MULTIOBJECTIVE OPTIMIZATION USING PARALLEL VECTOR EVALUATED PARTICLE SWARM OPTIMIZATION

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1 MULTIOBJECTIVE OPTIMIZATION USING PARALLEL VECTOR EVALUATED PARTICLE OPTIMIZATION K.E. Parsopoulos, D.K. Tasouls, M.N. Vrahats Department of Mathematcs, Unversty of Patras Artfcal Intellgence Research Center (UPAIRC), Unversty of Patras, GR 26 Patras, Greece emal: fkostasp, dtas, ABSTRACT Ths paper studes a parallel verson of the Vector Evaluated Partcle Swarm Optmzaton (VEPSO) method for multobjectve problems. Experments on well known and wdely used test problems are performed, amng at nvestgatng both the effcency of VEPSO as well as the advantages of the parallel mplementaton. The obtaned results are compared wth the correspondng results of the Vector Evaluated Genetc Algorthm approach, yeldng the superorty of VEPSO. KEY WORDS Partcle Swarm Optmzaton, Multobjectve Optmzaton, PVM Introducton Multobjectve optmzaton (MO) problems consst of several objectves that need to be acheved smultaneously. Such problems arse n many applcatons, where two or more, sometmes competng and/or ncommensurable objectve functons have to be mnmzed concurrently. Due to the multcrtera nature of MO problems, the optmalty of a soluton has to be redefned, gvng rse to the concept of Pareto optmalty. In contrast to the sngle objectve optmzaton case, MO problems are characterzed by trade offs and, thus, a multtude of Pareto optmal solutons. Tradtonal gradent based optmzaton technques can be used to detect Pareto optmal solutons. However, these technques suffer from two crtcal drawbacks; (I) the objectves have to be aggregated n a sngle objectve functon, and, (II) only one soluton can be detected per optmzaton run. The nherent dffculty to foreknow whch aggregaton of the objectves s approprate n addton to the heavy computatonal cost of gradent based technques, necesstates the development of more effcent and rgorous methods. Evolutonary Algorthms (EAs) seem to be partcularly suted to MO problems due to ther ablty to synchronously search for multple Pareto optmal solutons and perform better global exploraton of the search space [, 2, 3]. Moreover, EAs are easly parallelzed, thus, decreasng the computatonal load and the requred executon tme. The parallel computaton of many solutons may also result n a better representaton of the possble outcomes, enhancng the performance of the EA [4]. Partcle Swarm Optmzaton (PSO) s a swarm ntellgence method that roughly models the socal behavor of swarms [5]. PSO s characterzed by ts smplcty and straghtforward applcablty, and t has proved to be effcent on a plethora of problems n scence and engneerng. Several studes have been recently performed wth PSO on MO problems, and new varants of the method, whch are more sutable for such problems, have been developed [6, 7, 8, 9]. Vector Evaluated Partcle Swarm Optmzaton (VEPSO) s a mult swarm varant of PSO, whch s nspred by the Vector Evaluated Genetc Algorthm (VEGA) [3, 8]. In VEPSO, each swarm s evaluated usng only one of the objectve functons of the problem under consderaton, and the nformaton t possesses for ths objectve functon s communcated to the other swarms through the exchange of ther best experence. In ths paper, a study of the performance of VEPSO, usng more than two swarms, as well as a parallel mplementaton of ths approach, s presented. The effcency of the algorthm, as well as the advantages of the parallel mplementaton are nvestgated and the results are reported and compared wth the correspondng results of the VEGA approach. The rest of the paper s organzed as follows; n Secton 2 the basc MO concepts are descrbed, and, n Secton 3, the PSO and the VEPSO algorthms are brefly presented and, also, a descrpton of the parallel mplementaton s provded. Expermental results are reported n Secton 4, followed by conclusons n Secton 5. 2 Basc Concepts of Multobjectve Optmzaton Let S ρ R n be an n dmensonal search space and f (x) :S! R; =;:::;k; () be k objectve functons defned over S. Assumng, g j (x) 6 ; j =;:::;m; be m nequalty constrants, the MO problem can be stated as fndng a vector x Λ =(x Λ ;xλ 2 ;:::;xλ n ) 2 S;

