Cooperative Micro Differential Evolution for High Dimensional Problems

Transcription

1 Cooperatve Mcro Dfferental Evoluton for Hgh Dmensonal Problems Konstantnos E. Parsopoulos Department of Mathematcs Unversty of Patras GR Patras, Greece ABSTRACT Hgh dmensonal optmzaton problems appear very often n demandng applcatons. Although evolutonary algorthms consttute a valuable tool for solvng such problems, ther standard varants exhbt deteroratng performance as dmenson ncreases. In such cases, cooperatve approaches have proved to be very useful, snce they dvde the computatonal burden to a number of cooperatng subpopulatons. In contrast, Mcro evolutonary approaches consttute lght versons of the orgnal evolutonary algorthms that employ very small populatons of just a few ndvduals to address optmzaton problems. Unfortunately, ths property s usually accompaned by lmted effcency and proneness to get stuck n local mnma. In the present work, an approach that combnes the basc propertes of cooperaton and Mcro-evolutonary algorthms s presented for the Dfferental Evoluton algorthm. The proposed Cooperatve Mcro Dfferental Evoluton approach employs small cooperatve subpopulatons to detect subcomponents of the orgnal problem soluton concurrently. The subcomponents are combned through cooperaton of subpopulatons to buld complete solutons of the problem. The proposed approach s llustrated on hgh-dmensonal nstances of fve wdely used test problems wth very promsng results. Comparsons wth the standard Dfferental Evoluton algorthm are also reported and ther statstcal sgnfcance s analyzed. Categores and Subject Descrptors G.1.6 [Optmzaton]: Global optmzaton,unconstraned optmzaton; G.3 [Probablty and Statstcs]: Probablstc algorthms General Terms Algorthms, Performance, Expermentaton Keywords Dfferental Evoluton, Mcro Dfferental Evoluton, Evolu- Permsson to make dgtal or hard copes of all or part of ths work for personal or classroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commercal advantage and that copes bear ths notce and the full ctaton on the frst page. To copy otherwse, to republsh, to post on servers or to redstrbute to lsts, requres pror specfc permsson and/or a fee. GECCO 09, July 8 12, 2009, Montréal Québec, Canada. Copyrght 2009 ACM /09/07...$5.00. tonary Algorthms, Cooperatve Algorthms, Mcro Evolutonary Algorthms 1. INTRODUCTION Modern applcatons often nvolve the soluton of complex hgh dmensonal optmzaton problems. Evolutonary algorthms have been used wdely for solvng such problems, especally n cases where the underlyng objectve functons lack nce mathematcal propertes such as contnuty and dfferentablty. Usually, the ncreased dmensonalty of the problem poses obstacles on the employed algorthm, reducng ts performance sgnfcantly. Ths defcency s also known as the curse of dmensonalty and ts allevaton consttutes a subject of ongong research. Cooperatve Evolutonary Algorthms (CEAs) have proved to be a valuable tool n cases where the standard evolutonary algorthms fal [9]. CEAs consst of a number of cooperatve populatons that attack low dmensonal subcomponents of the orgnal problem and evolve them concurrently. Cooperaton among subpopulatons s responsble for brngng together ther dscoveres and buld complete solutons of the orgnal problem. Varous cooperatve approaches have been proposed and analyzed n lterature [2, 8, 9, 13]. A typcal example s the Cooperatve Coevolutonary Genetc Algorthm (CCGA) of Potter and De Jong [8], whch s based on Genetc Algorthms. Evoluton Strateges and Partcle Swarm Optmzaton have also been used as the basc algorthmc elements of cooperatve approaches [4, 14, 16]. Recently, two cooperatve approaches based on the Dfferental Evoluton (DE) algorthm were proposed [7, 12]. Extensve expermentaton has revealed also several defcences of cooperatve schemes, such as the deteroratng performance n problems wth correlated coordnate drectons and the ntroducton of new local mnma [8]. In contrast to CEAs, Mcro Evolutonary Algorthms (Mcro EAs) are nstances of the standard