E +
00) - 7.80E +
01 (5.02E +
01) - 1.21E +
02 (6.10E +
01) - 1.31E +
02 (5.58E +
01) -F21 2.08E +
02 (2.36E +
00) 2.07E +
02 (2.19E + = +
02 (2.24E +
00) - 2.07E +
02 (1.49E + + +
02 (3.74E +
00) - 2.27E +
02 (5.36E +
00) - 2.33E +
02 (4.79E +
00) -F22 1.00E +
02 (0.00E +
00) 1.00E +
02 (0.00E + = +
02 (0.00E + = +
02 (0.00E + = +
02 (0.00E + = +
02 (3.18E + = +
02 (0.00E + = F23 3.50E +
02 (3.29E +
00) 3.51E +
02 (3.47E + = +
02 (3.20E +
00) - 3.49E +
02 (2.70E + = +
02 (5.67E +
00) - 3.74E +
02 (6.00E +
00) - 3.78E +
02 (5.24E +
00) -F24 4.26E +
02 (2.45E +
00) 4.27E +
02 (2.10E + = +
02 (2.43E + = +
02 (1.67E + = +
02 (4.98E +
00) - 4.40E +
02 (4.50E +
00) - 4.44E +
02 (4.80E +
00) -F25 3.79E +
02 (0.00E +
00) 3.87E +
02 (0.00E +
00) - 3.87E +
02 (0.00E +
00) - 3.87E +
02 (0.00E +
00) - 3.87E +
02 (7.84E-01) - 3.87E +
02 (0.00E +
00) - 3.87E +
02 (0.00E +
00) -F26 9.33E +
02 (3.92E +
01) 9.38E +
02 (3.77E + = +
02 (3.60E + = +
02 (3.69E + = +
03 (1.53E +
02) - 1.18E +
03 (1.43E +
02) - 1.27E +
03 (6.27E +
01) -F27 4.79E +
02 (6.51E +
00) 4.98E +
02 (7.29E +
00) - 4.96E +
02 (5.97E +
00) - 5.04E +
02 (5.50E +
00) - 5.05E +
02 (8.12E +
00) - 5.03E +
02 (7.56E +
00) - 5.03E +
02 (5.76E +
00) -F28 3.02E +
02 (1.60E +
01) 3.04E +
02 (2.23E + = +
02 (2.23E + = +
02 (4.86E +
01) - 3.42E +
02 (5.80E +
01) - 3.42E +
02 (5.91E +
01) - 3.39E +
02 (5.59E +
01) -F29 4.14E +
02 (2.72E +
01) 4.46E +
02 (1.39E +
01) - 4.38E +
02 (1.87E +
01) - 4.34E +
02 (6.46E +
00) - 4.77E +
02 (3.53E +
01) - 4.82E +
02 (3.84E +
01) - 5.04E +
02 (3.34E +
01) -F30 1.04E +
03 (3.21E +
02) 1.97E +
03 (1.10E +
01) - 1.97E +
03 (1.05E +
01) - 1.98E +
03 (4.71E +
01) - 2.14E +
03 (1.66E +
02) - 2.35E +
03 (1.36E +
03) - 2.55E +
03 (2.81E +
03) - +/=/ - 1 / / / /
11 5 / /
13 1 / /
26 0 / /
27 0 / / = +
00 (1.77E +
00) - 4.18E-15 (6.53E-15) = +
02 (1.17E +
03) - 5.44E-04 (2.18E-03) -F2 1.34E +
13 (7.88E +
13) - 1.30E +
22 (3.78E +
22) - 1.34E +
12 (7.09E +
12) - 2.99E +
16 (2.10E +
17) - 1.32E +
11 (6.29E +
11) -F3 7.85E +
03 (1.39E + = +
04 (5.59E +
03) - 5.00E +
03 (9.68E +
03) - 7.12E +
03 (1.81E +
04) - 1.03E +
01 (3.02E +
01) -F4 5.67E +
01 (1.17E +
01) - 8.25E +
01 (5.37E +
00) - 2.91E +
01 (2.99E + = +
01 (4.13E + = +
01 (1.02E +
01) -F5 5.08E +
01 (5.83E +
00) - 1.22E +
02 (9.30E +
00) - 6.00E +
01 (1.05E +
01) - 3.58E +
01 (5.80E +
00) - 5.02E +
01 (3.81E +
01) -F6 2.10E-13 (1.85E-13) = = = +
01 (5.87E +
00) - 1.77E +
02 (1.24E +
01) - 9.68E +
01 (9.09E +
00) - 7.37E +
01 (7.63E +
00) - 1.38E +
02 (5.36E +
01) -F8 5.03E +
01 (5.36E +
00) - 1.27E +
02 (1.06E +
01) - 6.54E +
01 (9.78E +
00) - 3.54E +
01 (5.54E +
00) - 5.97E +
01 (4.63E +
01) -F9 0.00E +
00 (0.00E + = +
02 (6.54E +
01) - 1.60E-01 (3.81E-01) - 1.18E +
00 (1.57E +
00) - 0.00E +
00 (0.00E + = F10 3.40E +
03 (2.92E +
02) - 4.64E +
03 (2.54E +
02) - 3.65E +
03 (2.95E +
02) - 2.43E +
03 (2.85E +
02) - 5.65E +
03 (7.00E +
02) -F11 3.98E +
01 (1.89E +
01) - 1.09E +
02 (1.83E +
01) - 4.73E +
01 (3.52E +
01) - 2.18E +
01 (1.73E +
01) - 1.11E +
01 (4.25E +
00) -F12 1.12E +
03 (3.92E +
02) - 3.53E +
06 (1.08E +
06) - 6.31E +
03 (8.86E +
03) - 4.90E +
04 (9.99E +
04) - 9.10E +
03 (7.03E +
03) -F13 2.64E +
04 (3.92E +
04) - 8.67E +
02 (4.65E +
02) - 1.08E +
03 (3.96E +
03) - 3.35E +
03 (8.96E +
03) - 2.67E +
01 (7.65E +
00) -F14 9.48E +
03 (7.38E +
03) - 7.19E +
01 (7.68E +
00) - 7.49E +
01 (4.28E +
01) - 1.82E +
03 (5.27E + = +
01 (1.20E + = F15 2.22E +
04 (1.35E +
04) - 9.21E +
01 (2.18E +
01) - 1.14E +
02 (1.17E +
02) - 4.53E +
03 (8.10E +
03) - 9.69E +
00 (2.36E +
00) -F16 5.99E +
02 (1.28E +
02) - 5.87E +
02 (1.34E +
02) - 4.02E +
02 (1.21E +
02) - 3.67E +
02 (1.36E +
02) - 2.87E +
02 (2.32E +
02) -F17 1.41E +
02 (3.06E +
01) - 1.03E +
02 (3.52E +
01) - 8.43E +
01 (2.42E +
01) - 7.63E +
01 (1.25E +
01) - 3.29E +
01 (1.24E + + F18 1.13E +
05 (1.62E +
05) - 1.13E +
04 (9.46E +
03) - 5.36E +
02 (6.37E +
02) - 5.20E +
04 (7.55E +
04) - 2.18E +
01 (8.26E +
00) -F19 1.45E +
04 (1.15E +
04) - 4.09E +
01 (5.71E +
00) - 8.90E +
01 (6.57E +
01) - 9.46E +
02 (3.28E +
03) - 5.19E +
00 (1.62E +
00) -F20 1.91E +
02 (5.22E +
01) - 9.07E +
01 (6.49E +
01) - 1.24E +
02 (6.42E +
01) - 1.42E +
02 (5.83E +
01) - 1.57E +
01 (1.90E + + F21 2.50E +
02 (5.72E +
00) - 3.25E +
02 (9.45E +
00) - 2.63E +
02 (9.28E +
00) - 2.37E +
02 (4.91E +
00) - 2.41E +
02 (2.54E +
01) -F22 1.00E +
02 (0.00E + = +
02 (7.06E + = +
02 (3.92E-01) = +
02 (0.00E + = +
02 (0.00E + = F23 3.96E +
02 (6.41E +
00) - 4.63E +
02 (7.91E +
00) - 4.07E +
02 (9.73E +
00) - 3.80E +
02 (5.83E +
00) - 3.79E +
02 (9.86E +
00) -F24 4.60E +
02 (7.29E +
00) - 5.55E +
02 (1.16E +
01) - 4.76E +
02 (9.02E +
00) - 4.52E +
02 (7.69E +
00) - 4.54E +
02 (1.28E +
01) -F25 3.87E +
02 (0.00E +
00) - 3.87E +
02 (0.00E +
00) - 3.87E +
02 (9.92E-01) - 3.87E +
02 (0.00E +
00) - 3.87E +
02 (0.00E +
00) -F26 1.37E +
03 (6.61E +
01) - 2.10E +
03 (4.67E +
02) - 1.39E +
03 (2.40E +
02) - 1.23E +
03 (2.04E +
02) - 1.19E +
03 (1.33E +
02) -F27 5.02E +
02 (4.47E +
00) - 5.11E +
02 (3.89E +
00) - 5.01E +
02 (9.62E +
00) - 5.06E +
02 (5.76E +
00) - 4.85E +
02 (8.30E +
00) -F28 3.14E +
02 (3.80E + = +
02 (1.94E +
01) - 3.31E +
02 (5.15E +
01) - 3.44E +
02 (5.30E +
01) - 3.19E +
02 (3.96E +
01) -F29 5.39E +
02 (2.51E +
01) - 6.65E +
02 (8.67E +
01) - 5.05E +
02 (3.84E +
01) - 4.94E +
02 (4.48E +
01) - 4.23E +
02 (2.68E +
01) -F30 7.97E +
03 (8.80E +
03) - 8.92E +
03 (2.63E +
03) - 2.40E +
03 (8.19E +
02) - 8.23E +
03 (6.40E +
03) - 2.11E +
03 (8.36E +
01) - +/=/ - 0 / /
24 0 / /
29 0 / /
26 0 / /
26 2 / / The symbols “ +/=/ -” indicate that the corresponding algorithm performed significantly better ( + ), not significantly better or worse ( = ), or significantly worse ( − )compared to iLSHADE-RSP using the Wilcoxon rank-sum test with α = .
