Solving constrained optimization problems via Subset Simulation

Transcription

1 Solvng constraned optmzaton problems va Subset Smulaton Hong-Shuang L and Su-Ku Au Department of Buldng and Constructon, Cty Unversty of Hong Kong 83 Tat Chee Avenue, Kowloon, Hong Kong e-mal: sukuau@ctyu.edu.hk Abstract: Ths paper extends the applcaton of Subset Smulaton (SS), an advanced Monte Carlo algorthm for relablty analyss, to solve constraned optmzaton problems encountered n engneerng. The proposed algorthm s based on the dea that an extreme event (optmzaton problem) can be consdered as a rare event (relablty problem). The Subset Smulaton algorthm for optmzaton s a populaton-based stochastc global optmzaton approach realzed wth Markov Chan Monte Carlo and a smple evolutonary strategy, and so t does not requre ntal guess or gradent nformaton. The constrants are handled by a prorty-based ftness functon accordng to ther degree of volaton. Based on ths constrant ftness functon, a double-crteron sortng algorthm s used to guarantee that the feasble solutons are gven hgher prorty over the nfeasble ones. Four well studed constraned engneerng desgn problems n the lterature are studed to nvestgate the effcency and robustness of the proposed method. Comparson s made wth other well-known stochastc optmzaton algorthms, such as genetc algorthm, partcle swarm optmzaton and evolutonary strategy. Keywords: Subset Smulaton, constraned optmzaton, feasblty-based rule, Markov Chan Monte Carlo. Introducton Many constraned optmzaton problems arse from modern engneerng desgn process, wth systems often modeled wth nonlnear, multmodal or even dscontnuous behavor. They are complex and dffcult to solve by tradtonal optmzaton methods (Ravndran, Ragsdell, & Reklats, 006). Stochastc search algorthms have receved ncreasng attenton and have been successfully appled n many research felds n recently years (Spall, 003). Genetc Algorthm (GA) (Holland, 975) and Smulated Snnealng (SA) (Krkpatrck, Gelatt, & Vecch, 983) are amongst the most two successful heurstc technques. Other stochastc optmzaton algorthms nclude Ant Colony Optmzaton (ACO) (Manezzo, Dorgo, & Colorn, 996), Partcle Swarm Optmzaton (PSO) (Eberhart & Kennedy, 995; Kennedy & Eberhart, 995) etc. It s commonly beleved that there s no sngle unversal method capable of solvng all knds of global optmzaton problems effcently because each method has ts own defnton and heurstcs to mprove effcency. Subset Smulaton (SS) (Au & Beck, 00; Au & Beck, 003; Au, Chng, & Beck, 007) s an effcent Monte Carlo technque orgnally developed for structural relablty problems. The method takes advantage of Markov Chan Monte Carlo (MCMC) (Robert & Casella, 004) and a smple evolutonary strategy (Schuëller, 009). Based on the dea that an extreme event (optmzaton problem) can be consdered as a rare event (relablty problem), the am of ths paper s to explore applcaton of Subset Smulaton to solve constraned optmzaton problems. Motvated by Dong et al. (005), a modfed feasblty based rule s 4th Internatonal Workshop on Relable Engneerng Computng (REC 00) Edted by Mchael Beer, Raf L. Muhanna and Robert L. Mullen Copyrght 00 Professonal Actvtes Centre, Natonal Unversty of Sngapore. ISBN: Publshed by Research Publshng Servces. do:0.3850/