evolutonary algorthms wth very small populaton sze and ablty to handle smple ftness functons. Although Mcro EAs were prmarly used for very smple problems and educatonal purposes, several attempts have been made to use them n demandng applcatons. For example, Mcro Genetc Algorthms (Mcro GAs), also called Tny GAs, have been studed n mage processng problems [6]. Recently, Mcro Partcle Swarm Optmzaton was proposed for tacklng hgh dmensonal optmzaton problems [5], and a specal verson of Mcro Dfferental Evoluton (Mcro DE) wth opposton based operators was used for mage thresholdng [11]. The small populaton sze of Mcro EAs lmts ther ex- 531

2 ploraton capabltes, especally n complex envronments. Ths s usually caused due to the rapd convergence of the populaton to the most promsng detected solutons, whch decreases populaton dversty n early teratons and, consequently, deterorates effcency. As a countermeasure, dversty preservng schemes are usually ncorporated n the algorthm, along wth proper technques that prevent convergence to the same soluton [5]. The present work combnes the man concepts of CEAs and Mcro EAs to produce a Cooperatve Mcro Dfferental Evoluton (COMDE) algorthm. To the best of the author s knowledge, ths s the frst varant of DE that combnes these two approaches. The proposed scheme ams at solvng hgh dmensonal problems more effcently than the standard DE algorthm. To acheve ths, hgh dmensonal canddate solutons are dvded nto low dmensonal subcomponents, and each one s tackled wth a small, low dmensonal subpopulaton. Informaton sharng among subpopulatons allows the constructon of complete solutons for the evaluaton of each ndvdual wth the orgnal objectve functon. COMDE s tested aganst the standard DE algorthm on hgh dmensonal nstances of fve wdely used test problems from the relatve lterature. The remanng of the paper s organzed as follows: Secton 2 descrbes the DE algorthm, whle Secton 3 ntroduces the COMDE approach. Expermental results are reported and dscussed n Secton 4, and the paper concludes n Secton DIFFERENTIAL EVOLUTION The DE algorthm was developed by Storn and Prce [10, 15] as a populaton based stochastc optmzaton algorthm for numercal optmzaton problems. DE utlzes a populaton: P = {x 1, x 2,..., x N}, of N ndvduals to probe the search space. The populaton s ntalzed randomly n the search space, usually followng a unform dstrbuton. Each ndvdual s an n dmensonal vector: x = (x 1, x 2,..., x n), = 1, 2,..., N, and serves as a canddate soluton of the problem at hand. The populaton s evolved by applyng two operators, namely mutaton and recombnaton, whch produce new canddate solutons. Then, the old and the new populaton are merged, and selecton takes place to construct a new populaton that conssts of the N best ndvduals. These operators are appled teratvely untl a termnaton condton s met. The mutaton operator produces a new vector, v, for each ndvdual, x, = 1,2,..., N, by combnng some of the rest ndvduals of the populaton. Dfferent operators have been proposed for ths task, wth the followng consttutng the most common ones: v (t + 1)=x g(t) + F x r1 (t) x r2 (t), (1) v (t + 1)=x r1 (t) + F x r2 (t) x r3 (t), (2) v (t + 1)=x (t) + F x g(t) x (t) + x r1 (t) x r2 (t), (3) v (t + 1)=x g(t) + F x r1 (t) x r2 (t) + x r3 (t) x r4 (t), (4) v (t + 1)=x r1 (t) + F x r2 (t) x r3 (t) + x r4 (t) x r5 (t),(5) where t denotes the teraton counter; F s a fxed user defned parameter; g denotes the ndex of the best ndvdual n the populaton,.e., the one wth the smallest functon value; and r {1, 2,..., N}, = 1,2,...,5, are mutually dfferent randomly selected ndces that dffer also from the ndex. Thus, n order to be able to apply all mutaton operators, t must hold that N > 5. All vector operatons n Eqs. (1) (5) are performed componentwse. The fve operators wll be henceforth denoted as OP1 OP5, respectvely. After mutaton, a recombnaton operator s appled on the generated vectors, v, producng for each one a tral vector: u = (u 1, u 2,..., u n), = 1, 2,..., N, whch s defned as follows: j vj(t + 1), u j(t + 1) = x j(t), f R j CR or j = RI(), f R j > CR and j RI(), where j = 1,2,..., n; R j s the j th evaluaton of a unform random number generator n the range [0, 1]; CR [0, 1] s a user defned crossover constant; and RI() s an ndex randomly selected from the set {1, 2,..., n}. Fnally, n the selecton phase, the produced tral vectors, u, are compared aganst the correspondng ndvduals, x, and the best among them comprse the populaton n the next generaton,.e.: j u(t + 1), f f(u x (t + 1) = (t + 1)) < f(x (t)), x (t), otherwse, where f(x) s the objectve functon under consderaton. 