2 that satsfes the constrants and optmzes (wthout loss of generalty we consder only the mnmzaton case) the functon f (x) =[f (x);f 2 (x);:::;f k (x)] > : R n! R k : The objectve functons may be n conflct, thus, n most cases t s mpossble to obtan the global mnmum at the same pont for all the objectves. The goal of MO s to provde a set of Pareto optmal solutons to the aforementoned problem. Let u =(u ;:::;u k ), and v =(v ;:::;v k ),betwo vectors. Then, u domnates v f and only f u 6 v ; = ;:::;k, and u < v for at least one component. Ths property s known as Pareto domnance and t s used to defne the Pareto optmal ponts. Thus, a soluton x of the MO problem s sad to be Pareto optmal f and only f there does not exst another soluton y, such that f (y) domnates f (x). The set of all Pareto optmal solutons of an MO problem s called Pareto optmal set and t s denoted as P Λ. The set PF Λ Φ = f (x);:::;f k (x) j x 2 P Λ Ψ s called Pareto front. A Pareto front PF Λ s called convex f and only f there exsts w 2PF Λ, such that kuk +( )kvk > kwk; 8 u; v 2PF Λ ; 8 2 (; ): Respectvely, t s called concave f and only f there exsts w 2PF Λ, such that kuk +( )kvk 6 kwk; 8 u; v 2PF Λ ; 8 2 (; ): A Pareto Front can be convex, concave or partally convex and/or concave and/or dscontnuous. The last three cases present the greatest dffculty for most MO technques. 3 Partcle Swarm Optmzaton and Vector Evaluated Partcle Swarm Optmzaton Partcle Swarm Optmzaton (PSO) s a swarm ntellgence algorthm, nspred by the socal dynamcs and emergent behavor that arses n socally organzed colones [5, 8, ]. PSO s a populaton based algorthm,.e. t explots a populaton of ndvduals to probe promsng regons of the search space. In ths context, the populaton s called a swarm and the ndvduals (.e. the search ponts) are called partcles. Each partcle moves wth an adaptable velocty wthn the search space, and retans a memory of the best poston t ever encountered. In the global varant of PSO, the best poston ever attaned by all ndvduals of the swarm s communcated to all the partcles. In the local varant, each partcle s assgned to a topologcal neghborhood consstng of a prespecfed number of partcles. In ths case, the best poston ever attaned by the partcles that comprse the neghborhood s communcated among them [5]. In ths paper only the global varant s consdered. Assume an n dmensonal search space, S ρ R n, and a swarm consstng of N partcles. The th partcle s n effect an n dmensonal vector X =(x ;x 2 ;:::;x n ) > 2 S. The velocty of ths partcle s also an n dmensonal vector, V = (v ;v 2 ;:::;v n ) > 2 S. The best prevous poston encountered by the th partcle s a pont n S, denoted by P = (p ;p 2 ;:::;p n ) > 2 S. Assume g to be the ndex of the partcle that attaned the best prevous poston among all the partcles n the swarm (global verson), and t to be the teraton counter. Then, the swarm s manpulated by the equatons [, 2]: V (t +) = χ h wv (t) +c r P (t) X (t) + c 2 r 2 Pg (t) X (t) + ; (2) X (t +) = X (t) +V (t +); (3) where = ;:::;N; c and c 2 are two parameters called the cogntve and the socal parameter, respectvely, and they are used to bas the search of a partcle toward ts best experence and the best experence of the whole swarm, respectvely; r, r 2, are random numbers unformly dstrbuted wthn [; ]. The parameters χ and w are called the constrcton factor and the nerta weght, respectvely, and they are used alternatvely as mechansms for the control of the velocty s magntude, gvng rse to the two dfferent PSO versons. The selecton of the aforementoned parameters has been wdely dscussed and studed n the relatve lterature [, 3, 4]. The Vector Evaluated Partcle Swarm Optmzaton (VEPSO) algorthm [8] has been nspred by the concept of the Vector Evaluated Genetc Algorthm (VEGA) [3]. In VEGA, fractons of the next generaton or subpopulatons are selected from the prevous generaton accordng to each of the objectves, separately. After shufflng all these sub populatons together, crossover and mutaton are appled to generate the new populaton. These deas have been adopted and modfed to ft the PSO framework. Specfcally, n VEPSO, two or more swarms are employed to probe the search space