05 significance level.Table 4: Friedman test with Hochberg’s post hoc for test algorithms on CEC 2017 test suite in 30 dimension
Algorithm Average ranking z-value p-value Adj. p-value (Hochberg) Sig. Test statistics1 iLSHADE-RSP 2.852 LSHADE-RSP 3.47 -6.62.E-01 5.08.E-01 1.02.E +
00 No N 303 jSO 3.40 -5.91.E-01 5.55.E-01 5.55.E-01 No Chi-Square 183.614 L-SHADE 3.62 -8.24.E-01 4.10.E-01 1.23.E +
00 No df 115 SHADE 5.80 -3.17.E +
00 1.53.E-03 6.12.E-03 Yes p-value 1.84.E-336 JADE 7.50 -4.99.E +
00 5.89.E-07 3.53.E-06 Yes Sig. Yes7 EDEV 7.53 -5.03.E +
00 4.89.E-07 3.42.E-06 Yes8 MPEDE 9.15 -6.77.E +
00 1.31.E-11 1.31.E-10 Yes9 CoDE 10.90 -8.65.E +
00 5.28.E-18 5.81.E-17 Yes10 EPSDE 8.80 -6.39.E +
00 1.64.E-10 1.48.E-09 Yes11 SaDE 8.70 -6.28.E +
00 3.30.E-10 2.64.E-09 Yes12 dynNP-DE 6.28 -3.69.E +
00 2.26.E-04 1.13.E-03 Yes sor LSHADE-RSP, the proposed algorithm considerably out-performed on 8 test functions and underperformed it on 5 testfunctions. Additionally, Table 8 presents the Friedman test withHochberg’s post hoc, which supports the comparative analysisin Table 7 where iLSHADE-RSP ranked the first among the test algorithms, and the outperformance over SHADE, JADE,EDEV, MPEDE, CoDE, EPSDE, SaDE, and dynNP-DE wasstatistically significant.The convergence graphs of the proposed and comparison al-gorithms in 100 dimension are provided in Figs 3, 4, 5, 6, and 7.9 able 5: Means and standard deviations of FEVs of test algorithms on CEC 2017 test suite in 50 dimension iLSHADE-RSP LSHADE-RSP jSO L-SHADE SHADE JADE EDEVMEAN (STD DEV) MEAN (STD DEV) MEAN (STD DEV) MEAN (STD DEV) MEAN (STD DEV) MEAN (STD DEV) MEAN (STD DEV)F1 2.03E-14 (1.21E-14) 1.50E-14 (5.25E-15) = = = = F2 9.75E-14 (3.30E-13) 6.07E-14 (9.54E-14) = = + +
27 (2.91E +
28) -F3 2.13E-13 (7.51E-14) 1.30E-13 (3.61E-14) + + + +
04 (4.44E +
04) - 1.74E +
04 (4.16E + + F4 4.76E +
01 (4.52E +
01) 3.70E +
01 (3.38E +
01) - 4.77E +
01 (4.44E +
01) - 7.86E +
01 (5.11E +
01) - 3.26E +
01 (4.20E + = +
01 (4.90E + = +
01 (4.83E + = F5 1.68E +
01 (5.19E +
00) 1.45E +
01 (3.54E + + +
01 (3.92E + = +
01 (2.45E + + +
01 (6.17E +
00) - 5.44E +
01 (7.50E +
00) - 6.25E +
01 (7.84E +
00) -F6 7.10E-07 (1.04E-06) 2.16E-07 (3.91E-07) + = + + + F7 7.21E +
01 (8.24E +
00) 7.04E +
01 (5.54E + = +
01 (4.66E + = +
01 (2.04E + + +
01 (6.54E +
00) - 1.02E +
02 (7.24E +
00) - 1.14E +
02 (8.62E +
00) -F8 1.71E +
01 (5.65E +
00) 1.60E +
01 (4.54E + = +
01 (4.64E +
00) - 1.28E +
01 (2.10E + + +
01 (6.37E +
00) - 5.36E +
01 (7.35E +
00) - 6.29E +
01 (9.25E +
00) -F9 3.13E-14 (5.14E-14) 4.47E-15 (2.23E-14) + = = +
00 (1.20E +
00) - 1.03E +
00 (1.08E +
00) -F10 4.16E +
03 (6.32E +
02) 4.01E +
03 (5.78E + = +
03 (4.98E + + +
03 (3.05E + + +
03 (3.10E + + +
03 (2.90E + + +
03 (3.01E +
02) -F11 1.83E +
01 (4.20E +
00) 2.32E +
01 (3.51E +
00) - 2.78E +
01 (2.97E +
00) - 4.90E +
01 (7.59E +
00) - 1.01E +
02 (2.69E +
01) - 1.34E +
02 (3.48E +
01) - 9.08E +
01 (2.58E +
01) -F12 1.50E +
03 (3.25E +
02) 1.59E +
03 (4.62E + = +
03 (3.98E +
02) - 2.28E +
03 (4.84E +
02) - 4.55E +
03 (2.60E +
03) - 4.39E +
03 (2.44E +
03) - 6.15E +
03 (3.12E +
03) -F13 2.75E +
01 (1.84E +
01) 3.07E +
01 (2.01E + = +
01 (2.24E + = +
01 (3.16E +
01) - 3.65E +
02 (2.69E +
02) - 2.87E +
02 (1.77E +
02) - 5.45E +
02 (1.06E +
03) -F14 2.37E +
01 (2.08E +
00) 2.37E +
01 (1.89E + = +
01 (2.28E +
00) - 2.95E +
01 (3.15E +
00) - 2.18E +
02 (6.74E +
01) - 7.89E +
03 (4.05E +
04) - 1.84E +
02 (9.20E +
01) -F15 1.87E +
01 (2.10E +
00) 2.07E +
01 (2.02E +
00) - 2.32E +
01 (2.51E +
00) - 4.10E +
01 (9.71E +
00) - 3.27E +
02 (1.12E +
02) - 3.05E +
02 (1.42E +
02) - 1.86E +
02 (8.93E +
01) -F16 3.15E +
02 (1.44E +
02) 3.30E +
02 (1.69E + = +
02 (1.42E +
02) - 3.97E +
02 (1.32E +
02) - 7.83E +
02 (1.87E +
02) - 8.80E +
02 (1.76E +
02) - 9.97E +
02 (1.75E +
02) -F17 2.26E +
02 (9.30E +
01) 2.73E +
02 (1.14E + = +
02 (1.11E +
02) - 2.33E +
02 (6.86E + = +
02 (1.20E +
02) - 6.44E +
02 (1.34E +
02) - 6.94E +
02 (1.04E +
02) -F18 2.29E +
01 (1.48E +
00) 2.31E +
01 (1.39E + = +
01 (1.74E +
00) - 3.78E +
01 (1.08E +
01) - 1.88E +
02 (9.11E +
01) - 1.77E +
02 (1.07E +
02) - 8.04E +
04 (2.38E +
05) -F19 1.06E +
01 (2.46E +
00) 1.03E +
01 (2.15E + = +
01 (2.69E +
00) - 2.38E +
01 (6.74E +
00) - 1.37E +
02 (4.17E +
01) - 9.19E +
02 (3.40E +
03) - 1.04E +
02 (5.16E +
01) -F20 1.30E +
02 (5.18E +
01) 1.53E +
02 (9.32E + = +
02 (1.22E + = +
02 (8.29E +
01) - 3.12E +
02 (1.04E +
02) - 5.25E +
02 (1.37E +
02) - 6.00E +
02 (1.28E +
02) -F21 2.15E +
02 (4.96E +
00) 2.15E +
02 (4.60E + = +
02 (2.98E +
00) - 2.14E +
02 (2.60E + = +
02 (6.61E +
00) - 2.53E +
02 (8.65E +
00) - 2.63E +
02 (1.05E +
01) -F22 1.67E +
03 (2.19E +
03) 2.08E +
03 (2.18E + = +
03 (1.72E + = +
03 (1.78E +
03) - 3.87E +
03 (1.17E +
03) - 3.82E +
03 (1.41E +
03) - 3.92E +
03 (2.02E +
03) -F23 4.33E +
02 (6.63E +
00) 4.31E +
02 (6.16E + = +
02 (6.55E + = +
02 (3.54E + + +
02 (8.56E +
00) - 4.78E +
02 (9.98E +
00) - 4.91E +
02 (1.19E +
01) -F24 5.09E +
02 (4.29E +
00) 5.09E +
02 (3.51E + = +
02 (4.21E + = +
02 (2.49E + + +
02 (9.10E +
00) - 5.41E +
02 (8.68E +
00) - 5.45E +
02 (1.13E +
01) -F25 4.79E +
02 (8.41E-01) 4.80E +
02 (0.00E +
00) - 4.81E +
