Self-Adaptive Simulated Binary Crossover for Real-Parameter Optimization

Transcription

1 Self-Adaptve Smulated Bnary Crossover for Real-Parameter Optmzaton Kalyanmoy Deb Dept. of Mechancal Engneerng Indan Inst. of Tech. Kanpur Kanpur, PIN 20806, Inda S. Karthk Dept. of Mechancal Engneerng Indan Inst. of Tech. Kanpur Kanpur, PIN 20806, Inda Tatsuya Okabe Honda Research Insttute Japan 8- Honcho, Wako-sh Satama, , Japan ABSTRACT Smulated bnary crossover (SBX) s a real-parameter recombnaton operator whch s commonly used n the evolutonary algorthm (EA) lterature. The operator nvolves a parameter whch dctates the spread of offsprng solutons vs-a-vs that of the parent solutons. In all applcatons of SBX so far, researchers have kept a fxed value throughout a smulaton run. In ths paper, we suggest a self-adaptve procedure of updatng the parameter so as to allow a smooth navgaton over the functon landscape wth teraton. Some basc prncples of classcal optmzaton lterature are utlzed for ths purpose. The resultng EAs are found to produce remarkable and much better results compared to the orgnal operator havng a fxed value of the parameter. Studes on both sngle and multple objectve optmzaton problems are made wth success. Categores and Subject Descrptors I.2.8 [Computng Methodologes]: Problem Solvng, Control Methods, and Search General Terms Algorthms Keywords Self-adaptaton, smulated bnary crossover, real-parameter optmzaton, recombnaton operator.. INTRODUCTION Most real-world optmzaton problems nvolve decson varables whch are real-valued. Despte the dedcated realparameter EAs, such as evoluton strategy, dfferental evoluton etc., real-parameter GAs have ganed adequate popularty n the recent past. The man challenge n developng an effcent real-parameter GA les n devsng a recombnaton operator n whch two or more real-parameter vectors 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 07, July 7, 2007, London, England, Unted Kngdom. Copyrght 2007 ACM /07/ $5.00. must be blended to create two or more offsprng vectors of real numbers [, 2, 9, 7]. These recombnaton operators employ a non-unform probablty dstrbuton around the parent solutons to create an offsprng soluton. A theoretcal study [] attempted to fnd smlartes among these operators. In ths paper, we concentrate on a partcular recombnaton operator the smulated bnary crossover (SBX) operator [2]. The SBX operator uses two parent vectors and apply the blendng operator varable by varable to create two offsprng solutons. The operator nvolves a parameter, called the dstrbuton ndex (η c), whch s kept fxed to a non-negatve value throughout a smulaton run. If a large value of η c s chosen, the resultng offsprng solutons are close to the parent solutons. On the other hand, for a small value of η c, solutons away from parents are lkely to be created. Thus, ths parameter has a drect effect n controllng the spread of offsprng solutons. Snce a search process of fndng the mnmum soluton of a functon landscape largely depends on controllng the spread (or dversty) of offsprng solutons vs-a-vs the selecton pressure ntroduced by the chosen selecton operaton, fxng the η c parameter to an approprate value s an mportant task. Here, we suggest a self-adaptve procedure of updatng the η c parameter by usng the extenson-contracton concept n a classcal optmzaton algorthm. If the created chld soluton s better than the partcpatng parent solutons, the chld soluton s extended further n the hope of creatng even a better soluton. On the other hand, f a worse soluton s created, a contracton s performed. Ether task wll result n a update of η c, so that the newly-created extended or contracted offsprng soluton has an dentcal probablty of creaton wth an updated η c. Ths modfcaton procedure has been appled to three sngle-objectve and three twoobjectve optmzaton problems and compared wth correspondng GAs wth a fxed η c value. In all cases, better performance of the suggested self-adaptve procedure s observed. In the remander of the paper, we brefly descrbe the SBX operator. Thereafter, we suggest the self-adaptve η c update procedure and show smulaton results on sngle-objectve optmzaton problems. In each case, a parametrc study wth a parameter α s performed to fnd a sutable workng range of ths parameter. Then, a scale-up study by varyng the number of decson varables s performed to demonstrate the polynomal complexty of the suggested algorthm. Fnally, the self-adaptve update of η c s extended to mult- 87