3. THE PROPOSED APPROACH Mcro DE has the same structure and operatons wth standard DE. The only dfference s the populaton sze, whch s typcally very small. Thus, although t s recommended to use populatons of sze up to N = 10n [15], where n s the problem dmenson, Mcro DE uses the smallest possble number of ndvduals. Takng nto consderaton the restrcton, N > 5, that permts the applcaton of all mutaton operators, a populaton sze, N = 6, can be consdered a reasonable choce for Mcro DE. Moreover, Mcro DE s expected to converge rapdly due to the small number of ndvduals. Usually, the rato, n/n, s ndcatve of the dffculty met by an algorthm on a gven problem. Small values of ths rato (less than 1) correspond to populaton wth sze larger than ts dmenson. On the other hand, values hgher than 1 correspond to problem dmenson hgher than populaton sze. Emprcal evdence suggest that n most cases the hgher the rato s, the harder the problem becomes for the algorthm. Therefore, Mcro DE can be consdered as a promsng approach n rather low dmensonal problems. The aforementoned defcency can be addressed through the proposed COMDE approach, a cooperatve scheme for Mcro DE. To put t formally, let, n 1, n 2,..., n K, be K postve ntegers such that: n = KX n k, k=1 where n s the dmenson of the orgnal problem. Then, a canddate soluton vector of the orgnal problem can be dvded nto K subcomponents, each one addressed by a 532

3 Table 1: Pseudocode of the proposed COMDE approach. Input: K (number of subpopulatons); N (szes of subpopulatons); n (dmensons of subpopulatons); = 1, 2,..., K; M (buffer vector); f (objectve functon) Step 1. Intalze subpopulatons randomly wthn ther search spaces (subspaces of the orgnal one). Step 2. Intalze buffer vector, M, usng a randomly selected ndvdual from each subpopulaton. Step 3. Whle (termnaton condton not met) Step 4. Do (k = 1,..., K) Step 5. Do ( = 1,..., N k ) Step 6. Update the ndvdual x [k] wth the standard DE operatons. Step 7. Evaluate x [k] usng Eq. (6) and the buffer M. Step 8. Update the best poston x [k] g of the populaton P k. Step 9. If f < f(m) Then x [k] Step 10. Copy x [k] n the proper poston of the buffer M. Step 11. End If Step 12. End Do Step 13. End Do Step 14. End Whle Step 15. Prnt buffer M and f(m). Table 2: Dmenson and range for each test problem. Problem Dmenson (n) Range TP1 300, 600, 900, 1200 [ 100, 100] n TP2 300, 600, 900, 1200 [ 30, 30] n TP3 300, 600, 900, 1200 [ 5.12, 5.12] n TP4 300, 600, 900, 1200 [ 600, 600] n TP5 300, 600, 900, 1200 [ 20, 30] n Table 4: COMDE subpopulaton parameters. Parameter Descrpton Value N k subpopulaton sze 6 n k subpopulaton dmenson 5 t max maxmum teratons 10 3 F DE parameter 0.5 CR DE parameter 0.7 Table 3: The total number of ndvduals and subpopulatons of 6 ndvduals per problem dmenson. Problem Total number Number of Dmenson of ndvduals subpopulatons dfferent subpopulaton, P, of sze, N, and dmenson, n, = 1, 2,..., K. Thus, each subpopulaton s assgned the mnmzaton of ts correspondng subcomponent, whch has strctly smaller dmenson than n. The subpopulatons work n the same manner as for the orgnal DE algorthm descrbed n Secton 2. However, an apparent ssue arses regardng the evaluaton of ndvduals wth the objectve functon due to ther dfferent dmenson. Ths problem can be addressed by usng an nformaton sharng mechansm n the form of a common memory buffer for all subpopulatons, where they depost ther best ndvduals. Ths buffer s also called context vector and t s defned as an n dmensonal vector, M = (m 1, m 2,..., m n), where each subpopulaton deposts ts contrbuton. Hence, f: s [k] =, s [k] 1, s[k] 2,..., s[k] n k s the n k dmensonal vector (wth n k < n) contrbuted by the k th subpopulaton, P k, k = 1, 2,..., K, then the buffer vector s defned as: M = s [1] 1,..., s[1] [2] n {z 1, s1,..., } s[2] n {z 2 } s [1] of P 1 s [2] of P 2. 