02 (3.26E +
00) - 4.85E +
02 (1.37E +
01) - 5.30E +
02 (3.71E +
01) - 5.27E +
02 (3.41E +
01) - 5.18E +
02 (3.02E +
01) -F26 1.13E +
03 (4.85E +
01) 1.11E +
03 (5.10E + = +
03 (5.37E +
01) - 1.13E +
03 (4.86E + = +
03 (1.14E +
02) - 1.65E +
03 (1.13E +
02) - 1.71E +
03 (1.09E +
02) -F27 4.79E +
02 (6.50E +
00) 5.15E +
02 (1.42E +
01) - 5.08E +
02 (8.94E +
00) - 5.31E +
02 (1.67E +
01) - 5.47E +
02 (1.87E +
01) - 5.60E +
02 (2.98E +
01) - 5.57E +
02 (2.86E +
01) -F28 4.53E +
02 (7.30E +
00) 4.60E +
02 (6.86E +
00) - 4.59E +
02 (0.00E +
00) - 4.76E +
02 (2.36E +
01) - 4.96E +
02 (1.68E +
01) - 4.95E +
02 (2.59E +
01) - 4.89E +
02 (2.20E +
01) -F29 3.10E +
02 (2.02E +
01) 3.73E +
02 (1.76E +
01) - 3.73E +
02 (1.50E +
01) - 3.54E +
02 (1.07E +
01) - 4.74E +
02 (7.32E +
01) - 4.76E +
02 (8.27E +
01) - 5.09E +
02 (8.52E +
01) -F30 5.48E +
03 (6.12E +
03) 6.13E +
05 (4.60E +
04) - 6.04E +
05 (3.10E +
04) - 6.68E +
05 (9.38E +
04) - 6.40E +
05 (6.01E +
04) - 6.53E +
05 (7.43E +
04) - 6.52E +
05 (7.09E +
04) - +/=/ - 4 / / / /
17 9 / /
16 1 / /
28 2 / /
27 2 / / + +
06 (6.66E +
05) - 1.57E-07 (6.34E-07) - 1.05E +
03 (2.10E +
03) - 3.66E +
03 (3.85E +
03) -F2 1.77E +
03 (1.26E + = +
49 (5.70E +
49) - 7.43E +
31 (2.73E +
32) - 9.25E +
30 (5.12E +
31) - 1.32E +
31 (7.65E +
31) -F3 2.47E +
04 (4.05E +
04) - 1.02E +
05 (1.08E +
04) - 6.49E +
03 (1.04E +
04) - 7.48E +
03 (2.03E +
04) - 4.13E +
03 (2.06E +
03) -F4 7.28E +
01 (4.50E +
01) - 2.57E +
02 (2.15E +
01) - 6.68E +
01 (4.85E +
01) - 6.15E +
01 (5.48E + = +
01 (5.21E +
01) -F5 9.51E +
01 (9.36E +
00) - 3.11E +
02 (1.59E +
01) - 1.91E +
02 (1.99E +
01) - 8.30E +
01 (9.65E +
00) - 1.68E +
02 (1.14E +
02) -F6 1.78E-08 (7.52E-08) + + +
02 (1.01E +
01) - 4.22E +
02 (1.42E +
01) - 2.42E +
02 (1.45E +
01) - 1.51E +
02 (1.89E +
01) - 3.36E +
02 (5.43E +
01) -F8 9.75E +
01 (9.73E +
00) - 3.11E +
02 (1.59E +
01) - 1.88E +
02 (1.46E +
01) - 8.70E +
01 (1.06E +
01) - 1.91E +
02 (1.18E +
02) -F9 3.69E-01 (6.34E-01) - 2.22E +
03 (4.67E +
02) - 2.29E +
00 (4.36E +
00) - 4.73E +
00 (1.04E +
01) - 2.12E-02 (6.92E-02) = F10 6.40E +
03 (3.83E +
02) - 9.70E +
03 (3.04E +
02) - 8.47E +
03 (5.81E +
02) - 4.93E +
03 (4.41E +
02) - 1.12E +
04 (4.21E +
02) -F11 6.93E +
01 (1.05E +
01) - 2.09E +
02 (1.89E +
01) - 1.18E +
02 (6.24E +
01) - 1.97E +
02 (1.22E +
02) - 3.76E +
01 (5.95E +
00) -F12 9.67E +
03 (7.94E +
03) - 7.03E +
07 (1.58E +
07) - 1.09E +
04 (1.95E +
04) - 2.95E +
06 (2.14E +
06) - 9.27E +
04 (5.88E +
04) -F13 4.86E +
03 (1.91E +
04) - 4.02E +
04 (3.01E +
04) - 4.64E +
03 (5.64E +
03) - 2.36E +
03 (3.80E +
03) - 1.52E +
02 (1.52E +
02) -F14 3.96E +
04 (5.60E +
04) - 2.38E +
02 (1.03E +
02) - 3.18E +
02 (2.11E +
02) - 5.51E +
04 (7.61E +
04) - 3.95E +
01 (7.35E +
00) -F15 6.54E +
03 (1.06E +
04) - 1.82E +
03 (9.54E +
03) - 3.57E +
02 (1.70E +
02) - 1.65E +
03 (3.16E +
03) - 2.86E +
01 (4.16E +
00) -F16 1.16E +
03 (2.21E +
02) - 1.50E +
03 (2.56E +
02) - 8.75E +
02 (2.08E +
02) - 9.91E +
02 (2.18E +
02) - 9.11E +
02 (3.49E +
02) -F17 9.01E +
02 (1.21E +
02) - 9.28E +
02 (1.60E +
02) - 7.05E +
02 (1.54E +
02) - 6.18E +
02 (1.40E +
02) - 6.99E +
02 (3.36E +
02) -F18 1.65E +
05 (3.38E +
05) - 2.66E +
05 (2.14E +
05) - 1.52E +
05 (4.32E +
05) - 1.79E +
05 (3.79E +
05) - 8.54E +
02 (7.48E +
02) -F19 4.40E +
03 (6.08E +
03) - 1.66E +
02 (5.10E +
01) - 4.08E +
02 (1.97E +
03) - 2.35E +
03 (4.26E +
03) - 1.23E +
01 (2.78E +
00) -F20 7.18E +
02 (1.17E +
02) - 6.95E +
02 (1.57E +
02) - 4.77E +
02 (1.32E +
02) - 5.26E +
02 (1.39E +
02) - 3.99E +
02 (2.32E +
02) -F21 2.96E +
02 (9.02E +
00) - 5.13E +
02 (1.59E +
01) - 3.98E +
02 (1.87E +
01) - 2.79E +
02 (1.17E +
01) - 3.62E +
02 (1.14E +
02) -F22 3.72E +
03 (3.32E +
03) - 9.69E +
03 (2.45E +
03) - 6.89E +
03 (3.85E +
03) - 4.10E +
03 (2.41E +
03) - 4.49E +
03 (5.53E + = F23 5.15E +
02 (1.06E +
01) - 7.36E +
02 (1.62E +
01) - 6.12E +
02 (1.87E +
01) - 5.09E +
02 (1.09E +
01) - 4.98E +
02 (5.91E +
01) -F24 5.69E +
02 (1.17E +
01) - 8.38E +
02 (1.67E +
01) - 6.67E +
02 (2.22E +
01) - 5.81E +
02 (1.73E +
01) - 5.58E +
02 (4.26E +
01) -F25 5.32E +
02 (3.01E +
01) - 5.81E +
02 (1.82E +
01) - 5.33E +
02 (4.07E +
01) - 5.32E +
02 (3.39E +
01) - 4.82E +
02 (1.19E +
01) -F26 1.85E +
03 (9.48E +
01) - 4.16E +
03 (1.54E +
02) - 2.75E +
03 (1.74E +
02) - 1.97E +
03 (1.46E +
02) - 1.60E +
03 (1.63E +
02) -F27 5.35E +
02 (1.74E +
01) - 5.64E +
02 (2.53E +
01) - 6.04E +
02 (6.61E +
01) - 5.43E +
02 (2.60E +
01) - 5.06E +
02 (7.95E +
00) -F28 4.86E +
02 (2.46E +
01) - 4.79E +
02 (1.56E +
01) - 4.92E +
02 (1.99E +
01) - 4.81E +
02 (2.38E +
01) - 4.59E +
02 (0.00E +
00) -F29 5.38E +
02 (7.40E +
01) - 1.04E +
03 (1.34E +
02) - 5.62E +
02 (9.08E +
01) - 4.83E +
02 (9.23E +
01) - 3.80E +
02 (1.19E +
02) -F30 6.76E +
05 (1.18E +
05) - 7.20E +
05 (6.89E +
04) - 6.67E +
05 (7.95E +
04) - 6.21E +
05 (5.58E +
04) - 5.95E +
05 (1.68E +
04) - +/=/ - 2 / /
27 0 / /
30 1 / /
29 0 / /
29 0 / / The symbols “ +/=/ -” indicate that the corresponding algorithm performed significantly better ( + ), not significantly better or worse ( = ), or significantly worse ( − )compared to iLSHADE-RSP using the Wilcoxon rank-sum test with α = .