2 objectve optmzaton and results are dscussed. Fnally, conclusons from the study are made. 2. SIMULATED BINARY CROSSOVER (SBX) As the name suggests, the SBX operator [2, 6] smulates the workng prncple of the sngle-pont crossover operator on bnary strngs. In the above-mentoned studes, t was shown that ths crossover operator respects the nterval schemata processng [7], n the sense that common nterval schemata between the parents are preserved n the offsprng. The procedure of computng the offsprng x (,t+) and x (2,t+) from the parent solutons x (,t) and x (2,t) s descrbed as follows. A spread factor β s defned as the rato of the absolute dfference n offsprng values to that of the parents: β = x (2,t+) x (2,t) x (,t+) x (,t). () Frst, a random number u between 0 and s created. Thereafter, from a specfed probablty dstrbuton functon, the ordnate β q s found so that the area under the probablty curve from zero to β q s equal to the chosen random number u. The probablty dstrbuton used to create an offsprng s derved to have a smlar search power to that n a sngle-pont crossover n bnary-coded GAs and s gven as follows [2]: ( 0.5(ηc +)β ηc, f β ; P(β )= 0.5(η c +), otherwse. (2) β ηc+2 Fgure shows the above probablty dstrbuton wth η c = 2 and 5 for creatng offsprng from two parent solutons (x (,t) =2.0 andx (2,t) = 5.0). In the above expressons, the dstrbuton ndex η c s any non-negatve real number. A large value of η c gves a hgher probablty for creatng near-parent solutons (thereby allowng a focussed search) and a small value of η c allows dstant solutons to be selected as offsprng (thereby allowng to make dverse search). Probablty densty per offsprng η η c = 2 c= 5 o o Offsprng soluton Fgure : The probablty densty functon for creatng offsprng under an SBX-η c operator. 8 After obtanng β q from the above probablty dstrbuton, the offsprng are calculated as follows: h x (,t+) = 0.5 ( + β q )x (,t) h x (2,t+) = 0.5 ( β q )x (,t) +( β q )x (2,t) +(+β q )x (2,t), (3). (4) The SBX operator bases solutons near each parent more favorably than solutons away from the parents. Essentally, the SBX operator has two propertes:. The dfference between the offsprng s n proporton to the parent solutons. 2. Near-parent solutons are monotoncally more lkely to be chosen as offsprng than solutons dstant from parents. An nterestng aspect of ths crossover operator s that for afxedη c theoffsprnghaveaspreadwhch s proportonal to that of the parent solutons x (2,t+) x (,t+) = β q x (2,t) x (,t). (5) In ntal populatons, where the solutons are randomly placed, makng the dfference n parents ((x (2,t) x (,t) )) large, ths allows to create a chld soluton whch s also far away from the parents. However, when the solutons tend to converge due to the acton of genetc operators (thereby makng the parent dfference small), dstant solutons are not lkely to occur, thereby focusng the search to a narrow regon. As we dscuss n the followng secton that ths aspect of self-adaptve nature of SBX operator s not adequate alone n reachng near the optmum n large-szed and complex functons. 3. SELF-ADAPTIVE SBX The SBX operator dscussed above nvolves a parameter: the dstrbuton ndex η c. In most applcatons of SBX, a fxed value of η c = 2 s used for sngle-objectve optmzaton [2]. For a fxed value of η c, the dfference between the created chld soluton and the closest parent soluton depends on the net dfference between the two parent solutons, thereby causng a self-adaptve property of such a operator [5]. It has always been a research ssue whether such a self-adaptve property s adequate n solvng dffcult optmzaton problems. Past studes of real-coded genetc algorthms wth the SBX operator was not found to be sutable for mult-modal problems, such as Rastrgn s functon [4]. Here, we suggest a procedure for updatng the η c parameter adaptvely to solve such problems. To llustrate the modfed procedure, let us consder Fgure 2, n whch a typcal chld soluton c s shown to be created from two parent solutons p and p 2 by usng SBX wth a dstrbuton ndex of η c. Say, the random number usedforthspartcularcrossoveroperatonsu ( (0, )). Then, f β (> ) corresponds to the spread factor for the specfc case of crossover, we obtan the followng from the defnton of spread factor: By notng that Z β β =+ 2(c p2) p 2 p. (6) 0.5(η c +) β ηc+2 dβ =(u 0.5), 88