1,..., s [K] n {z K } [K],..., s s [K] of P K Then, the th ndvdual of the j th subpopulaton:, x [j] = x [j] 1, x[j] 2,..., x[j],n j s evaluated usng the buffer vector, M, by substtutng the components that correspond to the contrbuton of the j th populaton wth the actual components of x [j], whle the rest components of the buffer reman unaffected,.e.: f = f x [j] M [j], (6) where,, M [j] = s [1] 1,..., s[1] n 1,..., x [j] 1,..., x[j],n j,..., s [K] 1,..., s [K] n K {z } ndvdual x [j] = 1,2,..., N j; j = 1, 2,..., K. A straghtforward choce for the contrbuton of each subpopulaton s ts overall best poston,.e., s [k] = x [k] g, whch results n a buffer that contans all best postons of the sub- 533

4 Table 5: Results for TP1. Oper. Dm. K Mean StD OP e e e e e e e e e e e e e e e e + 04 OP e e e e e e e e e e e e e e e e + 04 OP e e e e e e e e e e e e e e e e + 04 OP e e e e e e e e e e e e e e e e + 04 OP e e e e e e e e e e e e e e e e + 04 populatons:. M = x [1] g1,..., x[1] g,n 1, x [2] g1,..., x[2] g,n 2,..., x [K] g1,..., x[k] g,n K {z } {z } {z } x [1] g of P 1 x [2] g of P 2 x [K] g of P K Therefore, by defnton, the buffer consttutes the best poston ever attaned by the algorthm,.e., t s the best obtaned approxmaton of the actual mnmzer. Instead of the best from each subpopulaton, a randomly selected ndvdual could be alternatvely used. Ths scheme would result n a COMDE approach wth slower convergence but hgher dversty. Clearly, the type of buffer update can affect the convergence propertes of the algorthm substantally. Also, n some approaches, a restart of the subpopulatons s performed to avod the rapd dversty loss caused by ther small szes. COMDE does not use restart because DE s a greedy algorthm that stores ts best postons n Table 6: Hypothess testng for TP1. Oper. Dm. Improvement p value Decson OP % e 10 Reject % e 11 Reject % e 11 Reject % e 11 Reject OP % e 11 Reject % e 11 Reject % e 11 Reject % e 11 Reject OP % e 06 Reject % e 11 Reject % e 11 Reject % e 11 Reject OP % e 11 Reject % e 11 Reject % e 11 Reject % e 11 Reject OP % e 11 Reject % e 11 Reject % e 11 Reject % e 11 Reject the populaton; thus, a populaton restart would destroy all nformaton obtaned n prevous teratons. The COMDE algorthm s reported n pseudocode n Table 1, and t s llustrated on hgh dmensonal nstances of wdely used benchmark problems n the followng Secton. 4. EXPERIMENTAL RESULTS COMDE was appled on hgh dmensonal nstances of the followng wdely used test problems: Test Problem 1 (TP1 - Sphere) [15]. Ths n dmensonal problem s defned as: nx f(x) = x 2. (7) It has a global mnmzer, x = (0,..., 0), wth f(x ) = 0. Test Problem 2 (TP2 - Generalzed Rosenbrock) [15]. Ths n dmensonal problem s defned as: =1 =1 n 1 X f(x) = 100 `x +1 x (x 1) 2. (8) It has a global mnmzer, x = (1,..., 1), wth f(x ) = 0. Test Problem 3 (TP3 - Rastrgn) [15]. Ths n dmensonal problem s defned as: f(x) = 10n + nx `x2 10 cos(2πx ). (9) =1 It has a global mnmzer, x = (0,..., 0), wth f(x ) = 0. Test Problem 4 (TP4 - Grewank) [15]. Ths n dmensonal problem s defned as: nx x 2 n f(x) = 4000 Y «x cos + 1. (10) =1 =1 534