05 significance level.Table 6: Friedman test with Hochberg’s post hoc for test algorithms on CEC 2017 test suite in 50 dimension
Algorithm Average ranking z-value p-value Adj. p-value (Hochberg) Sig. Test statistics1 iLSHADE-RSP 2.472 LSHADE-RSP 2.63 -1.79.E-01 8.58.E-01 8.58.E-01 No N 303 jSO 3.12 -6.98.E-01 4.85.E-01 9.70.E-01 No Chi-Square 220.474 L-SHADE 3.73 -1.36.E +
00 1.74.E-01 5.21.E-01 No df 115 SHADE 6.07 -3.87.E +
00 1.10.E-04 4.41.E-04 Yes p-value 4.12.E-416 JADE 6.93 -4.80.E +
00 1.60.E-06 8.01.E-06 Yes Sig. Yes7 EDEV 7.43 -5.34.E +
00 9.55.E-08 6.69.E-07 Yes8 MPEDE 8.80 -6.80.E +
00 1.02.E-11 8.19.E-11 Yes9 CoDE 11.40 -9.60.E +
00 8.32.E-22 9.15.E-21 Yes10 EPSDE 9.40 -7.45.E +
00 9.51.E-14 9.51.E-13 Yes11 SaDE 8.93 -6.95.E +
00 3.75.E-12 3.37.E-11 Yes12 dynNP-DE 7.08 -4.96.E +
00 7.08.E-07 4.25.E-06 Yes
Each convergence graph provides the median and interquartileranges (one-fourth and three-fourth) of the FEVs of the pro-posed and comparison algorithms. The first part of the conver-gence graphs ( F - F ) is given in Fig. 3. The second part of theconvergence graphs ( F - F ) is given in Fig. 4. The third part of the convergence graphs ( F - F ) is given in Fig. 5. Thefourth part of the convergence graphs ( F - F ) is given in Fig.6. The last part of the convergence graphs ( F - F ) is givenin Fig. 7. As can be seen from the figures, iLSHADE-RSP iscompetitive with the other test algorithms in terms of solution10 able 7: Means and standard deviations of FEVs of test algorithms on CEC 2017 test suite in 100 dimension iLSHADE-RSP LSHADE-RSP jSO L-SHADE SHADE JADE EDEVMEAN (STD DEV) MEAN (STD DEV) MEAN (STD DEV) MEAN (STD DEV) MEAN (STD DEV) MEAN (STD DEV) MEAN (STD DEV)F1 9.60E-09 (1.39E-08) 1.31E-10 (2.63E-10) + + + + + + F2 2.75E +
04 (1.50E +
05) 6.12E +
06 (4.09E + = +
04 (5.55E + + +
11 (1.35E +
12) - 7.28E +
24 (5.18E + = +
22 (1.08E + = +
26 (1.67E + + F3 8.60E-06 (7.87E-06) 2.22E-06 (3.42E-06) + + + + +
04 (1.41E + + +
05 (1.58E + + F4 2.00E +
02 (2.43E +
01) 2.00E +
02 (9.41E + + +
02 (2.46E + + +
02 (2.20E + + +
01 (6.52E + + +
01 (7.17E + + +
01 (6.67E + + F5 3.73E +
01 (1.26E +
01) 3.62E +
01 (9.22E + = +
01 (8.92E +
00) - 4.25E +
01 (6.57E +
00) - 1.42E +
02 (1.97E +
01) - 1.47E +
02 (1.86E +
01) - 1.50E +
02 (1.73E +
01) -F6 3.68E-05 (2.07E-05) 2.74E-05 (1.90E-05) + = + + F7 1.57E +
02 (2.10E +
01) 1.54E +
02 (1.82E + = +
02 (1.19E + = +
02 (4.24E + + +
02 (1.80E +
01) - 2.77E +
02 (2.37E +
01) - 2.69E +
02 (1.76E +
01) -F8 3.70E +
01 (1.39E +
01) 3.53E +
01 (9.66E + = +
01 (7.40E +
00) - 4.37E +
01 (4.86E +
00) - 1.42E +
02 (2.07E +
01) - 1.45E +
02 (1.91E +
01) - 1.49E +
02 (1.59E +
01) -F9 7.02E-03 (2.43E-02) 1.75E-03 (1.25E-02) = = +
01 (2.85E +
01) - 9.69E +
01 (8.40E +
01) - 6.84E +
01 (4.62E +
01) -F10 1.26E +
04 (1.09E +
03) 1.26E +
04 (1.05E + = +
04 (1.17E + + +
04 (4.65E + + +
03 (5.09E + + +
04 (5.77E + + +
04 (5.17E + + F11 7.96E +
01 (3.04E +
01) 7.55E +
01 (2.60E + = +
02 (3.18E +
01) - 4.87E +
02 (1.25E +
02) - 1.07E +
03 (2.34E +
02) - 3.70E +
03 (3.66E +
03) - 1.47E +
03 (1.77E +
03) -F12 1.66E +
04 (7.34E +
03) 1.35E +
04 (5.12E + + +
04 (7.60E + = +
04 (8.47E +
03) - 2.14E +
04 (1.20E + = +
04 (2.38E + = +
04 (2.55E +
04) -F13 1.28E +
02 (3.83E +
01) 1.28E +
02 (3.53E + = +
02 (3.87E +
01) - 4.14E +
02 (2.10E +
02) - 3.69E +
03 (3.81E +
03) - 2.24E +
03 (2.27E +
03) - 2.82E +
03 (2.97E +
03) -F14 4.44E +
01 (4.66E +
00) 4.53E +
01 (6.15E + = +
01 (9.03E +
00) - 2.51E +
02 (3.72E +
01) - 5.68E +
02 (1.87E +
02) - 6.37E +
02 (2.20E +
02) - 9.39E +
02 (2.12E +
03) -F15 1.10E +
02 (2.66E +
01) 1.19E +
02 (3.70E + = +
02 (3.66E +
01) - 2.54E +
02 (4.44E +
01) - 3.56E +
02 (1.23E +
02) - 3.63E +
02 (1.48E +
02) - 4.78E +
02 (2.93E +
02) -F16 1.56E +
03 (3.07E +
02) 1.70E +
03 (3.45E +
02) - 1.83E +
03 (3.42E +
02) - 1.70E +
03 (2.47E +
02) - 2.39E +
03 (3.55E +
02) - 2.53E +
03 (3.23E +
02) - 2.94E +
03 (3.18E +
02) -F17 1.09E +
03 (2.75E +
02) 1.26E +
03 (3.09E +
02) - 1.30E +
03 (2.50E +
02) - 1.14E +
03 (2.21E + = +
03 (2.30E +
02) - 1.89E +
03 (2.55E +
02) - 2.09E +
03 (2.42E +
02) -F18 1.41E +
02 (2.98E +
01) 1.47E +
02 (2.98E + = +
02 (3.16E +
01) - 2.34E +
02 (4.76E +
01) - 1.65E +
03 (1.10E +
03) - 1.96E +
03 (1.29E +
03) - 2.60E +
05 (9.85E +
05) -F19 6.06E +
01 (9.24E +
00) 6.10E +
01 (9.96E + = +
01 (1.90E +
01) - 1.71E +
02 (2.12E +
01) - 1.41E +
03 (1.66E +
03) - 1.60E +
03 (2.55E +
03) - 3.13E +
02 (3.23E +
02) -F20 1.28E +
03 (2.17E +
02) 1.66E +
03 (4.26E +
02) - 1.60E +
03 (3.16E +
02) - 2.02E +
03 (2.10E +
02) - 1.70E +
03 (2.50E +
02) - 2.10E +
03 (2.77E +
02) - 2.39E +
03 (2.04E +
02) -F21 2.56E +
02 (1.33E +
01) 2.55E +
02 (1.04E + = +
02 (7.59E +
00) - 2.64E +
02 (5.62E +
00) - 3.67E +
02 (1.74E +
01) - 3.69E +
02 (2.03E +
01) - 3.78E +
02 (2.21E +
01) -F22 1.35E +
04 (1.15E +
03) 1.31E +
04 (1.15E + = +
04 (1.08E + + +
04 (4.97E + + +
04 (1.61E + + +
04 (5.93E + + +
04 (5.52E + + F23 5.61E +
02 (8.82E +
00) 5.68E +
02 (1.02E +
01) - 5.67E +
02 (1.36E +
01) - 5.69E +
02 (9.34E +
00) - 6.39E +
02 (1.57E +
01) - 6.49E +
02 (1.59E +
01) - 6.73E +
02 (1.62E +
01) -F24 9.03E +
02 (6.82E +
00) 9.02E +
02 (7.36E + = +
02 (1.02E + = +
02 (7.84E +
00) - 1.01E +
03 (2.91E +
