Real Parameter Single Objective Optimization using Self-Adaptive Differential Evolution Algorithm with more Strategies

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1 Real Parameter Sngle Objectve Optmzaton usng Self-Adaptve Dfferental Evoluton Algorthm wth more Strateges Janez Brest, Borko Boškovć, Aleš Zamuda, Iztok Fster Insttute of Computer Scence, Faculty of Electrcal Engneerng and Computer Scence, Unversty of Marbor, Smetanova ul. 17, 2000 Marbor, Slovena Emal: Efrén Mezura-Montes Departamento de Intelgenca Artfcal Unversdad Veracruzana Sebastán Camacho 5, Centro, Xalapa, Veracruz, 91000, MEXICO Emal: Abstract A new dfferental evoluton algorthm for sngle objectve optmzaton s presented n ths paper. The proposed algorthm uses a self-adaptaton mechansm for parameter control, dvdes ts populaton nto more subpopulatons, apples more DE strateges, promotes populaton dversty, and elmnates the ndvduals that are not changed durng some generatons. The expermental results obtaned by our algorthm on the benchmark consstng of 25 test functons wth dmensons D = 10, D = 30, and D = 50 as provded for the CEC 2013 competton and specal sesson on Real Parameter Sngle Objectve Optmzaton are presented. Key words: sngle objectve optmzaton, dfferental evoluton, self-adaptaton. I. INTRODUCTION Ths paper presents an algorthm for Real Parameter Sngle Objectve Optmzaton (RPSOO) [1]. Sngle objectve optmzaton algorthms serve as a bass of the more complex optmzaton algorthms such as mult-objectve optmzatons algorthms, nchng algorthms, constraned optmzaton algorthms, and so on [1]. The real-parameter optmzaton problem can be defned as follows. We need to fnd the varables of vector x = x 1,x 2,...,x D }, where D denotes the dmensonalty of a problem, such that ther correspondng objectve functon f( x) s optmzed (mnmzed or maxmzed). Domans of the varables are defned by ther lower and upper bounds x j,low,x j,upp, where j = 1,2,...,D. Therefore, ths knd of optmzaton s also named as bound-constraned optmzaton. Dfferental Evoluton (DE) [2], [3], [4] s a well-known evolutonary algorthm for optmzaton n contnuous and dscrete domans [5], [6]. Recently, n the lterature, t can be found that ths algorthm s hghly compettve, especally n areas, where real-parameters are needed [7], [8], [9], [10]. DE s a populaton based evolutonary algorthms. Its ntal populaton s generated at random wth a Unform dstrbuton over the entre search space. Then DE repeatedly apples mutaton, crossover, and selecton operators over the populaton to generate a next populaton. The evolutonary process stops when stoppng crtera s met and then the DE algorthm reports the best soluton found. Three control parameters are used n the orgnal DE, proposed by Storn and Prce n 1995: the amplfcaton factor of the dfference vector F, the crossover control parameter CR, and populaton sze NP. All three control parameters are fxed durng the optmzaton process n the orgnal DE algorthm. A tunng of the control parameters can be used to fnd out good values for them before the actual optmzaton process starts. A tunng process mght be tme consumng, especally, when t s performed as hand-tunng. In order to overcome the tunng problems and to mprove algorthm s performance, adaptve and self-adaptve mechansms were appled on the DE control parameters [4], [3]. In the specalzed lterature, a lot of mproved versons of DE can be found that are dealng wth the control parameters F and CR. Adaptve and/or self-adaptve technques were ntroduced n [11], [12]. Both self-adaptve technques, known also as algorthms jde and SaDE, have a great research nfluence onto DE-based algorthms. Moreover, many other mechansms have been proposed [13]. Some years ago only a few researches were related wth the thrd DE control parameter,.e., populaton sze. But more recently, works dealng wth populaton sze.e. changng populaton sze durng the evolutonary process are ncreasng [14]. They outlned that NP plays also an mportant role among control parameters n the DE. A self-adaptve jde algorthm was ntroduced n 2006[11]. jde-based algorthms were appled to solve large-scale sngle objectve optmzaton problems: CEC 2008 [15], CEC 2010 [16], CEC 2012 [17], large-scale contnuous optmzaton problems [18]. However, varous mechansms are proposed n order to changng the DE control parameters adaptvely or selfadaptvely [19], [20], [17]. Qn and Suganthan n [12] proposed Self-adaptve Dfferental Evoluton algorthm (SaDE), where the choce of a learnng strategy and the two control parameters F and CR are gradually self-adapted accordng to the learnng experence. In the specalzed lterature, usually, control parameters F and CR are changng n an adaptve or self-adaptve manner [3], [21], [4]. In ths paper, an evaluaton of our self-adaptve jdesoo algorthm s performed on the benchmark functons provded for the CEC2013 specal sesson on RPSOO [1],.e., a successor of specal Sesson on Real-Parameter Optmzaton that has been occurred on CEC2005 [22]. The DE-based algorthms presented at ths competton, lke DE [23] and L-SADE [24], were one of the more compettve among other algorthms n such prevous competton [25].