3 0 0 0 η c 00 η 00 c u 000 u p p2 c c Fgure 2: A schematc dagram showng the η c update procedure. we obtan the followng relatonshp: η c = + log 2( u) log β η c «. (7) We now get nto four scenaros (resultng n equatons 9 to 2), as dscussed below. If ths chld soluton c s better than both parent solutons, we assume that the regon n whch the chld soluton s created s better than the regon n whch the parents solutons are and ntend to extend the chld soluton further away from the closest parent soluton (p 2, n the case shown n the fgure) by a factor α (> ). Ths dea s smlar to that followed n Nelder and Meade s smplex search method [0], n whch an ntermedate soluton s ether expanded or contracted or kept the same dependng on the functon value of the soluton compared to that of the prevously computed solutons. Rechenberg s /5th rule also keeps track of proporton of successful mutatons over a certan number of trals [3]. If the success happens too often, the mutaton strength s ncreased to create solutons away from the parent, else the mutaton strength s reduced. By usng α>, the dfference between the new chld soluton c wth closest parent p 2 s made α tmes than that between orgnal chld soluton c and p 2, such that (c p 2)=α(c p 2). We then modfy the dstrbuton ndex to a value η c such that wth ths value and havng the dentcal random number u, the modfed chld soluton c gets created from parent solutons p and p 2, thereby yeldng η c = + log 2( u) log β «. (8) Here, β s the correspondng dstrbuton ndex, gven by β = + 2(c p 2) (p 2 p, ) = +α(β ). Usng the above relatonshp between β and β and usng equatons 7 and 8, we obtan the followng update relatonshp for a chld soluton beng found to be better than the nearest parent soluton and the chld les outsde the regon bounded by parents: η c (ηc +)logβ = + log ( + α(β )). (9) To make the η c value meanngful and free from numercal underflow error, we restrct t wthn [0, 50], that s f η c < 0 s obtaned by the above equaton, we set t to be zero and f η c > 50 s used, we set t to be 50. Ths equaton gves us an update procedure of η c for provdng a dstrbuton ndex whch s able to create chldren solutons n the rght drecton away from the parents. It s nterestng to note that f a chld c was created to the left of p, a smlar update relatonshp wll also be acheved. If the chld soluton c s worse than both parent solutons, we would lke to move the modfed chld c to get closer to theparentsandwemayuse/α nstead of α n equaton 6: η c (η c +)logβ = + log ( + (β )/α). (0) However, f a chld soluton s created nsde the regon bounded by both parents, a dfferent update relatonshp wll be obtaned, snce the SBX probablty dstrbuton functon s dfferent n ths case. By followng the above arguments, we obtan β = β α and the relatonshp for an mproved chld soluton between η c and η c values s obtaned as follows: η c = +ηc. () α Once agan, we restrct ts value wthn [0, 50] to avod any computatonal error and to make η c meanngful. For a scenaro of creatng a worse chld, the above update relatonshp changes to η c = α( + η c). (2) If the chld soluton c has a functon value whch s wthn the functon value of the two parent solutons, we set η c = η c. To mplement the update concept, we assgn a η c value wthn [ηc L,ηc U ] n the ntal populaton. In all smulatons here, we use ηc L = ηc U = 2. For a chld soluton, one random number s created and a correspondng β s computed. Ths β s used for all n varables and a chld s created by varablewse applcaton of the SBX operator. Ths results n a lne- SBX operator whch produces a chld along the lne jonng the two parent vectors. Thereafter, dependng on the event of whether a better chld (than both parents) or a worse chld (than both parents) s created, η c s updated and s assgned to the correspondng chld soluton. We assgn a new η c value to each chld soluton separately. It s nterestng to note that f α = s used, all the above update of η c procedure result n η c = η c and the above procedure s dentcal to the fxed η c (orgnal SBX) procedure. 3. Effect of mutaton operator In addton to the above modfed SBX operator, a mutaton operator can be used to perturb the created chld c. In such an event, there s an extra soluton evaluaton per chld creaton. Soluton c gets evaluated durng the crossover operator and the mutated verson of the modfed soluton c s evaluated. In our mplementaton, we add such extra functon evaluatons n the computaton of performance of the modfed procedure. However, f the mutaton probablty s so low that no varable gets mutated by the mutaton operator, there s no extra evaluaton recorded. 4. SIMULATION RESULTS We now apply the modfed approach to three dfferent unconstraned functons whch are popularly used n the GA 89