01) - 1.02E +
03 (2.35E +
01) - 1.03E +
03 (2.47E +
01) -F25 7.13E +
02 (3.64E +
01) 7.36E +
02 (3.96E +
01) - 7.38E +
02 (3.85E +
01) - 7.48E +
02 (2.76E +
01) - 7.66E +
02 (6.51E +
01) - 7.43E +
02 (5.77E +
01) - 7.49E +
02 (5.16E +
01) -F26 3.20E +
03 (9.68E +
01) 3.19E +
03 (9.06E + = +
03 (1.02E +
02) - 3.28E +
03 (8.52E +
01) - 4.61E +
03 (2.92E +
02) - 4.51E +
03 (2.63E +
02) - 4.58E +
03 (2.43E +
02) -F27 5.65E +
02 (1.34E +
01) 5.77E +
02 (1.62E +
01) - 5.85E +
02 (2.04E +
01) - 6.27E +
02 (1.73E +
01) - 7.06E +
02 (3.94E +
01) - 7.34E +
02 (4.11E +
01) - 7.26E +
02 (3.81E +
01) -F28 5.24E +
02 (2.28E +
01) 5.23E +
02 (2.14E + = +
02 (2.50E + = +
02 (2.04E + = +
02 (4.03E + = +
02 (4.76E + = +
02 (3.81E + = F29 1.11E +
03 (2.24E +
02) 1.28E +
03 (2.14E +
02) - 1.30E +
03 (2.10E +
02) - 1.24E +
03 (1.73E +
02) - 2.13E +
03 (2.80E +
02) - 2.20E +
03 (2.79E +
02) - 2.28E +
03 (2.64E +
02) -F30 1.28E +
03 (3.42E +
02) 2.33E +
03 (1.96E +
02) - 2.29E +
03 (1.09E +
02) - 2.42E +
03 (1.54E +
02) - 2.69E +
03 (2.73E +
02) - 4.11E +
03 (1.88E +
03) - 3.92E +
03 (1.69E +
03) - +/=/ - 5 / / / /
18 6 / /
22 5 / /
22 6 / /
21 7 / / + +
08 (5.82E +
07) - 2.96E +
00 (1.60E +
01) - 4.26E +
03 (3.80E +
03) - 5.06E +
03 (4.61E +
03) -F2 3.62E +
07 (2.35E + + +
126 (1.65E + +
99 (4.23E + +
58 (1.39E +
59) - 1.03E +
72 (7.31E +
72) -F3 1.19E +
05 (1.51E +
05) - 3.66E +
05 (2.94E +
04) - 1.04E +
05 (6.28E +
04) - 4.16E +
04 (1.23E +
05) - 1.19E +
05 (1.56E +
04) -F4 4.29E +
01 (5.70E + + +
02 (7.32E +
01) - 1.45E +
02 (5.14E + + +
02 (5.59E + = +
02 (1.52E + = F5 2.39E +
02 (1.79E +
01) - 9.36E +
02 (2.38E +
01) - 6.59E +
02 (3.34E +
01) - 2.24E +
02 (3.26E +
01) - 4.88E +
02 (2.80E +
02) -F6 1.63E-02 (1.70E-02) - 3.88E +
00 (4.09E-01) - 3.73E-04 (1.23E-03) + + +
02 (3.14E +
01) - 1.30E +
03 (3.73E +
01) - 8.04E +
02 (3.69E +
01) - 4.33E +
02 (1.16E +
02) - 8.12E +
02 (2.56E +
01) -F8 2.23E +
02 (3.15E +
01) - 9.22E +
02 (3.09E +
01) - 6.47E +
02 (3.47E +
01) - 2.46E +
02 (3.67E +
01) - 5.73E +
02 (2.38E +
02) -F9 9.34E +
00 (6.82E +
00) - 1.84E +
04 (2.33E +
03) - 1.23E +
03 (1.05E +
03) - 6.67E +
02 (9.59E +
02) - 1.43E +
00 (1.28E +
00) -F10 1.62E +
04 (5.71E +
02) - 2.60E +
04 (6.51E +
02) - 2.44E +
04 (6.92E +
02) - 1.14E +
04 (1.33E + + +
04 (7.17E +
02) -F11 6.49E +
02 (2.07E +
02) - 4.12E +
03 (6.43E +
02) - 5.98E +
02 (2.93E +
02) - 2.64E +
03 (4.71E +
03) - 1.56E +
02 (3.77E +
01) -F12 3.00E +
04 (1.41E +
04) - 8.19E +
08 (1.77E +
08) - 8.85E +
04 (5.73E +
04) - 3.00E +
06 (6.39E +
06) - 4.20E +
05 (1.31E +
05) -F13 2.16E +
03 (1.52E +
03) - 4.81E +
05 (1.82E +
05) - 4.25E +
03 (5.16E +
03) - 2.00E +
03 (1.43E +
03) - 2.58E +
03 (3.84E +
03) -F14 1.75E +
05 (6.10E +
05) - 1.68E +
05 (1.18E +
05) - 3.51E +
03 (1.65E +
04) - 7.76E +
05 (1.50E +
06) - 4.02E +
02 (2.59E +
02) -F15 2.75E +
02 (4.89E +
01) - 2.04E +
04 (1.18E +
04) - 2.40E +
03 (2.84E +
03) - 9.07E +
02 (6.45E +
02) - 7.86E +
02 (1.08E +
03) -F16 3.33E +
03 (4.47E +
02) - 5.78E +
03 (3.22E +
02) - 2.88E +
03 (3.96E +
02) - 2.72E +
03 (4.05E +
02) - 4.02E +
03 (1.72E +
03) -F17 2.47E +
03 (3.01E +
02) - 3.39E +
03 (2.63E +
02) - 2.25E +
03 (2.35E +
02) - 2.14E +
03 (3.17E +
02) - 3.08E +
03 (1.08E +
03) -F18 6.63E +
04 (3.35E +
05) - 1.02E +
07 (2.54E +
06) - 2.19E +
06 (2.51E +
06) - 8.19E +
05 (1.35E +
06) - 8.12E +
04 (4.04E +
04) -F19 5.02E +
02 (2.11E +
03) - 8.23E +
04 (3.73E +
04) - 3.12E +
03 (3.38E +
03) - 7.53E +
02 (7.47E +
02) - 2.60E +
03 (3.36E +
03) -F20 2.74E +
03 (2.47E +
02) - 2.86E +
03 (2.48E +
02) - 2.27E +
03 (2.52E +
02) - 1.94E +
03 (3.33E +
02) - 3.26E +
03 (9.30E +
02) -F21 4.44E +
02 (2.56E +
01) - 1.17E +
03 (2.21E +
01) - 8.89E +
02 (3.67E +
01) - 5.39E +
02 (4.53E +
01) - 7.22E +
02 (2.66E +
02) -F22 1.70E +
04 (2.17E +
03) - 2.71E +
04 (5.62E +
02) - 2.53E +
04 (7.80E +
02) - 1.28E +
04 (1.56E + + +
04 (6.84E +
02) -F23 7.08E +
02 (1.42E +
01) - 1.23E +
03 (1.79E +
01) - 1.02E +
03 (2.58E +
01) - 6.70E +
02 (1.36E +
01) - 7.09E +
02 (1.23E +
02) -F24 1.08E +
03 (2.12E +
01) - 1.74E +
03 (3.12E +
01) - 1.47E +
03 (3.56E +
01) - 1.11E +
03 (4.70E +
01) - 9.95E +
02 (9.03E +
01) -F25 7.69E +
02 (5.20E +
01) - 1.77E +
03 (9.98E +
01) - 7.93E +
02 (5.84E +
01) - 7.98E +
02 (4.80E +
01) - 7.35E +
02 (4.19E +
01) -F26 4.93E +
03 (2.46E +
02) - 1.22E +
04 (3.32E +
02) - 9.23E +
03 (4.25E +
02) - 6.15E +
03 (8.63E +
02) - 4.04E +
03 (2.61E +
02) -F27 6.66E +
02 (2.42E +
01) - 8.95E +
02 (6.86E +
01) - 7.56E +
02 (5.17E +
01) - 6.91E +
02 (3.44E +
01) - 5.98E +
02 (1.73E +
01) -F28 5.35E +
02 (3.64E + = +
02 (3.36E +
01) - 5.50E +
02 (2.55E +
01) - 5.72E +
02 (3.70E +
01) - 5.49E +
02 (2.44E +
01) -F29 2.47E +
03 (2.45E +
02) - 4.23E +
03 (2.29E +
02) - 2.53E +
03 (2.90E +
02) - 2.24E +
03 (3.97E +
02) - 1.87E +
03 (7.05E +
02) -F30 3.58E +
03 (1.41E +
03) - 9.70E +
04 (2.23E +
04) - 3.52E +
03 (1.30E +
03) - 4.89E +
03 (2.47E +
03) - 3.19E +
03 (1.44E +
03) - +/=/ - 3 / /
26 0 / /
30 2 / /
28 3 / /
26 0 / / The symbols “ +/=/ -” indicate that the corresponding algorithm performed significantly better ( + ), not significantly better or worse ( = ), or significantly worse ( − )compared to iLSHADE-RSP using the Wilcoxon rank-sum test with α = .