2 Hong-Shuang L and Su-Ku Au presented to handle general constrants, whch segregates feasble and nfeasble solutons and optmzes of the objectve functon smultaneously n a natural way. Smulaton results of four well studed constraned engneerng desgn problems llustrate the robustness and effcency of the proposed method.. Connecton between optmzaton problem and relablty problem Consder the constraned optmzaton problem formulated as max hx s.t. g x0,,, L, xs () n where h: R R s the objectve functon; g : n R R,, L s the th nequalty constrant, L s n the number of nequalty constrants; and S R s the defnton doman or the search space. To vew an optmzaton problem n the context of a relablty problem, consder the nput varables x as random varables so that the objectve functon hx s now random and has ts own probablty densty functon (PDF), and cumulatve dstrbuton functon (CDF). Ths concept of augmentng determnstc varables as random ones s merely a computatonal technque for solvng complex problems takng advantage of Monte Carlo Smulaton (Au 005). From probablty theory, every CDF s monotone, non-decreasng and rghtcontnuous. Let h opt be the global maxmum of h, where x x opt. By defnton the CDF value at h hopt s unty. Consder the relablty problem of fndng the falure probablty P F defned as PF PF P( hx hopt ) () where the falure event s F hx h opt. Clearly, ths s zero because h opt s the global maxmum. In the context of an optmzaton problem, the falure probablty s of lttle concern, however; the attenton s on the pont or regon where the objectve functon attans the largest value(s). The rare falure regon n the relablty problem corresponds to the regon where the objectve functon attans ts global maxmum. Ths suggests the possblty of convertng a global optmzaton problem nto a rare event smulaton problem, where the feasble potentally optmal solutons are analogously the rare event samples close to falure n the relablty problem. The rare event smulaton problem s n turn handled by Subset Smulaton. 3. The proposed methodology Along the same sprt of Subset Smulaton for relablty analyss (Au & Beck, 00; Au & Beck, 003; Au et al., 007), Phx h opt can be expressed as a product of a sequence of condtonal probabltes. Durng Subset Smulaton a seres of ntermedate threshold values h :,, are generated that correspond to boundares consstent wth the specfed condtonal probabltes p : k, p Ph ( x h) P F (3) k 440 4th Internatonal Workshop on Relable Engneerng Computng (REC 00)

3 Solvng constraned optmzaton problems va Subset Smulaton p Ph ( x h h x h ) P F F, (4) where F s the ntermedate event determned by the objectve functon value h. By the defnton of condtonal probablty, we have P p (5) F We can generate samples (solutons) usng the modfed Metropols-Hastng algorthm (Au & Beck, 00) that progressvely towards the maxmum, whle the rare event regon s gradually explored. For an optmzaton problem wth at least one global pont, one can expect that h hopt as PF CONSTRAINT HANDLING One central problem for applyng stochastc optmzaton technques to the constraned optmzaton problem s how to handle dfferent types of constrants. A lot of research has been devoted to handlng constrants n Genetc Algorthm (GA) as an mportant step n the desgn process (Coello Coello, 00a; Mchalewcz, 996). Smulated Annealng (SA), on the other hand, has not yet ganed smlar popularty because t requres specal strateges to mantan dversty (Hedar & Fukushma, 006). That s, when the feasble doman conssts of several dscontnuous sub-feasble domans, SA may encounter dffculty n explorng the whole feasble doman. Smlar to SA, Subset Smulaton s also bult on Markov Chan Monte Carlo to generate canddate samples (solutons). However, SA s a typcally pont-to-pont algorthm, whle Subset Smulaton operates a populaton of samples. To a certan extent, Subset Smulaton may be regard as a populaton-based SA wth a smple evolutonary strategy for relablty problem. In ths work, we borrow constrant handlng technques n populaton-based stochastc optmzaton algorthms to develop approprate strateges for Subset Smulaton. The exstng constrant handlng technques n the lterature can be roughly classfed nto four groups: () rejecton method; () penalty functon method; (3) mult-objectve functon method; (4) specfed strategy. In the rejecton method, only the feasble solutons are kept n the search process and nfeasble solutons are dscarded. It s dffcult to approach the feasble regon f the feasble regon n the search space s comparatvely small. By constructon, the rejecton method cannot explore the nfeasble regon. In contrast, the penalty functon method explots also the nfeasble solutons n the searchng process by transformng a constraned problem nto an unconstraned one through the ncorporaton of a penalty functon nto the objectve functon. The man dsadvantage of ths technque s the dffculty n the selecton of the penalty functon because t s problem- and scale-dependent (Coello Coello, 00b). Multobjectve optmzaton concept has recently been adapted to handle constrants n GA (Mezura-Montes & Coello Coello, 006), whch converts a sngle objectve optmzaton problem nto a mult-objectve one and then apples mult-objectve technques to solve t. Ths strategy by-passes the selecton of penalty functon. There s, however, one major dfference n the optmzaton am between mult-objectve optmzaton and handlng constrants usng mult-objectve optmzaton concept. The former ams at fndng a Pareto optmal set n the searchng space, whle the later requres the assocated mult-objectve optmzaton problem to degenerate nto a sngle-objectve one n the feasble regon agan because the optmal value s ganed at one pont nstead of a pont set. In the forth group, the specfed strateges for dfferent heurstc algorthms have been developed, such as gene repar n GA and fly-back n PSO. However, these methods lack generalty. 4th Internatonal Workshop on Relable Engneerng Computng (REC 00) 44