4 lterature. Many studes n the lterature on the chosen test problems use a populaton whch s ntalzed symmetrcally around the global mnmum of the functons. When an algorthm wth such a populaton uses recombnaton and mutaton operators, whch have a tendency to create solutons n the central regon of the search space bounded by the populaton members, t fnds an easer tme convergng to the global mnmum. To avod any such undue advantage from the operators, n all smulatons here, the ntal populaton s bounded n the range x [0, 5] for all, such that the global mnmum s not bracketed n the ntal populaton. Ths provdes a stff test to an algorthm to frst get out of the regon bounded by the ntal populaton and then keep movng n the correct drecton so as to reach the global mnmum. 4. Sphere Functon The sphere functon s the smplest of the three functons used n ths study: f(x) = nx x 2. (3) Frst, we employ the real-coded GA wth the orgnal fxedη c based SBX operator on the 30-varable sphere functon wth followng parameter settngs: populaton sze = 50, p c = 0.9, η c = 2, p m = /30, and η m = 50. A run s termnated when a soluton havng a functon value equal to 0.00 s found. Eleven runs are made from random ntal populatons. Fgure 3 shows a typcal varaton of best populaton functon value wth the number of functon evaluatons. To nvestgate the effect of recombnaton operator alone, we make another set of runs wth p m =0andatypcal performance s also shown n the fgure. Both results ndcate that the fxed-η c based GA s unable to fnd the true optmum n any reasonable number of evaluatons for a 30-varable sphere functon. However, t s worthwhle to menton here that after a varable-wse creaton of two offsprng vectors, f the varables values are swapped randomly between the two offsprngs (smlar to a unform crossover operator or the crossover operator n dfferental evoluton [2]), a much qucker convergence wth a fxed η c =2can be obtaned for varable-wse separable functons, lke the sphere functon [5]. Ths s because varables can ndependently reach near the true optmum n dfferent populaton members and a unform swappng of such solutons can create a complete soluton havng near optmal varables values. However, ths unform-lke crossover operaton may not work well n rotated and more complex problems. In ths paper, we only concentrate on the blendng part of the operaton and nvestgate whether the fxed nature of η c or the selfadaptve nature of η c s more effectve wthout havng any crossover-lke swappng effect. Next, we employ our self-adaptve update procedure of η c. All ntal populaton members are ntalzed wth η c =2 and then allowed to change based on our update procedure descrbed above. All other parameters are the same as above and to nvestgate the effect of the self-adaptve recombnaton operator alone, we use p m = 0 and reduce the crossover probablty to p c =0.7. For the η c update we use α =.50. A typcal varaton of the best populaton functon value s shown n Fgure 3 and s found to have good convergng property. Table shows the best, medan and worst number of functon evaluatons to reach a soluton wth f = 0.00 = out of runs. It s clear that compared to the fxed-η c schemes, the self-adaptve scheme s able to steer the search towards the true mnmum soluton quckly and converge close to the mnmum. It s clear that even wth 300,000 functon evaluatons, the fxed η c scheme s not able to fnd a near-optmum soluton, whereas wth a medan of 84,050 functon evaluatons, a soluton wth three decmal places accuracy from the true optmum s found repeatedly n runs. 4.. Parametrc Study Fgure 4 does a parametrc study wth α n the range [.05, 2.00]. Recall that the parameter α sgnfes the extent of change n the offsprng soluton performed to recompute the η c parameter for an dentcal probablty event of creatng the modfed soluton. Once agan, runs are performed for every case and the best, medan and worst number of functon evaluatons are shown n the fgure. It can be sad that the effect of α s not sgnfcant. Although there s a