05 significance level.Table 8: Friedman test with Hochberg’s post hoc for test algorithms on CEC 2017 test suite in 100 dimension
Algorithm Average ranking z-value p-value Adj. p-value (Hochberg) Sig. Test statistics1 iLSHADE-RSP 2.532 LSHADE-RSP 2.67 -1.43.E-01 8.86.E-01 8.86.E-01 No N 303 jSO 3.33 -8.59.E-01 3.90.E-01 7.80.E-01 No Chi-Square 221.944 L-SHADE 3.77 -1.32.E +
00 1.85.E-01 5.56.E-01 No df 115 SHADE 5.77 -3.47.E +
00 5.14.E-04 2.06.E-03 Yes p-value 2.04.E-416 JADE 6.67 -4.44.E +
00 9.00.E-06 4.50.E-05 Yes Sig. Yes7 EDEV 7.33 -5.16.E +
00 2.52.E-07 1.51.E-06 Yes8 MPEDE 7.73 -5.59.E +
00 2.33.E-08 1.63.E-07 Yes9 CoDE 11.87 -1.00.E +
01 1.18.E-23 1.29.E-22 Yes10 EPSDE 9.63 -7.63.E +
00 2.41.E-14 2.41.E-13 Yes11 SaDE 8.33 -6.23.E +
00 4.66.E-10 3.73.E-09 Yes12 dynNP-DE 8.37 -6.27.E +
00 3.70.E-10 3.33.E-09 Yes accuracy, especially for the test functions F , F - F , F - F ,and F - F . In particular, iLSHADE-RSP can escape fromthe local optimum of the test functions F and F , while theother test algorithms cannot. We first discuss the comparative analysis between the pro-posed algorithm and each of the two state-of-the-art L-SHADEvariants, LSHADE-RSP and jSO. After that, we discuss the ex-perimental results with the proposed algorithm and the other11 a) F (b) F (c) F (d) F (e) F (f) F Figure 3: Convergence graphs of test algorithms on CEC 2017 test suites in 100 dimension ( F - F )(a) F (b) F (c) F (d) F (e) F (f) F Figure 4: Convergence graphs of test algorithms on CEC 2017 test suites in 100 dimension ( F - F ) test algorithms. • The performance di ff erence between the proposed al-gorithm and LSHADE-RSP is negligible in 10 dimen-sion. The proposed algorithm has only two improvements against zero deteriorations. However, the performance dif-ference is much di ff erent in 30, 50, and 100 dimensions.The proposed algorithm has six improvements against onedeterioration in 30 dimension, eight improvements against12 a) F (b) F (c) F (d) F (e) F (f) F Figure 5: Convergence graphs of test algorithms on CEC 2017 test suites in 100 dimension ( F - F )(a) F (b) F (c) F (d) F (e) F (f) F Figure 6: Convergence graphs of test algorithms on CEC 2017 test suites in 100 dimension ( F - F ) four deteriorations in 50 dimension, and eight improve-ments against five deteriorations in 100 dimension. Addi-tionally, we investigated the performance di ff erence withrespect to the characteristics of the test functions. We found out that the proposed algorithm has worse perfor-mance on the unimodal ( F - F ) and some of the sim-ple multimodal test functions ( F - F ) but better perfor-mance on the expanded multimodal ( F - F ) and the hy-13 a) F (b) F (c) F (d) F (e) F (f) F Figure 7: Convergence graphs of test algorithms on CEC 2017 test suites in 100 dimension ( F - F ) brid composition test functions ( F - F ). In a word, theproposed algorithm performs better than LSHADE-RSPon more complicated optimization problems. • The performance di ff erence between the proposed algo-rithm and jSO is negligible in 10 dimension. The proposedalgorithm has only five improvements against three dete-riorations. However, the performance di ff erence is muchdi ff erent in 30, 50, and 100 dimensions. The proposed al-gorithm has 11 improvements against six deteriorations in30 dimension, 17 improvements against two deteriorationsin 50 dimension, and 18 improvements against six deteri-orations in 100 dimension. Additionally, we investigatedthe performance di ff erence with respect to the characteris-tics of the test functions. We found out that the proposedalgorithm has worse performance on the unimodal ( F - F )and some of the simple multimodal test functions ( F - F )but better performance on the expanded multimodal ( F - F ) and the hybrid composition test functions ( F - F ).In a word, the proposed algorithm performs better thanjSO on more complicated optimization problems. • The proposed algorithm found more significantly accu-rate solutions compared with the test algorithms, includ-ing L-SHADE, SHADE, JADE, EDEV, MPEDE, CoDE,EPSDE, SaDE, and dynNP-DE, in all the dimensions.Note that, the experimental results with the proposed algo-rithm and LSHADE-RSP lend weight to the e ff ectiveness ofthe modified recombination operator of the proposed algorithm,which can increase the probability of finding an optimal solu- tion by adopting the long-tailed property of the Cauchy distri-bution, and thus, it can improve the optimization performanceof LSHADE-RSP significantly. As we mentioned earlier, the proposed algorithm alternatelyapplies one of the two recombination operators according to thejumping rate p j . The jumping rate determines the additional ex-ploration of the proposed algorithm. If the jumping rate is toohigh, the modified recombination operator is applied too often,and thus, the proposed algorithm might not be beneficial fromexisting candidate solutions. On the other hand, if the jumpingrate is too low, the modified recombination operator is appliedtoo rarely, and thus, the additional exploration of the proposedalgorithm might be negligible. We carried out experiments tofind out appropriate parameter values for the jumping rate. Ta-ble 9 shows the means and standard deviations of the FEVs ofthe proposed algorithm with di ff erent parameter values for thejumping rate. As can be seen from the table, the parameter val-ues p j ∈ [0 . , .
35] can lead to satisfactory results.
