An Efficient Pareto Set Identification Approach for Multi-objective Optimization on Black-box Functions

Transcription

1 . Abstract An Effcent Pareto Set Identfcaton Approach for Mult-objectve Optmzaton on Black-box Functons Songqng Shan G. Gary Wang Both multple objectves and computaton-ntensve black-box functons often exst smultaneously n engneerng desgn problems. Few of exstng mult-objectve optmzaton approaches addresses problems wth expensve black-box functons. In ths paper, a new method called the Pareto set pursung (PSP) method s developed. By developng samplng gudance functons based on approxmaton models, ths approach progressvely provdes a desgner wth a rch and evenly dstrbuted set of Pareto optmal ponts. Ths work descrbes PSP procedures n detal. From testng and desgn applcaton, PSP demonstrates consderable promses n effcency, accuracy, and robustness. Propertes of PSP and dfferences between PSP and other approxmaton-based methods are also dscussed. It s beleved that PSP has a great potental to be a practcal tool for mult-objectve optmzaton problems. Key words: Black-box functon, mult-objectve optmzaton, Pareto set, Pareto set pursung, samplng gudance functon. Introducton Modern desgn problems often nvolve multple objectves and are thus treated as mult-objectve optmzaton (MOO) problems. In general mult-objectve optmzaton (MOO) methods, excludng methods dealng wth uncertantes n MOO, can be roughly categorzed nto two man classes. The frst class of methods converts a MOO problem nto a sngle-objectve problem. Such converson s Correspondng author, Dept. of Mechancal and Manufacturng Engneerng, The Unversty of Mantoba, Wnnpeg, MB, Canada RT 5V6, Tel: , Fax: , Emal: gary_wang@umantoba.ca

2 usually acheved by usng ether explct or mplct weghts, preferences, utltes, or targets. These methods requre a pror selecton of weghts, preferences, or utltes for each of the objectve functons [-]. It can be very dffcult to decde whch weghtng factors should be used n practce for real problems. In addton, these weghts may be nadequate n capturng decson makers preferences. Ths class of methods provdes nformaton for only one desgn scenaro and lacks of a rgorous method for the weght selecton. It s also found that the weghted-sum method s unable to fnd Pareto ponts n non-convex regons n the performance space []. Ths class of methods s recently mproved by the physcal programmng method [4-6]. The second class of methods tres to dentfy a set of dscrete ponts as an approxmaton of the Pareto optmal fronter for decson makers wthout pre-assumng ther preferences. The most successful and wdely used approaches n ths category seem to be evolutonary algorthms [7-]. These methods can generate a large number of Pareto ponts for decson makers. However, they are usually computatonally expensve because a massve number of non-pareto set ponts have to be evaluated. A recent detaled survey of MOO methods can be found n Ref. []. For MOO problems nvolvng expensve analyss and smulaton processes such as fnte element analyss (FEA) and computatonal flud dynamcs (CFD), the use of approxmaton becomes more attractve. Snce FEA and CFD processes are based on complex and numerous smultaneous equatons, these processes are often treated as black-box functons, for whch only nputs and outputs are known. Because the complexty of these expensve processes seems to mantan ts pace wth computng advances, the challenge brought by these computatonally expensve black-box functons on optmzaton should be addressed. For MOO problems nvolvng expensve black-box functons, the computatonal burden aggravates. Recent approaches to solve MOO problems wth black-box functons are to approxmate each sngle objectve functon or drectly approxmate the Pareto optmal fronter [-5]. The accuracy of the Pareto optmal fronter depends on the accuracy of approxmaton