4 Hong-Shuang L and Su-Ku Au We adopt the constrant ftness prorty based rankng method frst proposed by Dong et al (005) to solve constrants n PSO. The method automatcally takes care of the constrants durng samples generaton and so t by-passes the need to modfy the objectve functon. In the context of the proposed algorthm, a new constrant ftness functon for handlng constrants s proposed n ths work. A constrant volaton functon s frst defned to handle constrants and rankng samples. For an nequalty constrant g x 0, t s defned as 0 f g 0 x (6) gx f g 0 The overall constrant ftness functon that accounts for all constrants s defned as: F x max (7) It s clear that F 0 con con x x x f and only f the tral soluton x belongs to the feasble doman. The expresson n Eq.(7) reveals that all solutons n the feasble doman have the same value of Fcon x (equal to zero) whle the worst volated constrant domnates the volaton level for an nfeasble soluton. It should be noted that the constrant ftness functon used here s dfferent from that used n Dong et al (005). The real volaton s used to reflect the magntude of volaton n ths work, whle a relatve measure was adopted n Dong et al (005). Another dfference s that the most serous volaton among all constrants s used to construct constrant ftness functon nstead of a sum of relatve rato wth random weghted factors. In the proposed algorthm for optmzaton problems, samples wth favorable values are selected from the populaton at the current level to provde seeds for the next smulaton level. The selecton process needs to consder the constrant ftness functon and objectve functon smultaneously. A double-crteron rankng method s used for ths purpose (Dong etal, 005). Specfcally, the samples are frst sorted accordng to ther constrant ftness functon values F x F x F x (8) con con con N Those samples that satsfy the constrants wll appear at the top of the lst wth the same value of equal to zero. These samples are then sorted agan accordng to ther objectve functon value. The frst rankng based on the constrant ftness functon s desgned to search the feasble doman, whle the second rankng based on objectve functon looks for the optmal soluton. In ths manner the searchng processes for the feasble regon and the optmal soluton proceed together. Ths method preserves the advantage that the constrant ftness functon value of a feasble soluton s always better than that of an nfeasble one and t by-passes the need to trade off between the objectve and penalty functon (Dong et al, 005), whch can be nontrval due to dfferent scalng. 3.. PROCEDURE The proposed algorthm s descrbed as follows: Step 0 Select the dstrbuton parameters of nput varables. In the orgnal problem, each nput varable s Fcon x 44 4th Internatonal Workshop on Relable Engneerng Computng (REC 00)