As we mentioned earlier, the proposed algorithm employs amodified recombination operator, which calculates a perturba-tion of a target vector with the Cauchy distribution. The Cauchydistribution is a special case of the L´evy α -stable distribution.Therefore, it is interesting to test the other cases of the L´evy α -stable distribution for the modified recombination operator. The14 able 9: Means and standard deviations of FEVs of iLSHADE-RSP with di ff erent jumping rates on CEC 2017 test suite in 50 dimension iLSHADE-RSP pj = . pj = . pj = . pj = . pj = . pj = . pj = . pj = . + = = = = = = = = = = = F3 2.01E-13 (8.60E-14) 1.39E-13 (4.14E-14) + + = +
01 (5.32E +
01) 5.61E +
01 (4.73E + = +
01 (5.10E + = +
01 (5.32E + = +
01 (5.03E + + +
01 (5.20E + + +
01 (5.44E + + +
01 (4.96E + + F5 1.51E +
01 (5.06E +
00) 1.50E +
01 (3.86E + = +
01 (3.78E + = +
01 (4.47E + = +
01 (4.96E + = +
01 (5.66E + = +
01 (5.10E + = +
01 (5.35E + = F6 9.98E-07 (1.80E-06) 2.61E-07 (5.54E-07) + = = = +
01 (7.22E +
00) 6.96E +
01 (5.97E + + +
01 (5.70E + = +
01 (6.27E + = +
01 (8.33E + = +
01 (7.89E + = +
01 (9.80E + = +
01 (1.00E + = F8 1.62E +
01 (4.46E +
00) 1.64E +
01 (5.05E + = +
01 (3.86E + = +
01 (5.44E + = +
01 (5.45E + = +
01 (5.56E + = +
01 (5.28E + = +
01 (4.91E + = F9 4.25E-14 (5.57E-14) 6.71E-15 (2.71E-14) + = = = = = = F10 4.23E +
03 (5.00E +
02) 4.13E +
03 (5.09E + = +
03 (6.67E + = +
03 (6.18E + = +
03 (6.39E + = +
03 (6.33E + = +
03 (5.26E + = +
03 (5.96E + = F11 1.77E +
01 (3.42E +
00) 1.97E +
01 (4.26E +
00) - 1.75E +
01 (3.65E + = +
01 (4.07E + = +
01 (3.28E + = +
01 (2.67E + = +
01 (2.74E + = +
01 (2.76E +
00) -F12 1.42E +
03 (4.59E +
02) 1.55E +
03 (4.16E + = +
03 (4.04E + = +
03 (4.25E + = +
03 (4.15E + = +
03 (4.64E + = +
03 (3.50E + = +
03 (3.96E + = F13 3.03E +
01 (2.09E +
01) 2.38E +
01 (1.68E + = +
01 (2.44E + = +
01 (2.34E + = +
01 (2.42E + = +
01 (1.93E + = +
01 (1.73E + = +
01 (1.91E + = F14 2.36E +
01 (1.88E +
00) 2.40E +
01 (2.17E + = +
01 (2.02E + = +
01 (2.07E + = +
01 (2.09E + = +
01 (2.00E + = +
01 (1.82E + = +
01 (1.76E + = F15 1.87E +
01 (1.89E +
00) 1.91E +
01 (1.86E + = +
01 (1.61E +
00) - 1.88E +
01 (1.63E + = +
01 (2.48E + = +
01 (1.72E + = +
01 (2.14E + = +
01 (1.71E + = F16 2.94E +
02 (1.29E +
02) 3.04E +
02 (1.39E + = +
02 (1.28E + = +
02 (1.37E + = +
02 (1.12E + = +
02 (1.18E + = +
02 (1.32E +
02) - 3.35E +
02 (1.16E +
02) -F17 2.27E +
02 (8.62E +
01) 2.34E +
02 (9.75E + = +
02 (1.06E + = +
02 (9.24E + = +
02 (8.94E + = +
02 (8.92E + = +
02 (1.06E + = +
02 (9.12E + = F18 2.26E +
01 (1.27E +
00) 2.27E +
01 (1.27E + = +
01 (1.41E + = +
01 (1.18E + = +
01 (1.46E + = +
01 (1.49E + = +
01 (1.32E + = +
01 (1.40E + = F19 1.03E +
01 (2.46E +
00) 1.13E +
01 (2.01E +
00) - 1.06E +
01 (2.31E + = +
01 (2.85E + = +
01 (2.12E + = +
01 (2.26E + = +
01 (2.43E + = +
01 (2.35E + = F20 1.18E +
02 (4.79E +
01) 1.40E +
02 (7.51E + = +
02 (5.10E + = +
02 (2.78E + = +
02 (4.85E + = +
02 (4.78E + = +
02 (4.14E + = +
02 (5.64E +
01) -F21 2.15E +
02 (4.35E +
00) 2.15E +
02 (4.13E + = +
02 (3.70E + = +
02 (5.24E + = +
02 (4.93E + = +
02 (5.49E + = +
02 (5.17E + = +
02 (4.96E + = F22 1.86E +
03 (2.25E +
03) 1.91E +
03 (2.21E + = +
03 (2.22E + = +
03 (2.13E + = +
03 (2.25E + = +
03 (2.01E + = +
03 (2.13E + = +
03 (2.14E + = F23 4.33E +
02 (7.17E +
00) 4.31E +
02 (5.46E + = +
02 (6.28E + = +
02 (7.21E + = +
02 (6.43E + = +
02 (6.17E + = +
02 (6.92E + = +
02 (6.58E + = F24 5.09E +
02 (4.10E +
00) 5.08E +
02 (3.48E + = +
02 (4.42E + = +
02 (3.67E + = +
02 (4.09E + = +
02 (4.15E + = +
02 (4.31E + = +
02 (3.48E + = F25 4.79E +
02 (9.87E-01) 4.80E +
02 (1.41E +
00) - 4.80E +
02 (7.77E-01) - 4.80E +
02 (8.57E-01) = +
02 (8.08E-01) = +
02 (9.61E-01) = +
02 (9.51E-01) = +
02 (7.84E-01) + F26 1.13E +
03 (4.73E +
01) 1.15E +
03 (5.73E + = +
03 (5.44E + = +
03 (5.75E + = +
03 (5.27E + = +
03 (5.25E + + +
03 (5.84E + = +
03 (5.18E + = F27 4.77E +
02 (6.46E +
00) 4.89E +
02 (9.29E +
00) - 4.83E +
02 (9.11E +
00) - 4.78E +
02 (5.88E + = +
02 (7.93E + = +
02 (5.64E + = +
02 (5.90E + + +
02 (8.15E + + F28 4.52E +
02 (3.37E-01) 4.53E +
02 (6.69E-01) - 4.53E +
02 (5.94E-01) - 4.52E +
02 (3.82E-01) = +
02 (4.81E + = +
02 (4.76E-01) + +
02 (4.76E-01) + +
02 (4.93E-01) + F29 3.06E +
02 (1.93E +
01) 3.28E +
02 (3.69E +
01) - 3.17E +
02 (2.52E +
01) - 3.12E +
02 (2.03E + = +
02 (2.04E + = +
02 (1.75E + + +
02 (1.84E + = +
02 (1.61E + + F30 4.19E +
03 (5.95E +
03) 1.76E +
04 (4.90E +
04) - 3.97E +
03 (6.15E + = +
03 (3.05E + = +
03 (2.65E + = +
03 (3.31E + = +
03 (7.45E + = +
03 (6.91E +
03) - +/=/ - 5 / / / / / / / / / / / / / / The symbols “ +/=/ -” indicate that the corresponding algorithm performed significantly better ( + ), not significantly better or worse ( = ), or significantly worse ( − )compared to iLSHADE-RSP with p j = . α = .
05 significance level.Table 10: Means and standard deviations of FEVs of iLSHADE-RSP with di ff erent stability parameters on CEC 2017 test suite in 50 dimension iLSHADE-RSP α = . α = . α = . α = . α = . α = . α = . α = . = = = = = = = F2 6.18E-14 (1.07E-13) 4.21E-11 (3.00E-10) = = = = = = = F3 2.29E-13 (6.66E-14) 2.17E-13 (7.92E-14) = + = + + = + F4 5.93E +
01 (5.19E +
01) 7.70E +
01 (5.40E +
01) - 6.12E +
01 (5.24E +
01) - 6.44E +
01 (5.28E +
01) - 5.09E +
01 (4.67E + + +
01 (5.08E + = +
01 (5.30E + = +
01 (4.87E + = F5 1.50E +
01 (5.08E +
00) 1.55E +
01 (4.26E + = +
01 (5.91E + = +
01 (3.98E +
00) - 1.48E +
01 (4.27E + = +
01 (4.74E + = +
01 (3.66E + = +
01 (4.07E + = F6 4.24E-07 (5.09E-07) 4.60E-07 (6.69E-07) = = = = = = = F7 7.33E +
01 (8.29E +
00) 7.45E +
01 (8.12E + = +
01 (7.05E + = +
01 (9.08E + = +
01 (6.44E + = +
01 (7.06E + = +
01 (6.10E + = +
01 (5.93E + = F8 1.57E +
01 (4.72E +
00) 1.61E +
01 (4.59E + = +
01 (5.85E + = +
01 (5.91E + = +
01 (4.67E + = +
01 (4.37E + = +
01 (4.78E + = +
01 (3.97E + = F9 4.02E-14 (5.50E-14) 2.68E-14 (4.88E-14) = = = = = = = F10 4.19E +
03 (6.37E +
02) 4.66E +
03 (6.18E +
02) - 4.55E +
03 (6.74E +
02) - 4.34E +
03 (5.03E + = +
03 (6.17E + + +
03 (6.12E + + +
03 (5.35E + + +
03 (6.22E + + F11 1.84E +
01 (3.32E +
00) 1.91E +
01 (4.69E + = +
01 (4.16E +
00) - 1.83E +
01 (3.70E + = +
01 (3.70E + = +
01 (3.99E + = +
01 (4.17E + = +
01 (3.91E +
00) -F12 1.41E +
03 (3.66E +
02) 1.08E +
03 (3.01E + + +
03 (4.28E + + +
03 (4.26E + = +
03 (3.63E + = +
03 (3.89E + = +
03 (3.42E +
02) - 1.32E +
03 (3.84E + = F13 2.49E +
01 (1.69E +
01) 2.75E +
01 (2.14E + = +
01 (1.91E + = +
01 (1.63E + = +
01 (2.39E + = +
01 (1.85E + = +
01 (2.10E + = +
01 (1.87E +
01) -F14 2.42E +
01 (2.59E +
00) 2.37E +
01 (2.06E + = +
01 (1.95E + = +
01 (2.22E + = +
01 (1.95E + = +
01 (1.93E + = +
01 (2.21E + = +
01 (1.99E + = F15 1.80E +
01 (1.55E +
00) 1.86E +
01 (1.92E + = +
01 (1.73E + = +
01 (1.99E + = +
01 (1.93E +
00) - 1.97E +
01 (2.06E +
00) - 2.06E +
01 (2.30E +
00) - 2.06E +
01 (1.84E +
00) -F16 3.58E +
02 (1.49E +
02) 3.37E +
02 (1.49E + = +
02 (1.26E + = +
02 (1.28E + = +
02 (1.42E + = +
02 (1.14E + + +
02 (1.28E + = +
02 (9.99E + = F17 2.59E +
02 (1.06E +
02) 2.82E +
02 (1.39E + = +
02 (1.30E + = +
02 (7.98E + = +
02 (9.65E + = +
02 (8.00E + = +
02 (8.77E + + +