5 Table 7: Results for TP2. Oper. Dm. K Mean StD OP e e e e e e e e e e e e e e e e + 07 OP e e e e e e e e e e e e e e e e + 07 OP e e e e e e e e e e e e e e e e + 07 OP e e e e e e e e e e e e e e e e + 07 OP e e e e e e e e e e e e e e e e + 07 It has a global mnmzer, x = (0,...,0), wth f(x ) = 0. Test Problem 5 (TP5 - Ackley) [1]. Ths n dmensonal problem s defned as: 0 v 1 u f(x) = 20 + exp(1) t 1 nx x 2 A n =1! 1 nx exp cos(2πx ). (11) n =1 It has a global mnmzer, x = (0,...,0), wth f(x ) = 0. Each test problem was consdered for dmensons, n = 300, 600, 900, and The correspondng n dmensonal search spaces are reported n Table 2. COMDE dvdes canddate soluton vectors n K, 5 dmensonal subcomponents and uses a subpopulaton of 6 ndvduals on each. Hence, usng Table 8: Hypothess testng for TP2. Oper. Dm. Improvement p value Decson OP % e 04 Reject % e 11 Reject % e 11 Reject % e 11 Reject OP % e 11 Reject % e 11 Reject % e 11 Reject % e 11 Reject OP % e 01 Accept % e 11 Reject % e 11 Reject % e 11 Reject OP % e 11 Reject % e 11 Reject % e 11 Reject % e 11 Reject OP % e 11 Reject % e 11 Reject % e 11 Reject % e 11 Reject the notaton of Secton 3, t follows that n k = 5 and N k = 6 for all k = 1,2,..., K, and K = n/5; therefore, the rato n k /N k remans smaller than 1 for all subpopulatons. For example, n the 600 dmensonal case, the problem s dvded n 600/5 = 120 subcomponents; thus, 120 subpopulatons are used, each consstng of sx 5 dmensonal ndvduals. The number of subpopulatons, as well as the total number of ndvduals used per problem dmenson are reported n Table 3. All DE operators, OP1 OP5, defned by Eqs. (1) (5), were consdered n the experments. A maxmum number of 1000 teratons was allowed for each subpopulaton n all cases. We must notce that teratons are performed concurrently for all subpopulatons. Thus, COMDE has sgnfcant parallelzaton capabltes, snce each subpopulaton can be assgned to a dfferent processor, whle the buffer update can be ether synchronous or asynchronous. Regardng the DE parameters, the common settng, F = 0.5, CR = 0.7, was used for all subpopulatons. All parameter values are summarzed n Table 4. We must notce that parameters were arbtrarly set to reasonable values wthout any further fne tunng that could enhance the algorthm s performance. For each test problem, operator, and dmenson, 30 ndependent experments were performed. At each experment, the best soluton acheved after 1000 teratons was recorded along wth ts functon value. The obtaned functon values were analyzed statstcally, n terms of ther mean value and standard devaton averaged over the 30 experments. For comparson purposes, the experments were repeated also for the standard DE, usng a sngle populaton wth the same parameters as COMDE. In order to have far comparsons, the populaton sze of standard DE was set equal to the total number of ndvduals employed by all subpopulatons n COMDE per case. Ths number s reported n the second column of Table 3. DE was allowed to perform the same number of teratons as COMDE, and ts performance was also statstcally analyzed. 535

6 Table 9: Results for TP3. Oper. Dm. K Mean StD OP e e e e e e e e e e e e e e e e + 02 OP e e e e e e e e e e e e e e e e + 02 OP e e e e e e e e e e e e e e e e + 02 OP e e e e e e e e e e e e e e e e + 02 OP e e e e e e e e e e e e e e e e + 02 In addton, hypothess tests were conducted to ensure statstcal sgnfcance of the derved conclusons. Therefore, for each test problem, COMDE was compared aganst DE usng the nonparametrc Wlcoxon rank sum test [3] wth the null hypothess that the two samples of functon values, obtaned by COMDE and DE n 30 experments, come from dentcal contnuous dstrbutons wth equal medans, aganst the alternatve of dfferent medans. The decson for acceptance or rejecton of the null hypothess n a 95% level of sgnfcance, as well as the correspondng p value, were recorded for each test problem. Besdes that, the performance mprovement percentage between COMDE and DE, n terms of the obtaned soluton values averaged over the 30 experments, was computed for all cases. All results and statstcal tests are reported n Tables More specfcally, for each test problem, operator, and dmenson, the mean value and standard devaton of the Table 10: Hypothess testng for TP3. Oper. Dm. Improvement p value Decson OP % e 02 Accept % e 11 Reject % e 11 Reject % e 11 Reject OP % e 11 Reject % e 11 Reject % e 11 Reject % e 11 Reject OP % e 03 Reject % e 11 Reject % e 11 Reject % e 11 Reject OP % e 11 Reject % e 11 Reject % e 11 Reject % e 11 Reject OP % e 