5 Solvng constraned optmzaton problems va Subset Smulaton determnstc parameter but t s augmented as a random varable n the presented algorthm wth an artfcal probablty densty functons (PDF) f x. Step Generate N ndependent and dentcally dstrbuted samples x, x,, xn by drect Monte Carlo Smulaton accordng to the artfcal dstrbutons. Each sample x,, N has n components,.e. T n j x x, x,, x where x ( j,, n) are generated from f jxj( j,, n). Calculate the constrant ftness functon values and the objectve functon values of the samples, and then sort them accordng to the double-crteron rankng algorthm descrbed n Secton 3.. Refer x N as the best soluton and x as the worst. From the sequence, obtan the N p th sample x N p wth correspondng h, N p and Fcon x N p, assumng that p and N are chosen such that N p s an nteger. The subscrpt here denotes smulaton level. Note that the sample estmate of PF : hh F, N p con Fcon x s by defnton p N p. In ths manner, the event F s adaptvely chosen. Due to the choce of x N p, there are Np samples whose objectve functon values are large than h, N p and constrant ftness functon values are large than Fcon x N p among all samples, and the correspondng samples belong to the event F. These samples provde the seeds samples for next smulaton level. Step The modfed Metropols-Hastng algorthm (Au & Beck, 00; Au & Beck, 003) s employed for generatng condtonal samples. In the k-th smulaton level, startng from each of samples n Fk, a Markov chan can be generated wth the same condtonng. Snce the ntal samples obey the condtonal dstrbuton, all these Markov chans are automatcally n statonary state and samples n these Markov chans dstrbute accordng to condtonal dstrbuton. Note that the length of each Markov Chan s / pk k, where pk s the level probablty n last smulaton level. Evaluate and sort the samples agan, determne the N pk th sample xn pk from the sequence. Agan, select the samples whose objectve functon values are larger than hkn, pk and constrant ftness functon values are larger than Fcon xn p to provde seeds for samplng n the next smulaton level. k The procedure s repeated for hgher smulaton levels untl a convergence crteron s met. Note that the m total number of samples s equal to N pk N where m s the total smulaton levels for a problem. k 4th Internatonal Workshop on Relable Engneerng Computng (REC 00) 443

6 Hong-Shuang L and Su-Ku Au 4.. ARTFICIAL PDFS FOR INPUT VARIABLES 4. Computatonal ssues The choce of the dstrbuton for the nput varables drectly affects the doman where feasble solutons are searched. A truncated Gaussan dstrbuton may be used to handle smple bound constrants on ndvdual desgn varable x f x;,, xu, xl (9) xu xl where s the probablty densty functon of the standard Gaussan dstrbuton, cumulatve dstrbuton functon of the standard Gaussan dstrbuton and the defnton doman x x x x. The mean should be chosen close to the global optmum. If no pror nformaton : l u s the on the problem s avalable, one may locate t at the center of the defnton doman. The standard devaton of the artfcal dstrbuton controls the range to be explored and t has an nfluence on the effcency. If t s too small, most of samples wll cluster n a small regon and then the sequence of objectve functon wll ncrease slowly. If t s too large, the samples wll scatter over a large regon and t would requre a longer process to converge to the global optmum. One strategy s to use the three sgma lmts n relablty engneerng, settng the dstance from samplng center to the upper or lower bound equal to three standard devatons,.e. L (0) 6 where L s the nterval length of defnton doman of the th nput varable x, and s the correspondng artfcally standard devaton. 4.. CONVERGENCE CRITERION In ths paper, we use the standard devaton of samples n each smulaton level to check the convergence of the searchng process. In order to elmnate the effects of dfferent scalng of desgn varables, the nterval length of defnton doman s used as a reference. The convergence crteron s k () x x u l where k s the estmator of standard devaton of samples n k-th smulaton level, and s a specfed tolerance. The dea of ths crteron stems from the fact that when the searchng procedure approaches the global optmum, more repeated samples would be found n samples and then the estmator of standard devaton of samples wll tend to zero. Here, we adopt =0-5 ~ th Internatonal Workshop on Relable Engneerng Computng (REC 00)