2 The structure of the paper s as follows. Secton II gves background for ths work, where an overvew of DE, a descrpton of the self-adaptve control parameters, and outlnes of our prevous algorthm are gven. Secton III presents a new varant of the algorthm, called jdesoo. In Secton IV expermental results of the jdesoo algorthm on sngle objectve benchmark functons are presented. Secton V concludes the paper wth some fnal remarks. II. BACKGROUND In ths secton, we gve some backgrounds that are needed n Secton III. A. The Dfferental Evoluton DE s a populaton-based algorthm. It uses mutaton, crossover and selecton operators to generate a next populaton from the current populaton. Let us brefly explan the orgnal DE algorthm [2]. The populaton n generaton G conssts of NP vectors: x (G) = (x (G),1,x(G),2,...,x(G),D ), = 1,2,...,NP, 1) Mutaton: A mutant vector v (G) s created by usng one of the DE mutaton strateges. The most popular s rand/1 strategy, whch can be descrbed as follows: v (G) = x (G) r 1 +F ( x (G) r 2 x (G) r 3 ). Indexes r 1,r 2, and r 3 denote random ntegers wthn the set 1,NP} and r 1 r 2 r 3. F s a mutaton scale factor wthn the range [0,2], usually less than 1. x best,g denotes the best vector n generaton G. The other useful DE strateges [3], [4] are: rand/1 : v,g = x r1,g +F( x r2,g x r3,g), best/1 : v,g = x best,g +F( x r1,g x r2,g), current to best/1 : v,g = x,g +F( x best,g x,g )+F( x r1,g x r2,g), best/2 : v,g = x best,g +F( x r1,g x r2,g)+f( x r3,g x r4,g), rand/2 : v,g = x r1,g+f( x r2,g x r3,g)+f( x r4,g x r5,g), wherethendexesr 1 r 5 representtherandomandmutually dfferent ntegers generated wthn the range 1,NP} and also dfferent from ndex. u (G) 2) Crossover: A crossover operator forms a tral vector as follows: u (G) v (G),j,j =, f rand(0,1) CR or j = j rand, x (G),j, otherwse, for = 1,2,...,NP and j = 1,2,...,D. The crossover parameter CR s wthn the range [0,1) and presents the probablty of creatng components for tral vector from a mutant vector. Index j rand 1,NP} s a randomly chosen nteger that s responsble for the tral vector contanng at least one component from the mutant vector. If the control parameters from the tral vector are out of bounds, the proposed solutons found n the lterature [2], [26] are: they are reflected nto bounds, set on bounds or used as they are (out of bounds). The selecton mechansm for a mnmzaton problem s defned as follows: x (G+1) u (G), f f( u (G) ) < f( x (G) ), =, otherwse. x (G) The DE has a greedy selecton, whle other evolutonary algorthms have a more sophstcate selecton operaton. Ftness values between the populaton vector and ts correspondng tral vector are compared n DE. Ths better vector wll survve and become a member of the populaton for the next generaton. B. Self-Adaptaton of F and CR The jde algorthm [11] uses a self-adaptng mechansm of two control parameters. Each ndvdual s extended wth control parameter values F and CR as follows: x (G) = (x (G),1,x(G),2,...,x(G),D,F(G),CR (G) ). New control parameters F (G+1) and CR (G+1) are calculated before the mutaton operator as follows [11]: F (G+1) F l +rand 1 F u, f rand 2 < τ 1, = F (G), otherwse, CR (G+1) rand 3, f rand 4 < τ 2, = CR (G), otherwse, where rand j, for j 1,2,3,4} are unform random values wthn the range [0,1]. In [11] parameters τ 1,τ 2,F l,f u are fxed to values 0.1, 0.1, 0.1, 0.9, respectvely. III. A NEW VARIANT OF THE ALGORITHM In ths secton our new algorthm jdesoo for solvng real-parameter sngle objectve optmzaton s presented. The jdesoo algorthm ncludes some good characterstcs and mechansms that are smlar to our prevous algorthms jdelsgo[17] and jdelscop [18] that were used for solvng large-scale global optmzaton (D was upto 1000). The pseudo-code of new jdesoo algorthm s presented n Algorthm 1. The dea of how to dvde a populaton nto sub-populaton s depcted n Fgure 1. The new proposed algorthm uses jde as a bass and updated t wth the followng features: Dvdng the whole populaton nto subpopulatons, Applyng a dfferent DE strategy n each subpopulaton,

3 1: pop... populaton} 2: NP... populaton sze} 3: mn... ndex of currently best ndvdual} 4: x... -th ndvdual of populaton} 5: MaxFEs... maxmum number of functon evaluatons} 6: rand(0, 1)... unformly dstrbuted random number [0, 1)} 7: Intalzaton() 8: NP = 30 9: tage = : 11: ** Generate unformly dstrbuted random populaton wthn search space **} 12: for (t = 0; t < MaxFEs; t = t+1) do 13: = t mod NP mod... modulo operaton} 14: 15: ** Perform one teraton of the jde usng one of three strateges. BIN crossover s used. **} 16: 17: f (rand(0,1) < 0.5 and < b 1 ) then 18: ** jdebest ** } 19: F l = : u =jdebest(mn, pop) 21: else 22: f ( < b 2 ) then 23: ** jdebn **} 1 NP ; F u = 1.0; CR l = 0.7; CR u = 0.9; 24: 25: F l = 0.1; F u = 1.0; CR l = 0.0; CR u = NP ; u =jdebn(, pop) 26: else 27: ** jdebnsec **} 28: 29: F l = 0.8; F u = 1.0; CR l = 0.8; CR u = NP ; r 1 = 0.7 NP+0.3 NP rand(0,1); 30: u =jdebnsec(, pop) 31: end f 32: end f 33: Bound check; f volatle then set on bound} 34: f( u) Functon evaluaton} 35: Selecton} 36: 37: ** apply agng at every tage teraton **} 38: f (t mod tage == (tage 1)) then 39: for every ndvdual test f ts age s greater tha 50 then rentalze s wth probablty 0.1} 40: end f 41: end for Algorthm 1: jdesoo algorthm Ensurng the populaton dversty durng generatons, Agng mechansm. The ntal populaton s generated at random wth a Unform dstrbuton between the lower x j,low and upper x j,upp bounds defned for each varable x j. If an ndvdual was not mproved for some generatons (we set the value for checkng ths every 1000 teratons t s approx. 33 generatons) then t s rentalzed wth probablty of 0.1. We set the value for agng to 50. Please note that every 1000 teratons the whole populaton s tested f agng crtera s fulfll. In ths paper we used three smple DE strateges DE/rand/1/best, DE/rand/1/bn, and, once agan DE/rand/1/bn, and we set b 1 = 0.1 NP and b 2 = 0.9 NP. The strateges used ther own self-adaptve control parameters F and CR. They have dfferent bound values of F l, F u, CR l, and CR u (see Algorthm 1), whch can be calculated outsde the teraton loop snce they are constant n the whole search process. The reasons of employng DE/rand/1/best, DE/rand/1/bn strateges were made based on our prevous experences and the suggeston from lterature. The best strategy has fast convergence ablty (explotaton), whle DE/rand/1/bn has good exploraton ablty. We used DE/rand/1/bn strategy two-tmes, one wth wde range of CR parameter, whle the second wth hgher value for CR and also F (CR l = 0.8, F l = 0.8).