11 Reject % e 11 Reject % e 11 Reject % e 11 Reject obtaned soluton values after 1000 teratons, n the 30 ndependent experments, are reported both for the COMDE (table rows wth K > 1) and the standard DE (table rows wth K = 1). Also, the p values and decson of hypothess testng are reported per problem and operator, along wth the mprovement attaned by COMDE, wth negatve values denotng worse performance of COMDE aganst DE. As a frst observaton, we can see that COMDE has sgnfcantly mproved performance for all operators. Especally for OP4 and OP5, there was a tremendous mprovement over 90% n three test problems (TP1, TP2, and TP4), whle, for the rest problems, ther mprovement remaned the hghest among all operators. Remarkable mprovement was observed also for the OP2 operator. If we take a closer look at the aforementoned three operators, defned n Eqs. (2), (4), and (5), we wll observe that they employ the hghest number of randomly selected ndvduals from the populaton. Indeed, OP2 conssts of one dfference vector that combnes three randomly selected ndvduals, n contrast to the smlar operator OP1, whch also has one dfference vector but wth two randomly selected ndvduals. Smlarly, both OP4 and OP5 consst of two dfference vectors, nvolvng four and fve randomly selected ndvduals, respectvely, whle OP3, whch also uses two dfference vectors, nvolves only two randomly selected ndvduals. Ths ndcates the ncreasngly benefcal effect of COMDE when the number of nvolved randomly selected ndvduals n the operators s ncreased. As a second observaton, we see that the two most benefted operators, OP4 and OP5, of COMDE also exhbt the hghest overall performance n all test problems and dmensons, n terms of the reported mean values. Indeed, OP4 s the best for all dmensons n TP2 and TP4, whle the same holds for OP5 n TP3 and TP5. Only n TP1, OP5 was the best among all operators for the 300 dmensonal case, whle OP4 was the best for all other dmensons. Ths verfes the nstrumental contrbuton of the COMDE approach to these operators. 536

7 Table 11: Results for TP4. Oper. Dm. K Mean StD OP e e e e e e e e e e e e e e e e + 02 OP e e e e e e e e e e e e e e e e + 02 OP e e e e e e e e e e e e e e e e + 02 OP e e e e e e e e e e e e e e e e + 02 OP e e e e e e e e e e e e e e e e + 02 Table 12: Hypothess testng for TP4. Oper. Dm. Improvement p value Decson OP % e 10 Reject % e 11 Reject % e 11 Reject % e 11 Reject OP % e 11 Reject % e 11 Reject % e 11 Reject % e 11 Reject OP % e 08 Reject % e 11 Reject % e 11 Reject % e 11 Reject OP % e 11 Reject % e 11 Reject % e 11 Reject % e 11 Reject OP % e 11 Reject % e 11 Reject % e 11 Reject % e 11 Reject devatons among all operators for both DE and COMDE. For the rest dmensons of TP2, COMDE under OP3 was the most robust. However, ths s not the case for TP3 and TP5, wth OP3 achevng n many cases the best standard devatons among all COMDE operators but not overall. Summarzng the results, t s shown expermentally that COMDE can be a very promsng approach, producng for all operators superor results than standard DE. There are only two exceptons to ths observaton, namely the 300 dmensonal cases of OP3 n TP2 and OP1 n TP3. Both these operators nvolve the best ndvdual of the populaton, whch seems to be benefcal for the specfc problems, although ths result s not statstcally sgnfcant n a 95% level. Nevertheless, n hgher dmensonal cases even ths advantage was surpassed by COMDE, whch has shown the potental to occupy a salent place among the alternatves for hgh dmensonal problems. We shall also note that the null hypothess was rejected n all but two cases, namely the 300 dmensonal cases of OP3 n TP2 and OP1 n TP3. These two exceptons are both characterzed by a slght worsenng of the COMDE performance compared to the correspondng standard DE wth respect to the reported mean values, although ths s not accompaned by statstcal sgnfcance. Regardng ther robustness, as t s expressed by the reported standard devatons, OP3 can be dstngushed as the most robust operator, especally for