7 Solvng constraned optmzaton problems va Subset Smulaton When the convergence crteron s satsfed, the largest value n the objectve functon sequence s taken as the optmal value, and the correspondng sample as the optmal soluton LEVEL PROBABILITY The level probablty p k s a parameter that regulates the convergence of the optmzaton process. If a small value s used, the algorthm would have a low probablty of reachng a global optmum. The level probabltes must be hgh enough to permt the locally developed Markov Chan samples to move out of a local optmum n favor of fndng a global optmal, especally n early smulaton levels. However, hgh level probabltes would ncrease the number of smulaton levels. Here, we adopt a decreasng strategy to handle ths dffculty. Frst, we set p 0.5 at ntal smulaton level and then reduce to 0. n the (+)th smulaton level when the largest estmator of all desgn varables 0. when j 0.0 n order to accumulate the convergent speed. s less than 0., and further reduce to 5. Applcaton examples The proposed algorthm s appled to four well-studed engneerng desgn problems wth dfferent knds of constrants. In each example, 30 ndependent runs are carred out to nvestgate the performance of the algorthm n a statstcal sense. 5.. WELDED BEAM DESIGN The frst problem consders the optmal desgn of a welded beam, whch s taken from Coello Coello h x ncludng setup cost, weldng labor cost (000). The objectve s to mnmze the total cost functon and materal cost wth constrants on shear stress x, bendng stress n the beam on the bar P x, end deflecton of the beam x, sde constrants g, g, g x, bucklng load c on desgn varables. There are four desgn varables as shown n Fgure : x, x, x, x h,,, l t b x x x, and lmts th Internatonal Workshop on Relable Engneerng Computng (REC 00) 445

8 Hong-Shuang L and Su-Ku Au h P l t L b Fgure The welded beam desgn problem The problem can be formulated as mn h x.047x x 0.048x x 4.0 x () Subject to where x 0 x x g x x (3) max 0 g x x (4) max 0 g x x x (5) g x 0.047x 0.048x x 4.0 x (6) g x (7) 5 g6 max 0 (8) g7 x P P c x 0 (9) 0. x, 0. x 0, 0. x 0, 0. x (0) 3 4 ' ' '' '' x () ' P '' MR x,, M P L xx J R x x x3 4 J x x x x x 3 () (3) (4) 446 4th Internatonal Workshop on Relable Engneerng Computng (REC 00)

9 Solvng constraned optmzaton problems va Subset Smulaton 3 6PL 4PL x, x (5) x x Ex x E x3x4 36 x3 E Pc x (6) L L 4 G 6 6 P 6000lb, L4n, E 300 ps, G 0 ps (7) max 3600ps, max 30000ps, max 0.5n (8) Ths problem has been attempted prevously by several stochastc optmzaton algorthms: GA based on a co-evoluton model (GA) (Coello Coello, 000), GA through the use of domnance-based tournament selecton (GA) (Coello Coello & Montes, 00), evolutonary programmng wth a cultural algorthm (EP) (Coello Coello & Becerra, 004), co-evolutonary partcle swarm optmzaton (CPSO) (He & Wang, 007a), hybrd partcle swarm optmzaton (HPSO) wth a feasblty-based rule (He & Wang, 007b) and hybrd Nelder Mead smplex search method and partcle swarm optmzaton (NM PSO) (Zahara & Kao, 009). Ther best solutons are compared wth that obtaned by the proposed algorthm, and are lsted n Table I. It should be noted that the result produced by NM-PSO (Zahara & Kao, 009) s an nfeasble soluton because the thrd constrant g3 x had been slghtly volated. From Table I, the best feasble soluton obtaned by Subset Smulaton s compettve to the results produced by EP (Coello Coello & Becerra, 004) and HPSO (He & Wang, 007b) and s better than the results obtaned by other optmzaton technques. Table II summares the average optmum-locatng performance and computatonal effort spent by dfferent methods over 30 ndependent runs. It can be seen that the mean of the objectve functon yelded by Subset Smulaton s better that of other algorthms although the worst case s some what average among others. Based on 30 ndependent runs, the average teraton number was and the average number of functon evaluatons was 83,703. Comparng wth the functon evaluatons requred by other methods the proposed algorthm can be an effcent choce for ths problem. Table I Comparson of the best soluton for welded beam desgn problem by dfferent methods Best soluton GA GA EP CPSO HPSO NM-PSO SS x x x x x g g g g g g h th Internatonal Workshop on Relable Engneerng Computng (REC 00) 447