degradaton of the medan performance wth ncreasng α, the performance s best n the range α [.05,.50]. Thus, despte the ntroducton of a new parameter (α) for updatng an exstng parameter (η c), the effect of the parameter α s not sgnfcant Scale-up Study Fnally, we perform a scalablty study by varyng the number of varables (n) from 20 to 200. In ths study, we have used the followng update of parameter due to the ncrease n number of varables: populaton sze = 5n. All other parameters are kept the same as before and we use α =.50. Fgure 5 shows that () the real-coded GA wth self-adaptve recombnaton operator s able to fnd a soluton close to the true mnmum (wthn a tolerance of 0.00 n the functon value) and () the ncrease n number of functon evaluatons s polynomal to the ncrease n number of varables (O(n 2.2 )). The GA wth the self-adaptve SBX operator suggested here does not explot the varable separablty of the objectve functons. Ths s the reason why the proposed GA takes more number of evaluatons than other approaches whch favor the varable separablty and unmodalty of the sphere functon [5, 4]. 4.2 Rosenbrock Functon Next, we consder the Rosenbrock s functon: n X f(x) = 00(x 2 x +)+(x ) 2. (4) = Ths functon has the global mnmum at x =forall wth a functon value equal to zero. Near the mnmum regon, ths functon has a very small slope towards the mnmum. Ths property of the landscape causes an algorthm to have a slow convergence to the mnmum. Once agan, we frst employ the orgnal SBX operator wth a fxed η c = 2 and ntalze x [0, 5] for all. Fgure 6 shows a typcal varaton of populaton-best functon value wth the number of functon evaluatons for the 30-varable Rosenbrock functon wth followng parameter settngs: populaton sze = 50, p c =0.9, p m =/30, and η m = 50. The algorthm s termnated when a soluton wth a mnmum functon value of 0.00 s found. The procedure s not able to come close to the true mnmum. When we use 90

5 Functon Value Org. (p_c=0.9,p_m=0.033) Org. (p_c=0.9,p_m=0) Self adaptve Functon Evaluatons Functon Evaluatons e+08 e+07 e Slope = Functon Evaluatons α Number of Varables Fgure 3: Varaton of populatonbest functon value wth number of functon evaluatons for the 30- varable sphere functon. Fgure 4: Parametrc study of α for the 30-varable sphere functon. Fgure 5: Scale-up study wth number of varables for the sphere functon. Table : Performance of real-coded GAs wth fxed and self-adaptve η c update on 30-varable sphere functon. Method Optmzed functon value (func. eval.) Orgnal (p c =0.9, p m =0.033).24e03 (300,000) 4.6e03 (300,000) 4.30e03 (300,000) Orgnal (p c =0.9, p m =0).2e03 (300,000) 4.3e03 (300,000) 4.24e03 (300,000) Self-adp.(p c =0.7, p m =0) 0 3 (5,800) 0 3 (84,050) 0 3 (23,450) p m = 0, a somewhat better performance s observed, but the procedure s unable to reach near to the global mnmum. Now, we apply the self-adaptve recombnaton operator startng wth η c = 2 to all ntal populaton members. We use α =.5 for the update procedure. Fgure 6 shows a typcal varaton n the populaton-best functon value. The self-adaptve property of the recombnaton operator s able to adjust the η c adequately to navgate through the ftness landscape to reach near the global mnmum. Table 2 shows the best, medan and worst performance out of runs of orgnal and self-adaptve GAs. For the self-adaptve case, we have shown results wth α =.4, whch produces the best result. Wth as many as 0 mllon functon evaluatons, the fxed η c scheme s not able to fnd a near-optmum soluton (n fact, the best soluton has a functon value of 6.85(0 6 )), whereas wth a medan of about 6.9 mllon functon evaluatons, a soluton wth three decmal places accuracy from the true optmum s found repeatedly Parametrc Study A parametrc study on α s shown n Fgure 7. Although there s a upward trend n number of functon evaluatons wth ncreasng α, wth α [.05,.50] the performance s better. Interestngly, n ths problem also we fnd that the effect of α n a good range of values s not sgnfcant Scale-up Study A scale-up study s made next by varyng the number of varables from 20 to 200. Fgure 8 shows that the number of functon evaluatons needed to reach up to three decmal places of accuracy vares as O(n (log n) 9.47 ) by the proposed self-adaptve procedure. 