3 models. Wlson et al. [] used the surrogate approxmaton n leu of the computatonally expensve analyses to explore the mult-objectve desgn space and dentfy Pareto optmal ponts, or the Pareto set, from the surrogate. L et al. [] used a hyper-ellpse surrogate to approxmate the Pareto optmal fronter for b-crtera convex optmzaton problems. If the approxmaton s not suffcently accurate, then the Pareto optmal fronter obtaned usng the surrogate approxmaton wll not be a good approxmaton of the actual Pareto optmal fronter. A latest work by Yang et al. [5] proposed the frst framework (as clamed by the authors) managng approxmaton models n MOO. In the framework, a GA based method s employed wth a sequentally updated approxmaton model. It dffers from [] by updatng the approxmaton model n the optmzaton process. The fdelty of the dentfed fronter solutons, however, s stll bult upon the accuracy of the approxmaton model. The work n Ref. [5] also suffers from the problems of the GA-based MOO algorthm,.e., the algorthm has dffculty n fndng fronter ponts near the extreme ponts (the mnmum obtaned by consderng only one objectve functon). Ths work proposes a Pareto set dentfcaton method by ntellgent progressve samplng. Wth no a pror preference s needed, ths method could automatcally sample near the Pareto optmal fronter and rapdly converge to the Pareto set. Before ntroducng the proposed method, some basc yet mportant concepts of MOO are revewed n the next secton. The proposed method wll be descrbed and ts propertes wll be dscussed n Secton 4. Secton 5 wll present numercal studes on a few well-known MOO problems. The proposed method s then appled to an engneerng desgn problem, whch wll be presented n Secton 6. Secton 7 wll dscuss the propertes PSP and related ssues. Secton 8 gves conclusons and future work.

4 . Basc Concepts of Mult-Objectve Optmzaton Mult-objectve Optmzaton Mult-objectve optmzaton s a vector optmzaton problem. A general mult-objectve optmzaton problem s to fnd desgn varables that optmze a vector objectve functon subject to a number of constrants and bounds. The mult-objectve optmzaton s often formulzed as follows: Mnmze F(x) = {f (x), f (x),,f (x),, f m (x)} Subject to: h j (x) = 0, j =,, q g k (x) 0, k =,, p () x l r u x x, r =,, n r r Where the components of the multple objectve functon vector, F(x) = {f (x), f (x),, f m (x)}, are usually n conflct wth one another wth respect to ther own optma. The desgn varable vector, x = l u [x, x,, x n ], conssts of all desgn varables n the problem bounded n x x x, r =,, n. r r r h j (x) are equalty constrants and g k (x) are nequalty constrants. In ths work for the ease of llustraton, we defne feasble desgn space as the set of all desgn ponts represented by desgn varables that satsfy constrants. Those ponts wthn the feasble desgn space nclusvely are called feasble desgn ponts or feasble desgns. We defne feasble performance space as a set of all objectve functon values wth respect to every feasble desgn. Pareto Set Pareto set [5] s defned as a set of ponts of Pareto optmalty. A vector of x* s Pareto optmal f there exsts no feasble vector x that would decrease some objectve functon wthout causng a smultaneous ncrease n at least one objectve functon. Mathematcally, the Pareto optmalty s defned as follows: a vector of x* s a Pareto optmum ff, for any x and, * * f ( x) f ( x ), j =,..., m; j, f ( x) f ( x ) () j j 4

5 In general for MOO, the task s to dentfy Pareto set ponts. The queston s how to judge f a pont s n the Pareto set. To address ths queston, the ftness functon s ntroduced. Ftness Functon For a gven set of desgn ponts, a ftness functon s defned as [7]: G = ))] Where, G denotes the ftness value of the th desgn; j j j l [ max(mn( f s f s, f s f s,, f sm f sm () j f sk s the scaled k th objectve functon value of the th desgn, k =,, m; and l s called the fronter exponent. In ths work, l s taken as a constant. The max n Eq. () s over all other desgns j n the set and the mn s over all the objectves. The objectves f s, f s,, f sm n the Eq. () are scaled to a range [0, ]. For example for f s, f rawf rawf s rawf,,mn = rawf,max (4),mn Where, rawf, denotes un-scaled value of the frst objectve for the th desgn; rawf, max denotes the maxmum un-scaled value of the frst objectve among all desgns; and rawf, mn denotes the mnmum un-scaled value of the frst objectve among all desgns. In case that an objectve functon s a constant, the scaled objectve functon value f s s taken as n ths work. As the ftness functon measures relaton between ponts n the performance space and all the objectve functon values are scaled to [0, ], t s not hard to fnd that for a gven set of ponts followng statements hold true: Pareto set ponts should have a ftness functon value n the range of [, ]. Non-Pareto set ponts have a ftness value n [0, ); and the hgher the ftness functon value, the closer the pont to the Pareto fronter. When Pareto set ponts are closely and evenly dstrbuted, the ftness value of all Pareto set ponts tends to be. 5