10 Hong-Shuang L and Su-Ku Au Table II Statstcal performance of dfferent methods for welded beam desgn problem Methods Best Mean Worst Std. Functon evaluatons GA ,000 GA ,000 EP N/A CPSO ,000 HPSO ,000 SS , TENSION-COMPRESSION STRING DESIGN Consder the tenson-compresson strng desgn problem taken agan from Coello Coello (000), as shown n Fgure. The objectve s to mnmze the strng weght under constrants on deflecton, shear stress, surge frequency, lmts on outsde dameter and on desgn varables. There are three desgn varables n ths problem: the wre dameter d, the mean col dameter D and the number of actve cols P,.e. x, x, x d, D, P. 3 P P D Fgure The tenson-compresson strng desgn problem d The problem can be formulated as mn hx x3 xx (9) Subject to 3 xx 3 g x 0 4 (30) 7785x g g xx 508 x x 4x x x x 0 (3) 3 xx x x 0 (3) x x g4 x 0.5 (33) 0.05 x.0, 0.5 x.3,.0 x 5.0 (34) 3 Ths problem has been attempted prevously by GA, GA, EP, CPSO, HPSO, and NM PSO. A comparson of best solutons obtaned by these methods and Subset Smulaton s shown n Table III. The result produced by NM-PSO s nfeasble because the frst and second constrants are volated by a relatve 448 4th Internatonal Workshop on Relable Engneerng Computng (REC 00)

11 Solvng constraned optmzaton problems va Subset Smulaton 3 magntude of 0. As shown n Table III, HPSO gves the most favorable soluton, although Subset Smulaton s stll compettve compared to other methods. The statstcal results of 30 ndependent runs are lsted n Table IV. It can be seen that Subset Smulaton also produced compettve results to other compared methods. Table III Comparson of tenson-compresson strng desgn problem by dfferent methods Methods x x x 3 h GA GA EP CPSO HPSO NM-PSO SS On the other hand, the mean of the teraton number for Subset Smulaton was 69 and the average number of functon evaluatons was only 40,07. Consderng both qualtes of soluton and computatonal effort spent, the proposed method s qute favorably. Table IV Statstcal performance of dfferent methods for tenson-compresson strng desgn problem Methods Best Mean Worst Std. Functon evaluatons GA e ,000 GA e ,000 EP e-004 N/A CPSO e ,000 HPSO e-005 8,000 SS e , AN OPTIMIZATION PROBLEM WITH DISJOINT FEASIBLE REGION The search space of ths problem has 9 3 dsjont spheres, and t can be stated as follows (Coello Coello & Montes, 00): max h x 0.0x 5 x 5 x3 5 (35) Subject to gx x x j x3 k (36) where 0 x, x, x3 0 and, j, k,,,9. The optmal soluton of ths problem s f x wth x { , , }. Table V shows the statstcal results obtaned by EP, GA, HPSO, NM-PSO, Subset Smulaton and the homomorphous mappng (HM) (Kozel & Mchalewcz, 999), the stochastc rankng (SR) (Runarsson & Yao, 000). GA, HPSO, NM-PSO, SR and Subset Smulaton are very robust n reachng the optmal soluton for ths problem, showng low varablty of results. Wth respect to effcency, NM-SPO s the clear wnner as t needs only 93 functon evaluatons. The functon evaluatons requred by other methods 4th Internatonal Workshop on Relable Engneerng Computng (REC 00) 449

12 Hong-Shuang L and Su-Ku Au are also summarzed n Table V. For ths problem, the average teraton umber for Subset Smulaton was 34 and the average number of functon evaluatons was,093. Ths s better than other methods except NM-PSO. Note, however, that NM-PSO requres the explct gradent nformaton of constrants to repar nfeasble solutons, whch may not be avalable n realstc applcatons. Table V Statstcal performance of dfferent methods for optmzaton problem wth dsjont feasble regon Methods Best Mean Worst Std. Functon evaluatons HM N/A 40,000 SR e ,000 GA e ,000 EP e-003 N/A HPSO e-05 8,000 NM-PSO e SS e-06, TEN-BAR PLANE TRUSS DESIGN The ten-bar plane truss shown n Fgure 3 s taken from Coello Coello & Montes (00) as an example of structural optmzaton wth mplct constrants. The modulus of elastcty E s.004ks ( MPa) whle the mass densty s 0.0lb/n 3 ( kg/m 3 ). The structure s desgned for a sngle loadng condton: 00kps ( kg) applyng n the negatve y-drecton at nodes and n 360n n 6 P 4 P Fgure 3 The ten-bar plane truss desgn problem The objectve s to mnmze the weght of the truss consderng stress and dsplacement related constrants. The objectve functon s gven by h 0 x AjLj (37) j 450 4th Internatonal Workshop on Relable Engneerng Computng (REC 00)