4.3 Rastrgn s Functon Next, we consder the Rastrgn s functon whch has many local optma and one global mnmum at x =0for =, 2,...,n: f(x) = nx x 2 +0( cos(2πx )). (5) = In an earler study [4], ths functon was dffcult to solve for global optmalty usng the real-coded GA wth a parent centrc recombnaton operator, partcularly when the ntal populaton dd not bracket the global optmum. In ths study, we ntalze a populaton wth each x created randomly n [0, 5], away from the global mnmum soluton. In a sngle dmenson, an algorthm has to overcome at least 0 dfferent local optma to reach to the global mnmum. Wth a larger varable sze, exponentally more local optma must be overcome to reach the global mnmum. Frst, we apply the real-coded GA wth the orgnal SBX operator on the 20-varable Rastrgn functon wth a standard parameter settng: populaton sze = 00, p c =0.9, η c =2,p m =/20, η m = 50. GAs are run from dfferent ntal populatons tll a maxmum of 40,000 generatons or tll a soluton havng a functon value of 0 4 s obtaned. Table 3 shows the best functon values obtaned by the procedure. In ths case, the best obtaned functon value (n runs) wth four mllon functon evaluatons s , whereas the globally best soluton has a functon value equal to zero. It s clear that no run s able to fnd a soluton close to the global mnmum. Wth p c =0.7 andp m =0.0, we obtan slghtly better performance, but even now no soluton close to the global mnmum s found. Next, we apply the real-coded GA wth our proposed selfadaptve SBX operator wth p m =0.7 andp m =0.0. All ntal populaton members are ntalzed wth η c =2and then allowed to change usng the proposed η c update procedure descrbed earler. We use α =.5 here. Table 3 shows 9

6 Functon Value e+08 Org. (p_c=0.9, p_m=0.033) e+06 Org. (p_c=0.9,p_m=0) Self adaptve e+06 4e+06 6e+06 8e+06 e+07 Functon Evaluatons Functon Evaluatons.2e+07.e+07 e+07 9e+06 8e+06 7e+06 6e+06 5e+06 4e+06 3e+06 2e α Functon Evaluatons e+09 e+08 e+07 e Number of Varables 200 Fgure 6: Varaton of populatonbest functon value wth number of functon evaluatons for the 30- varable Rosenbrock s functon. Fgure 7: Parametrc study of α for the 30-varable Rosenbrock s functon. Fgure 8: Scale-up study wth number of varables for the Rosenbrock s functon. Table 2: Performance of real-coded GAs wth fxed and self-adaptve η c update on 30-varable Rosenbrock s functon. Method Optmzed functon value (func. eval.) Orgnal (p c =0.9, p m =0.033) 6.85e06 (0M) 7.44e06 (0M) 8.30e07 (0M) Orgnal (p c =0.9, p m =0) 9.94e06 (0M) 4.94e07 (0M) 5.9e07 (0M) Self-adp.(p c =0.7, p m =0) 0 3 (2,200,650) 0 3 (6,832,950) 0 3 (7,836,300) that n all cases the self-adaptve update of η c s able to fnd a soluton wth desred accuracy n a fracton of total evaluatons used n the case of fxed-η c procedures. Despte beng started far away from the global mnmum, the procedure s able to converge to the correct globally mnmum soluton. Fgure 9 shows the decrease n best functon value wth number of functon evaluatons for a typcal smulaton run wth a fxed η c = 2 procedure (p c =0.7, p m =0.0) and wth the self-adaptve procedure. The fgure clearly shows that the fxed η c run s poor n ts performance, whereas the self-adaptve procedure steadly fnds better and better solutons wth functon evaluatons. f(x) or eta_c e 04 eta_c Orgnal SBX Adaptve SBX e 05 0 e05 2e05 3e05 4e05 5e05 Functon Evaluatons Fgure 9: Decrease n best populaton functon value wth number of functon evaluatons for the 20- varable Rastrgn s functon. To understand the effect of the proposed self-adaptve update of η c,werecordtheη c value of the top 5 percentle 6e05 solutons n terms of ther functon values and show ts varaton wth number of functon evaluatons n Fgure 9. The y- axs of ths fgure s made logarthmc. Although, the above η c value seems to end at 0 5 n the fgure, the actual value recorded by the procedure s η c = 0. An nterestng aspect s that the η c value seems to take a value zero at varous stages of the smulaton. The lne correspondng to the best functon value (marked as Adaptve SBX ) ndcates that there are a number of functon values, especally wthn f = 0 to f =, n whch the algorthm seems to get stuck for a large number of evaluatons before fndng a better soluton. Wthn ths range of functon values, ths functon has one local mnmum at every nteger value of the functon, thereby havng a possblty to get stuck 0 tmes. Interestngly, every tme the best populaton member gets stuck at a local mnmum, the η c of the best 5 percentle soluton gets updated to zero, thereby ncreasng