and nformaton s exchanged among them [8]. Each swarm s exclusvely evaluated wth one of the objectve functons, but, nformaton comng from other swarm(s) s used to nfluence ts moton n the search space. The best poston attaned by each partcle (the partcle s memory) separately as well as the best among these postons are the man gudance mechansms of the swarm. Thus, exchangng ths nformaton among swarms can lead to Pareto optmal ponts. Let the problem at hand consst of k objectve functons, f l (x), l =;:::;k, as defned n Eq. (), and assume that M swarms, S ; S 2 ;:::;SM, of sze N, are employed to address t. Each swarm s evaluated accordng to one of the objectve functons. Let also X [j] ;V [j], and P [j], =;:::;N, j =;:::;M, be the current poston, the velocty and the best prevous poston of the th partcle n the j th swarm, respectvely, at a gven tme. Assumng that g [j] denotes the ndex of the partcle that attaned the best prevous poston n the j th swarm, then the VEPSO

3 2 6 CPUs 3 M 4 SERVER 5 Fgure. The rng mgraton scheme. Fgure 2. The PVM system used. swarms are, n general, manpulated accordng to the equatons [8]: V [j] (t +)=χ [j] h w [j] V [j] (t) +c [j] r [j] P (t) X [j] (t) + X [j] + c [j] 2 r2 [s] P g [s] (t +)=X [j] (t) +V [j] (t) X[j] (t) ; (4) (t +); (5) where = ;:::;N; j = ;:::;M; c [j] and c [j] 2 are the cogntve and socal parameters of the j th swarm; r, r 2, are random numbers unformly dstrbuted wthn [; ]; χ [j] and w [j] are the constrcton factor and the nerta weght of the j th swarm, respectvely; and s s an ndex takng values n f;:::;j ;j +;:::;Mg,.e. the veloctes of the j th swarm are updated usng the best prevous poston of another (the s th) swarm. The case of two swarms wth two objectve functons has been presented and nvestgated n [8]. The procedure of exchangng nformaton among swarms can be clearly vewed as a mgraton scheme n the parallel computaton framework. The parameter s can be selected n a number of ways resultng n dfferent VEPSO varants. For example, selectng s accordng to s = ρ M; f j =; j ; f j =2;:::;M; corresponds to the rng mgraton topology [5], whch s depcted n Fg. An alternatve choce s to select s randomly. Further constrants may also be posed on the selecton of s, e.g. allow the best partcle of a swarm to mgrate only to one swarm (ths holds for the rng topology but not for the random selecton). Ths paper ams at nvestgatng the effcency of the VEPSO method as well as possble benefts obtaned by ts parallel mplementaton. Specfcally, the man goals are to nvestgate VEPSO s performance usng dfferent numbers of swarms on a sngle machne, as well as the tme acceleraton obtaned f more than one machnes are used. (6) Characterstc Descrpton Number of CPUs 2to CPU Type Intel Celeron 9-MHz Memory 256-MB per machne Operatng System Red Hat Lnux 8. Communcaton Network Fast Ethernet -Mbps Communcaton Lbrary PVM Table. The characterstcs of the system used for the parallel experments. The VEPSO approach can be straghtforwardly parallelzed by dstrbutng the swarms n many machnes and allowng mgraton from node to node. For ths purpose the Parallel Vrtual Machne (PVM) has been used [6]. The key characterstcs of the system used n the parallel mplementaton of VEPSO are reported n Table and ts topology s depcted n Fg. 2. In addton to the reported hardware, a Pentum III machne wth 52-MB of memory, runnng under Red Hat Lnux 8., has been used as a server. The sngle machne experments have also been performed on one of the aforementoned systems. Regardng VEPSO s parallelzaton parameters, a rng mgraton topology has been selected, wth mgraton takng place at each teraton (synchronzed swarms move), employng from 2 up to swarms. For the mantenance of the Pareto optmal set, the archvng technque descrbed n [7] has been used. The obtaned results are evaluated usng two establshed measures, the C metrc [8, 9], and the V measure [8, 2]. The metrc C(A; B) measures the fracton of members of the Pareto front B that are domnated by members of the Pareto front A, whle V(A; B) s the fracton of the volume of the mnmal hypercube contanng both fronts, that s strctly domnated by members of A but s not domnated by members of B [8].