13 where x s the canddate soluton, second moment of area of bar j), Solvng constraned optmzaton problems va Subset Smulaton A s the cross-sectonal area of the j-th bar ( Aj I j j j, where I j s the L s the length of j-th bar. The second moment of area I, j,,,0 are taken as desgn varables wth a range of 0. I j n 4. There are 0 stress-related constrants and 8 nodal dsplacement-related constrants. The allowable stress of each bar s 5ks (7.4MPa). The dsplacement of free nodes -4 n both drecton x and y must be less than n (5.08cm). Ths problem has been attempted by GA (Coello Coello & Montes, 00), and ts best solutons s compared aganst those obtaned by Subset Smulaton n Table VI. From Table VI, t s can be seen that a better optmal soluton of f lb has been found by Subset Smulaton. Table VII shows the statstcal results based on 30 ndependent runs. From Table VII, t can be seen that the average searchng qualty of Subset Smulaton s also superor to those of GA for ths problem. Moreover, GA needed 80,000 functon evaluatons for ths problem, whle Subset Smulaton only needs 58,74 functon evaluatons. Table VI Comparson of best solutons obtaned by GA and Subset Smulaton Best soluton GA SS x x x x x x x x x x h j Table VII Statstcal performance of dfferent methods for ten-bar structure Methods Best Mean Worst Std. Functon evaluatons GA ,000 SS ,74 6. Conclusons Ths paper descrbes a new applcaton of Subset Smulaton for optmzng engneerng desgn problems under general constrants. The proposed algorthm s based on the dea that extreme events (optmzaton problems) can be consdered rare events (relablty problems). A feasblty-based rule s used to guarantee that feasble solutons are always better than nfeasble solutons n term of constrant ftness functon values. The rule employs a modfed constrant ftness functon to evaluate the constrant ftness of a soluton and a double-crteron rankng method to select the best solutons accordng to both constrant 4th Internatonal Workshop on Relable Engneerng Computng (REC 00) 45

14 Hong-Shuang L and Su-Ku Au ftness functon values and objectve functon values. Four benchmarks of engneerng desgn are shown to demonstrate the robustness and effcency of the proposed algorthm compared wth exstng ones. The proposed algorthm s found to be compettve n explotng the feasble regons and provdng optmal desgns n complex problems. Current research focuses on further testng ts performance on structural optmzaton desgn and mprovng ts effcency by combnng wth local search strateges. Acknowledgements The work presented n ths paper s supported by the Hong Kong Research Grant Councl (HKRGC) through General Research Fund (GRF) (Project No ). The support s gratefully acknowledged. References Au, S. K. Relablty-based desgn senstvty by effcent smulaton. Computers and Structures, 83:048 06, 005. Au, S. K., and J. L. Beck. Estmaton of small falure probabltes n hgh dmensons by subset smulaton. Probablstc Engneerng Mechancs, 6(4), 63-77, 00. Au, S. K., and J. L. Beck. Subset smulaton and ts applcaton to sesmc rsk based on dynamc analyss. Journal of Engneerng Mechancs, 9(8), 90-97, 003. Au, S. K., J. Chng, and J. L. Beck. Applcaton of subset smulaton methods to relablty benchmark problems. Structural Safety, 9(3), 83-93, 007. Coello Coello, C. A. Use of a self-adaptve penalty approach for engneerng optmzaton problems. Computers n Industry, 4(), Coello Coello, C. A. Theoretcal and numercal constrant-handlng technques used wth evolutonary algorthms: A survey of the state of the art. Computer Methods n Appled Mechancs and Engneerng, 9(-), 45-87, 00a. Coello Coello, C. A. Theoretcal and numercal constrant-handlng technques used wth evolutonary algorthms: A survey of the state of the art. Computer Methods n Appled Mechancs and Engneerng, 9(-), b. Coello Coello, C. A., and R. L. Becerra. Effcent evolutonary optmzaton through the use of a cultural algorthm. Engneerng Optmzaton, 36(), 9-36, 004. Coello Coello, C. A., and E. M. Montes. Constrant-handlng n genetc algorthms through the use of domnancebased tournament selecton. Advanced Engneerng Informatcs, 6(3), 93-03, 00. Dong, Y., et al. An applcaton of swarm optmzaton to nonlnear programmng. Computers and Mathematcs wth Applcatons, 49(-), , 005. Eberhart, R. C., and J. Kennedy. A new optmzer usng partcle swarm theory. Proceedngs of the Sxth Internatonal Symposum on Mcro Machne and Human Scence, 995. He, Q., and L. Wang. An effectve co-evolutonary partcle swarm optmzaton for constraned engneerng desgn problems. Engneerng Applcatons of Artfcal Intellgence, 0(), 89-99, 007a. He, Q., and L. Wang. A hybrd partcle swarm optmzaton wth a feasblty-based rule for constraned optmzaton. Appled Mathematcs and Computaton, 86(), 407-4, 007b. 45 4th Internatonal Workshop on Relable Engneerng Computng (REC 00)