the spread of created solutons by the SBX operator. Snce an η c = 0 wll be the smallest possble η c whch provdes the maxmum spread n created solutons, the algorthm fnds that the best way to counteract a local stats s to ncrease the dversty of created solutons to the extent possble. However, as soon as a better soluton s found, the η c s mmedately updated to a value close to one, thereby provdng a more focussed search around populaton members. Wth an assgned fxed value of η c over the entre smulaton run, such a varaton n spread n solutons n an offsprng populaton s not possble. The proposed procedure seems to employ ths prncple adaptvely and multply n as many tmes as the algorthms get stuck to a locally optmal soluton and mprove from such a stuaton. In the absence of such an adaptve update of η c wth the orgnal SBX operator, the correspondng GA was not able to mprove ts performance effcently every tme t gets stuck to a locally optmal soluton. 92

7 Table 3: Performance of real-coded GAs wth fxed and self-adaptve η c update on 20-varable Rastrgn s functon. Method Optmzed functon value (func. eval.) Orgnal (p c =0.9, p m =0.05) (4M) (4M) (4M) Orgnal (p c =0.7, p m =0.0) (4M) (4M) (4M) Self-adp.(p c =0.7, p m =0.0) 0 4 (287,822) 0 4 (429,5) 0 4 (569,597) 4.3. Parametrc Study A parametrc study of α to nvestgate ts effect on the performance of the proposed procedure s made next. For the 20-varable Rastrgn s functon and wth above parameter settngs, dfferent runs are made. Fgure 0 shows number of functon evaluatons needed to fnd a soluton wth a functon value equal to or smaller than 0 4. It s nterestng to note that for a large range of values of α ( [.05, 8.00]), the performance of the proposed procedure remans farly ndependent of α. However, the best performance seems to happen for α =3wthbest,medanand worst number of functon evaluatons of 22,082, 342,74, and 78,680, respectvely. Functon Evaluatons e alpha Fgure 0: Effect of parameter α on number of functon evaluatons to obtan the global mnmum wth four decmal places of accuracy for the 20-varable Rastrgn s functon Scale-up Study Motvated by the success of the self-adaptve SBX procedure on the 20-varable Rastrgn s functon, we now try to solve larger sze Rastrgn s functon wth n varyng n [20, 200]. Wth an ncrease n n, the number of local mnma n a partcular range of the varable values ncrease exponentally and the resultng problem s lkely to provde more dffculty to an optmzaton algorthm. Snce a functon wth a larger number of varables should deally requre a larger populaton sze for a GA ntalzed wth a randomly created populaton [8], we use the followng parametrc update for dfferent n: populaton sze=5n, p m =0.2/n, andp c =0.7. Here also, we ntalze populaton wth x [0, 5], so as to not bracket the global mnmum n the ntal populaton. We also use α =.5 for all n. We run tll a soluton wth a functon value equal to or smaller than 0 4 s obtaned. Fgure shows that the requred number of functon evaluatons ncreased polynomally (O(n.807 )) wth n over the entre range of number of varables used n ths study. It s Number of Functon Evaluatons e+08 e+07 e+06 e Number of Varables Slope=.807 Fgure : A polynomal ncrease n functon evaluatons wth number of decson varables for the Rastrgn s functon usng proposed algorthm. noteworthy that the proposed procedure s able to overcome exponentally many local mnma to converge to the globally mnmum soluton wth polynomally ncreasng number of functon evaluatons to as complex as a 200-varable Rastrgn s functon. 5. SELF-ADAPTIVE SBX FOR MULTI-OBJECTIVE OPTIMIZATION Lke the way we made the SBX operator self-adaptve for sngle-objectve optmzaton, we extend the dea here for mult-objectve optmzaton. The man dffculty arses n decdng when a chld soluton s better than a parent. Here, we smply use the dea of non-domnaton to decde on ths matter. Let us consder Fgure 2. For the two parent objectve vectors shown n the fgure, f a chld les on the non-domnated shaded regon (marked wth A ), we call that the chld to be better than the parents and we use the η c update equatons descrbed earler. On the other hand, f the chld les on the regon marked as B n the fgure, we call that the chld s worse than the parents and we use the approprate equaton (descrbed earler) to update η c. If, however, the chld les on the regon marked as C n thefgure,wedonotupdateη c. The remanng part of the NSGA-II algorthm [3] s used as usual