02 (7.52E + = F18 2.27E +
01 (1.32E +
00) 2.29E +
01 (1.49E + = +
01 (1.37E + = +
01 (1.40E + = +
01 (1.71E + = +
01 (1.40E + = +
01 (1.36E + = +
01 (1.04E + = F19 1.04E +
01 (2.15E +
00) 9.22E +
00 (2.37E + + +
00 (1.80E + = +
01 (1.96E + = +
01 (2.00E + = +
01 (2.67E + = +
01 (2.42E + = +
01 (2.72E + = F20 1.18E +
02 (3.65E +
01) 1.88E +
02 (1.17E +
02) - 1.53E +
02 (6.84E +
01) - 1.34E +
02 (7.27E + = +
02 (3.39E + + +
02 (3.91E + = +
02 (3.58E + = +
02 (1.78E + + F21 2.15E +
02 (5.23E +
00) 2.16E +
02 (4.81E + = +
02 (5.51E + = +
02 (4.97E + = +
02 (4.61E + = +
02 (4.24E + = +
02 (4.74E + = +
02 (4.27E + = F22 1.65E +
03 (2.15E +
03) 1.73E +
03 (2.34E + = +
03 (2.18E + = +
03 (2.28E + = +
03 (2.24E + = +
03 (2.00E + = +
03 (1.91E + = +
03 (1.83E + = F23 4.34E +
02 (6.62E +
00) 4.34E +
02 (7.70E + = +
02 (7.11E + = +
02 (8.02E + = +
02 (6.21E + + +
02 (6.96E + = +
02 (6.45E + + +
02 (7.17E + = F24 5.09E +
02 (3.67E +
00) 5.09E +
02 (4.60E + = +
02 (3.73E + = +
02 (4.74E + = +
02 (3.49E + = +
02 (3.75E + = +
02 (3.62E + = +
02 (3.63E + = F25 4.80E +
02 (1.20E +
00) 4.80E +
02 (1.83E +
00) - 4.80E +
02 (1.48E +
00) - 4.79E +
02 (9.18E-01) = +
02 (1.10E + = +
02 (6.66E-01) = +
02 (7.00E-01) - 4.80E +
02 (7.84E-01) -F26 1.12E +
03 (4.92E +
01) 1.14E +
03 (5.97E + = +
03 (5.17E + = +
03 (4.77E + = +
03 (5.25E + = +
03 (4.12E + = +
03 (6.13E + = +
03 (5.02E + = F27 4.79E +
02 (6.08E +
00) 5.00E +
02 (0.00E +
00) - 5.00E +
02 (0.00E +
00) - 4.86E +
02 (1.02E +
01) - 5.03E +
02 (1.17E +
01) - 5.08E +
02 (9.27E +
00) - 5.11E +
02 (1.19E +
01) - 5.10E +
02 (8.80E +
00) -F28 4.52E +
02 (2.44E-01) 4.99E +
02 (5.23E-01) - 4.97E +
02 (1.89E +
00) - 4.52E +
02 (5.05E-01) - 4.52E +
02 (3.25E-01) = +
02 (3.11E-01) = +
02 (4.40E-01) - 4.54E +
02 (6.70E +
00) -F29 3.15E +
02 (2.82E +
01) 3.03E +
02 (1.85E + + +
02 (1.35E + + +
02 (1.70E + + +
02 (1.95E +
01) - 3.70E +
02 (1.78E +
01) - 3.73E +
02 (1.70E +
01) - 3.71E +
02 (2.10E +
01) -F30 4.06E +
03 (4.03E +
03) 5.75E +
02 (1.42E + + +
03 (3.43E + + +
03 (2.93E + = +
05 (5.34E +
04) - 5.54E +
05 (5.52E +
04) - 5.89E +
05 (3.07E +
04) - 5.98E +
05 (2.60E +
04) - +/=/ - 4 / / / / / / / / / / / / / / The symbols “ +/=/ -” indicate that the corresponding algorithm performed significantly better ( + ), not significantly better or worse ( = ), or significantly worse ( − )compared to iLSHADE-RSP with α = . α = .
05 significance level.
L´evy α -stable distribution can be defined by the characteristicfunction, derived by L´evy [65] and Hall [66] as follows. log ø( t ) = (cid:40) − c α | t |{ − i β sign ( t ) tan πα } + i µ t if α (cid:44) − c | t |{ + i β sign ( t ) π log | t |} + i µ t if α = sign ( t ) = t >
00 if t = − t < α -stable distribution denoted by S α ( β, c , µ ) hasfour parameters: the stability parameter α ∈ (0 , β ∈ [ − , c ∈ (0 , ∞ ), andthe location parameter µ ∈ ( −∞ , ∞ ). The L´evy α -stable distri-bution has three special cases as follows.15 The Gaussian distribution: S ( β, c , µ ). • The Cauchy distribution: S (0 , c , µ ). • The L´evy distribution: S . (1 , c , µ ).The modified recombination operator, which calculates a per-turbation of a target vector with the L´evy α -stable distributioncan be defined as follows. u ji , g = x ji , g + F w · ( x jpbest , g − x ji , g ) if rand ji < CR or j = j rand + F · ( x jpr , g − ˜ x jpr , g ) S α (0 , . , x ji , g ) otherwise (21)We used Chambers-Mallows-Stuck method [67, 68, 69, 70]to generate L´evy α -stable random numbers as follows. Step 1.
Generate a random number from the uniformdistribution V ∈ [ − π , π ] and a random number fromthe exponential distribution W with mean 1. Step 2. If α (cid:44) X = S α,β · sin { α ( V + B α,β ) }{ cos ( V ) } α · (cid:34) cos { V − α ( V + B α,β ) } W (cid:35) (1 − α ) α (22)where B α,β = arctan ( β tan πα ) α (23) S α,β = (cid:110) + β tan (cid:18) πα (cid:19) (cid:111) α ) (24) Step 3. If α = X = π · (cid:18) π + β V (cid:19) tanV − π · β ln (cid:32) π WcosV π + β V (cid:33) (25) Step 4. If X ∼ S α ( β, ,
0) then: Y = (cid:40) cX + µ if α (cid:44) cX + π β clogc + µ if α = S α ( β, c , µ ).Table 10 shows the means and standard deviations of theFEVs of the proposed algorithm with di ff erent stability param-eters for the L´evy α -stable distribution with the jumping rate p j = .
2. For simplicity’s sake, we only considered symmet-ric distributions. The L´evy α -stable distribution has a short andwide PDF if the stability parameter is high, while a tall and nar-row PDF if the stability parameter is low. As can be seen fromthe table, the parameter values α = [1 , .
5] can lead to sat-isfactory results. In other words, the modified recombination operator with from the Cauchy distribution ( α =
1) to the L´evy α -stable distribution ( α = .
5) works best for the proposed al-gorithm. We chose the Cauchy distribution for the modified re-combination operator because generating Cauchy random num-bers is much easier than generating L´evy α -stable random num-bers.
7. Conclusion Di ff erential evolution is a popular evolutionary algorithmfor multidimensional real-valued functions. Like other evo-lutionary algorithms, it is important for di ff erential evolutionto establish a balance between exploration and exploitation tobe successful. Recently, a state-of-the-art DE variant calledLSHADE-RSP was proposed. Although it has shown excel-lent performance, the greediness of LSHADE-RSP may causepremature convergence in which all the candidate solutions fallinto the local optimum of an optimization problem and cannotescape from there.To mitigate the problem, we have devised a modified recom-bination operator for LSHADE-RSP, which perturbs a targetvector with the Cauchy distribution. Therefore, the modifiedrecombination operator can increase the probability of find-ing an optimal solution by adopting the long-tailed propertyof the Cauchy distribution. We called the resulting algorithmiLSHADE-RSP, which alternately applies the original and mod-ified recombination operators according to a jumping rate.The proposed algorithm has been tested on the CEC 2017test suite in 10, 30, 50, and 100 dimensions. Our experimen-tal results verify that the improved LSHADE-RSP significantlyoutperformed not only its predecessor LSHADE-RSP but alsoseveral cutting-edge DE variants in terms of convergence speedand solution accuracy. In particular, the proposed algorithmperforms better than the comparison algorithms on more com-plicated optimization problems, such as expanded multimodaltest functions and hybrid composition test functions, in all thedimensions.Possible directions for future work include 1) testing the pro-posed algorithm for constrained optimization problems; 2) test-ing the proposed algorithm for large-scale optimization prob-lems; 3) applying the proposed algorithm to various real-worldproblems. Acknowledgement
This work was supported by the National Research Foun-dation of Korea(NRF) grant funded by the Korea govern-ment(MSIT) (No. NRF-2017R1C1B2012752). The correspon-dence should be addressed to Dr. Chang Wook Ahn.
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