hgher dmensons. As we can see n Table 5 for TP1, the COMDE verson of OP3 has the smallest standard devatons among all operators of both DE and COMDE for n 600. In the case of n = 300, ts standard DE counterpart was the most robust, whle for COMDE, OP5 had the smallest standard devaton. The same holds for TP2, as reported n Table 7, except the case of n = 300, where OP3 exhbted the smallest standard 5. CONCLUSIONS COMDE, an approach that combnes cooperatve wth Mcro DE was ntroduced and expermentally assessed on wdely used test problems for dmensons rangng from 300 up to The proposed approach was also compared to the standard DE algorthm under the fve most common DE operators. Prelmnary results are very encouragng, exhbtng sgnfcant mprovement n performance of all operators, especally as problem dmenson ncreases. Further research s needed to fully reveal the potental of COMDE and dentfy possble drawbacks n cases where typcal cooperatve approaches meet obstacles, such as the case of problems wth hghly correlated coordnate drectons. Nevertheless, COMDE has shown to be a valuable tool n hgh dmensonal cases regardless of the employed operator. Dfferent DE parameter settngs shall also be consdered n future works to determne possble effects on the algorthm s performance. 537

8 Table 13: Results for TP5. Oper. Dm. K Mean StD OP e e e e e e e e e e e e e e e e 01 OP e e e e e e e e e e e e e e e e 01 OP e e e e e e e e e e e e e e e e 01 OP e e e e e e e e e e e e e e e e 01 OP e e e e e e e e e e e e e e e e REFERENCES [1] D. H. Ackley. A Connectonst Machne for Genetc Hllclmbng. Kluwer, Boston, [2] H. G. Cobb. Is the genetc algorthm a cooperatve learner? In Foundatons of Genetc Algorthms 2, pages Morgan Kaufmann, [3] W. J. Conover. Practcal Nonparametrc Statstcs. Wley, [4] M. El-Abd. Cooperatve Models of Partcle Swarm Optmzers. PhD thess, Dept. Elect. Comput. Eng., Unv. Waterloo, Waterloo, Ontaro, Canada, [5] T. Huang and A. S. Mohan. Mcro partcle swarm optmzer for solvng hgh dmensonal optmzaton problems. Appled Mathematcs and Computaton, 181(2): , [6] M. Köppen, K. Franke, and R. Vcente-Garca. Tny GAs for mage processng applcatons. IEEE Table 14: Hypothess testng for TP5. Oper. Dm. Improvement p value Decson OP % e 11 Reject % e 11 Reject % e 11 Reject % e 11 Reject OP % e 11 Reject % e 11 Reject % e 11 Reject % e 11 Reject OP % e 07 Reject % e 11 Reject % e 11 Reject % e 11 Reject OP % e 11 Reject % e 11 Reject % e 11 Reject % e 11 Reject OP % e 11 Reject % e 11 Reject % e 11 Reject % e 11 Reject Computatonal Intellgence Magazne, 1(2):17 26, [7] O. Olorunda and A. P. Engelbrecht. Dfferental evoluton n hgh dmensonal search spaces. In Proc. IEEE CEC 07, pages , Sngapore, [8] M. A. Potter and K. De Jong. A cooperatve coevolutonary approach to functon optmzaton. In Y. Davdor and H.-P. Schwefel, edtors, Proc. PPSN 94, pages Sprnger Verlag, [9] M. A. Potter and K. De Jong. Cooperatve coevoluton: An archtecture for evolvng coadapted subcomponents. Evol. Comput., 8(1):1 29, [10] K. V. Prce, R. M. Storn, and J. A. Lampnen. Dfferental Evoluton: A Practcal Approach to Global Optmzaton. Sprnger Verlag, Berln, [11] S. Rahnamayan and H. R. Tzhoosh. Image thresholdng usng mcro opposton-based dfferental evoluton (mcro-ode). In Proc. IEEE CEC 08, pages , Hong Kong, [12] Y.-J. Sh, H.-F. Teng, and Z.-Q. L. Cooperatve co evolutonary dfferental evoluton for functon optmzaton. In Lecture Notes n Computer Scence, volume 3611, pages Sprnger, [13] R. E. Smth, S. Forrest, and A. S. Perelson. Searchng for dverse, cooperatve populatons wth genetc algorthms. Evol. Comput., 1(2): , [14] D. Sofge, K. De Jong, and A. Schultz. A blended populaton approach to cooperatve coevoulton for decomposton of complex problems. In Proc. IEEE CEC 02, pages , [15] R. Storn and K. Prce. Dfferental evoluton a smple and effcent heurstc for global optmzaton over contnuous spaces. J. Glob. Opt., 11: , [16] F. Van den Bergh and A. P. Engelbrecht. A cooperatve approach to partcle swarm optmzaton. IEEE Trans. Evol. Comput., 8(3): ,