8 000 f2 B parents C A f Fgure 2: A sketch showng dfferent regons n whch the chld may le for an approprate update of η c. 6. SIMULATION RESULTS Here, we show the workng of NSGA-II wth the selfadaptve SBX on three test problems: 30-varable ZDT and ZDT2 and 2-varable DTLZ2 problems. 6. ZDT, ZDT2 and DTLZ2 Problems We use the hyper-volume measure to ndcate the performance of a procedure. We compare the performance of the self-adaptve procedure wth NSGA-II havng the fxed-η c based SBX procedure wth standard settng: η c =5and η m = 20. We use the followng parameter settng for selfadaptve EA: populaton sze = 00, p c =0.9, p m =/30, p m = 0 and maxmum number of generatons = 500. We ntalze all populaton members wth η c = 2, as before. Table 4 shows the hyper-volume measure for varous α values wth the self-adaptve procedure. We observe that wth α values near.70, the performance of Table 4: Performance the self-adaptve NSGA- (hyper-volume) comparson II s better than of self-adaptve NSGA-II the fxed-η c based wth fxed-η c based SBX on NSGA-II. Table 4 ZDT and ZDT2. also shows the performance for 30-varable α Best Medan Worst ZDT2 problem. Here, ZDT we observe that for Orgnal, fxed η c =5 α values larger than the performance Self-Adaptve SBX of the self-adaptve NSGA-II s better For the three-objectve DTLZ2 problem, selfadaptve EA obtaned ZDT2 better hyper-volume Orgnal, fxed η c =5 values of , and , Self-Adaptve SBX compared to , and as best, medan and worst values obtaned usng orgnal fxedη c EA. 7. CONCLUSIONS In ths paper, we have suggested a self-adaptve procedure for updatng the dstrbuton ndex η c used n the smulated bnary crossover or SBX operator whch s a commonly-used real-parameter recombnaton operator. The update procedure mmcs the extenson-contracton concept n Nelder and Meade s smplex search procedure and also follows, n prncple, Rechenberg s /5-th update rule. On three sngleobjectve optmzaton problems and on three two-objectve optmzaton problems, the suggested procedure s found to be perform much better than the orgnal SBX procedure. Further nvestgatons are now needed for solvng problems havng a lnkage among varables and problems havng more than two objectves. A smlar self-adaptve dea can also be used wth other real-parameter recombnaton operators. 8. ACKNOWLEDGMENT Ths study s supported by a research grant from Honda R&D, Japan. 9. REFERENCES [] H.-G. Beyer and K. Deb. On the desred behavor of self-adaptve evolutonary algorthms. In Parallel Problem SolvngfromNatureVI(PPSN-VI), pages 59 68, [2] K.DebandR.B.Agrawal.Smulatedbnarycrossoverfor contnuous search space. Complex Systems, 9(2):5 48, 995. [3] K.Deb,S.Agrawal,A.Pratap,andT.Meyarvan.Afast and eltst mult-objectve genetc algorthm: NSGA-II. IEEE Transactons on Evolutonary Computaton, 6(2):82 97, [4] K. Deb, A. Anand, and D. Josh. A computatonally effcent evolutonary algorthm for real-parameter optmzaton. Evolutonary Computaton Journal, 0(4):37 395, [5] K. Deb and H.-G. Beyer. Self-adaptaton n real-parameter genetc algorthms wth smulated bnary crossover. In Proceedngs of the Genetc and Evolutonary Computaton Conference (GECCO-99), pages 72 79, 999. [6] K. Deb and A. Kumar. Real-coded genetc algorthms wth smulated bnary crossover: Studes on mult-modal and mult-objectve problems. Complex Systems, 9(6):43 454, 995. [7] L. J. Eshelman and J. D. Schaffer. Real-coded genetc algorthms and nterval-schemata. In Foundatons of Genetc Algorthms 2 (FOGA-2), pages , 993. [8] D.E.Goldberg,K.Deb,andJ.H.Clark.Genetc algorthms, nose, and the szng of populatons. Complex Systems, 6(4): , 992. [9] T. Hguch, S. Tsutsu, and M. Yamamura. Theoretcal analyss of smplex crossover for real-coded genetc algorthms. In Parallel Problem Solvng from Nature (PPSN-VI), pages , [0] J. A. Nelder and R. Mead. A smplex method for functon mnmzaton. Computer Journal, 7:308 33, 965. [] I. Ono and S. Kobayash. A real-coded genetc algorthm for functon optmzaton usng unmodal normal dstrbuton crossover. In Proceedngs of the Seventh Internatonal Conference on Genetc Algorthms (ICGA-7), pages , 997. [2] K. V. Prce, R. Storn, and J. Lampnen. Dfferental Evoluton: A Practcal Approach to Global Optmzaton. Sprnger-Verlag, Berln, [3] I. Rechenberg. Evolutonsstratege: Optmerung Technscher Systeme nach Prnzpen der Bologschen Evoluton. Stuttgart: Frommann-Holzboog Verlag,