4 TABLE I. PROPERTIES OF THE CEC 2013 BENCHMARK FUNCTIONS [1] No. Functons f = f (x ) 1 Sphere Functon Unmodal 2 Rotated Hgh Condtoned Ellptc Functon Functons 3 Rotated Bent Cgar Functon Rotated Dscus Functon Dfferent Powers Functon Rotated Rosenbrock s Functon Rotated Schaffers F7 Functon Rotated Ackley s Functon Rotated Weerstrass Functon Rotated Grewank s Functon Rastrgn s Functon -400 Basc 12 Rotated Rastrgn s Functon -300 Multmodal 13 Non-Contnuous Rotated Rastrgn s Functon -200 Functons 14 Schwefel s Functon Rotated Schwefel s Functon Rotated Katsuura Functon Lunacek B Rastrgn Functon Rotated Lunacek B Rastrgn Functon Expanded Grewank s plus Rosenbrock s Functon Expanded Scaffer s F6 Functon Composton Functon 1 (n=5, Rotated) Composton Functon 2 (n=3, Unrotated) Composton Functon 3 (n=3, Rotated) 900 Composton 24 Composton Functon 4 (n=3, Rotated) 1000 Functons 25 Composton Functon 5 (n=3, Rotated) Composton Functon 6 (n=5, Rotated) Composton Functon 7 (n=5, Rotated) Composton Functon 8 (n=5, Rotated) 1400 Search Range: [-100,100] D x 1. x b x b+1. x c x c+1. x NP strategy1 strategy2 strategy3 Fg. 1. Populaton s dvded nto more sub-populaton. Each subpopulaton uses ts DE strategy. A. Benchmark Functons IV. RESULTS The jdesoo algorthm was tested on a set of 28 benchmark functons [1]. The benchmark functons are scalable. The dmensons of benchmark functons were D = 10, D = 30, and D = 50, respectvely, and 51 runs of algorthm were needed for each functon. The optmal values are known for all benchmark functons. We used our algorthm as black-box optmzer as requred for ths specal sesson. The general features of these functons are presented n Table I. B. Expermental Results Notce, that n ths competton, error values smaller than 10 8 are taken as zero. In the experments, parameters of the jdesoo algorthm were set as follows: F was self-adaptve,

5 TABLE II. EXPERIMENTAL RESULTS WITH DIMENSION D = 10. Func. Best Worst Medan Mean Std e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e+01 CR was self-adaptve, NP was fxed durng the optmzaton process, NP nt = 30, agng was set to 50. The obtaned results (error values f( x) f( x )) are presented n Tables II, III, IV for dmensons 10, 30, and 50, respectvely. PC Confgure: System: GNU/Lnux, CPU: 2.5 GHz, RAM: 4 GB, Language: C/C++, Algorthm: jdesoo, Compler: GNU Compler (g++). TABLE V. COMPUTATIONAL COMPLEXITY D T0 T1 T2 (T2 T1)/T s s s s s s s The algorthm complexty s presented n Table V as requred n [1]. V. CONCLUSIONS The jdesoo algorthm was presented n ths paper. The performance of the algorthm was evaluated on the set of benchmark functons provded for CEC 2013 specal sesson on real-parameter sngle objectve optmzaton. The algorthm uses a relatvely small populaton sze that remaned fxed for all 28 benchmark functons and dmensons (D = 10, D = 30, D = 50). The performance of ths algorthm aganst other algorthm wll be performed by organsers of ths specal sesson. ACKNOWLEDGMENT Ths work was supported n part by the Slovenan Research Agency under program P REFERENCES [1] J. J. Lang, B.-Y. Qu, P. N. Suganthan, and A. G. Hernández- Díaz, Problem Defntons and Evaluaton Crtera for the CEC 2013 Specal Sesson and Competton on Real-Parameter Optmzaton, Computatonal Intellgence Laboratory, Zhengzhou Unversty, Zhengzhou Chna and Techncal Report, Nanyang Technologcal Unversty, Sngapore, Tech. Rep , [Onlne]. Avalable:

6 TABLE III. EXPERIMENTAL RESULTS WITH DIMENSION D = 30. Func. Best Worst Medan Mean Std e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-04 [2] R. Storn and K. Prce, Dfferental Evoluton A Smple and Effcent Heurstc for Global Optmzaton over Contnuous Spaces, Journal of Global Optmzaton, vol. 11, pp , [3] S. Das and P. Suganthan, Dfferental evoluton: A survey of the stateof-the-art, IEEE Transactons on Evolutonary Computaton, vol. 15, no. 1, pp , [4] F. Ner and V. Trronen, Recent advances n dfferental evoluton: a survey and expermental analyss, Artfcal Intellgence Revew, vol. 33, no. 1 2, pp , [5] I. Fster and J. Brest, Usng dfferental evoluton for the graph colorng, n IEEE SSCI2011 symposum seres on computatonal ntellgence. Pscataway: IEEE, 2011, pp [6] I. Fster, M. Mernk, and B. Flpč, Graph 3-colorng wth a hybrd self-adaptve evolutonary algorthm, Computatonal Optmzaton and Applcatons, vol. 54, pp , [7] S. Amrabad, S. Kabr, R. Vakl, D. Iranshah, and M. R. Rahmpour, Dfferental Evoluton Strategy for Optmzaton of Hydrogen Producton va Couplng of Methylcyclohexane Dehydrogenaton Reacton and Methanol Synthess Process n a Thermally Coupled Double Membrane Reactor, Industral & Engneerng Chemstry Research, vol. 52, no. 4, pp , JAN [8] K. Opara and J. Arabas, Decomposton and Metaoptmzaton of Mutaton Operator n Dfferental Evoluton, n Swarm and Evolutonary Computaton, ser. Lecture Notes n Computer Scence, Rutkowsk, L and Korytkowsk, M and Scherer, R and Tadeusewcz, R and Zadeh, LA and Zurada, JM, Ed., vol. 7269, 2012, Proceedngs Paper, pp [9] R. Polakova and J. Tvrdk, A Comparson of Two Adaptaton Approaches n Dfferental Evoluton, n Swarm and Evolutonary Computaton, ser. Lecture Notes n Computer Scence, Rutkowsk, L and Korytkowsk, M and Scherer, R and Tadeusewcz, R and Zadeh, LA and Zurada, JM, Ed., vol. 7269, 2012, Proceedngs Paper, pp [10] S. M. Elsayed, R. A. Sarker, and D. L. Essam, An Improved Self- Adaptve Dfferental Evoluton Algorthm for Optmzaton Problems, IEEE Transactons on Industral Informatcs, vol. 9, no. 1, pp , FEB [11] J. Brest, S. Grener, B. Boškovć, M. Mernk, and V. Žumer, Self- Adaptng Control Parameters n Dfferental Evoluton: A Comparatve Study on Numercal Benchmark Problems, IEEE Transactons on Evolutonary Computaton, vol. 10, no. 6, pp , [12] A. K. Qn, V. L. Huang, and P. N. Suganthan, Dfferental evoluton algorthm wth strategy adaptaton for global numercal optmzaton, IEEE Transactons on Evolutonary Computaton, vol. 13, no. 2, pp , [13] M. Weber, F. Ner, and V. Trronen, Shuffle or update parallel dfferental evoluton for large-scale optmzaton, Soft Computng, vol. 15, pp , [Onlne]. Avalable: /s [14] I. Fajfar, T. Tuma, J. Puhan, J. Olensek, and A. Burmen, Towards Smaller Populatons n Dfferental Evoluton, INFORMACIJE MIDEM-JOURNAL OF MICROELECTRONICS ELECTRONIC COM- PONENTS AND MATERIALS, vol. 42, no. 3, pp , [15] J. Brest, A. Zamuda, B. Boškovć, M. S. Maučec, and V. Žumer, Hgh-dmensonal Real-parameter Optmzaton Usng Self-adaptve Dfferental Evoluton Algorthm wth Populaton Sze Reducton, n 2008 IEEE World Congress on Computatonal Intellgence. IEEE Press, 2008, pp [16] J. Brest, A. Zamuda, B. Boškovć, I. Fster, and M. S. Maučec, Large

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