15 Solvng constraned optmzaton problems va Subset Smulaton Hedar, A. R, and M. Fukushma. Dervatve-free flter smulated annealng method for constraned contnuous global optmzaton. Journal of Global Optmzaton, 35(4), 5-549, 006. Holland, J. H. Adaptaton n natural and artfcal systems. Ann Arbor, Mchgan: Unversty of Mchgan Press, 975. Kennedy, J., and R. C. Eberhart. Partcle swarm optmzaton. Proceedngs of IEEE Internatonal Conference on Neural Networks, 4, , 995. Krkpatrck, S., C. Gelatt, and M. Vecch. Optmzaton by smulated annealng. Scence, 0(4598), , 983. Kozel, S., and Z. Mchalewcz. Evolutonary algorthms, homomorphous mappngs, and constraned parameter optmzaton. Evolutonary Computaton, 7(), 9-44, 999. Manezzo, V., M. Dorgo, and N. Colorn. The ant system: Optmzaton by a colony of cooperatng agents. IEEE Transactons on Systems, Man and Cybernetcs - Part B, 6(), 9-4, 996. Mezura-Montes, E., and C. A. Coello Coello. Use of multobjectve optmzaton concepts to handle constrants n genetc Algorthms. In A. Abraham, L. Jan & R. Goldberg (Eds.), Evolutonary multobjectve optmzaton, Sprnger, 006. Mchalewcz, Z. Evolutonary algorthms for constraned parameter optmzaton problems. Evolutonary Computaton, 4(), -3,996. Ravndran, A., K. M. Ragsdell, and G. V. Reklats. Engneerng optmzaton: Methods and applcatons (nd Edton ed). New Jersey: John Wley & Sons, 006. Robert, C. P., and G. Casella. Monte carlo statstcal merhods (Second Edton ed.). New York: Sprnger, 004. Runarsson, T. P., and X. Yao. Stochastc rankng for constraned evolutonary optmzaton. IEEE Transactons on Evolutonary Computaton, 4(3), 84-94, 000. Schuëller, G. I. Effcent Monte Carlo smulaton procedures n structural uncertanty and relablty analyss - recent advances. Structural Engneerng and Mechancs, 3(), -0, 009. Spall, J. C. Introducton to stochastc search and optmzaton: Estmaton, smulaton, and control. New York: Wley, 003 Zahara, E., and Y. T. Kao. Hybrd Nelder-mead smplex search and partcle swarm optmzaton for constraned engneerng desgn problems. Expert Systems wth Applcatons, 36( PART ), , th Internatonal Workshop on Relable Engneerng Computng (REC 00) 453