4 NUMBER OF S NUMBER OF S NUMBER OF S NUMBER OF S Fgure 3. Results for the Test Problem NUMBER OF S NUMBER OF S NUMBER OF S NUMBER OF S Fgure 4. Results for the Test Problem 2. 4 Expermental Results The followng well known benchmark problems have been used to llustrate the performance of VEPSO: TEST PROBLEM. [9] Ths problem has a convex Pareto front: f (x ) = x ; (7) g(x 2 ;:::;x n ) = + 9 x ; (8) n =2 s f h(f ;g) = g ; (9) wth n =3and x 2 [; ]. TEST PROBLEM 2. [9] Ths s the nonconvex counterpart to the Test Problem : nx f (x ) = x ; () g(x 2 ;:::;x n ) = + 9 n wth n =3and x 2 [; ]. nx =2 x ; () 2 f h(f ;g) = ; (2) g TEST PROBLEM 3. [9] Ths Pareto front conssts of several convex parts: f (x ) = x ; (3) g(x 2 ;:::;x n ) = + 9 x =(n ); (4) n =2 s f h(f ;g) = g f g sn(ßf ); (5) wth n =3and x 2 [; ]. TEST PROBLEM 4. [9] Ths test problem has 2 9 local Pareto fronts: nx f (x ) = x ; (6) g(x 2 ;:::;x n ) = +(n ) + nx + x 2 cos(4ßx ) ; (7) =2 s f h(f ;g) = g ; (8) wth n =, x 2 [; ], and x 2 ;:::;x n 2 [ 5; 5]. In all experments, the global varant of the constrcton factor PSO has been used. The PSO parameters have been the same for each swarm and for all problems, equal to: χ = :729, c = c 2 = 2:5 [].

5 NUMBER OF S NUMBER OF S NUMBER OF S NUMBER OF S Fgure 5. Results for the Test Problem NUMBER OF S NUMBER OF S NUMBER OF S NUMBER OF S Fgure 6. Results for the Test Problem 4. The results obtaned through VEPSO, usng 2, 4, 6, 8, and swarms, have been compared wth the results obtaned through VEGA that are freely avalable at the web page For ths purpose, partcles have been used, n total, for each experment, dvded n 2 up to sub swarms. For each case, 3 ndependent experments have been performed. The maxmum number of teratons of the VEPSO algorthm for each experment has been set equal to 25. All results are statstcally dsplayed wth boxplots n Fgs The boxplots are based on the two metrcs, C and V, wth respect to the number of swarms that have been used. Each boxplot represents the dstrbuton of the C or V values for the ordered par (VEPSO,VEGA) and vce versa (notce that both the C and the V metrc are nether symmetrcal n ther arguments nor satsfy the trangle nequalty, thus, n general, C(A; B) 6= C(B;A)). Each box of the boxplot has lnes at the lower quartle, medan, and upper quartle values. The lnes that extend from each end of the box are the whskers, and they show the extent of the rest of the data. The outlers le beyond the ends of the whskers and they are denoted wth crosses. The notches represent a robust estmate of the uncertanty about the medans for box to box comparson. The obtaned results support the clam that the VEPSO algorthm outperforms the VEGA algorthm n all cases. The C(VEPSO; VEGA) and the V(VEPSO; VEGA) assume relatvely hgh values n all test problems whle C(VEGA; VEPSO) and V(VEGA; VEPSO) are n almost all cases equal to zero. Moreover, t seems that usng more than 2 swarms n some cases mproves the performance of VEPSO. However, too many swarms do not offer sgnfcant performance advantages. Perhaps ths happens due to the small sze of each swarm, whch, for the case of swarms, s equal to partcles per swarm. Increasng the swarm sze may further enhance the algorthm s performance. The aforementoned experments were performed both serally and n parallel. The parallel mplementaton resulted n an mprovement of the performance n terms of the requred executon tme, whch s depcted n Fg. 7. As can be seen, ncreasng the number of swarms from 2 up to 6 swarms, there s a sgnfcant gan n tme. However, usng more than 6 swarms results n ncreased tme due to the heavy network overhead. 5 Conclusons The VEPSO approach, whch s based on the PSO method, for MO problems has been appled on four well known test problems. Both sngle node and parallel mplementatons of the algorthm have been developed and appled wth very promsng results. Two wdely used metrcs have been used for the evaluaton of the results and for comparsons wth the correspondng results of the VEGA approach. VEPSO

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