6 4. Pareto Set Pursng (PSP) Methodology The goal of PSP s to drectly sample many Pareto solutons (ponts) to approxmate the entre Pareto optmal fronter n an economcal manner for MOO problems wth computatonally expensve blackbox functons. Assumng at the frst teraton we start from random samplng for the unknown problem, t s desrable at the next teraton to sample more ponts closer to the Pareto fronter than further away. If the trend contnues, we can sample rght on or very close to the fronter. Fgure llustrates such a desred samplng scheme, whch s to be acheved by the proposed PSP method. Fgure An llustraton of the desred samplng scheme for PSP. Before we ntroduce detals of PSP, the samplng gudance functon s defned frst. How to construct a samplng gudance functon to realze the desred samplng scheme as shown n Fgure s a key to PSP. Samplng Gudance Functon As we know, there exsts many methods n Statstcs to generate a sample from a gven probablty densty functon (PDF) [6]. These methods nclude nverse transformaton, acceptance-rejecton 6

7 technque, Markov Chan Monte Carlo (MCMC), mportance samplng, and so on. A recent work s gven by Fu and Wang [7], whch can be consdered as a Cumulatve Densty Functon (CDF) nverse method. The goal of such samplng s to generate sample ponts that conform to a gven PDF,.e. more ponts n the area that has hgh probablty and fewer ponts n the area that has low probablty, as defned by the PDF. Inspred by such samplng, the authors developed a Mode Pursung Samplng (MPS) method before [8], whch used a varaton of the objectve functon to act as a PDF so that more ponts are generated n areas havng lower objectve functon value and fewer n other areas. In bref, for an expensve black-box objectve functon, MPS frst constructs an approxmaton model from a few sample ponts. It then generates a large number of ponts from the approxmaton model, sorts the ponts, and constructs a cumulatve functon analogous to CDF by addng up all the functon values before the current pont n the sorted pont set. A sample s then drawn from the pont set accordng to ths cumulatve functon. As a result, more new sample ponts are generated around the current mnmum and less n other regons n the desgn space. MPS uses Fu and Wang s method [7] to generate a sample from a PDF and s an teratve process. It s n essence a dscrmnatve samplng method wth approved convergence. To generalze the samplng dea of MPS, ths work gves the defnton of a samplng gudance functon. A samplng gudance functon of a dscrete random varable X s descrbed as a functon z ˆ( x ) = P[ X = x ] whch satsfes: () z ˆ( ) 0, for each value x of X x () () (v) representng the nature of the problem reflectng one s samplng goal, and expressng pror nformaton f used teratvely Requrement () s to ensure all of the gudance functon values are larger than 0, smlar to a PDF. Apparently, a samplng gudance functon should represent the nature of the problem and reflect one s 7

8 samplng goal. Thus requrements () and () are then stpulated. If the samplng gudance functon s to be used n an teratve process, another desrable feature of a samplng gudance functon s to express pror nformaton and be adaptve, whch s Requrement (v). One can note that unlke a PDF, the samplng gudance functon doesn t have to satsfy z ˆ( ) =. A PDF can thus be consdered as a specal samplng gudance functon. The varaton of the objectve functon n MPS [8] s also a samplng gudance functon. k = x In ths work, two types of samplng gudance functon are developed. One s for the samplng of cheap ponts from the approxmaton model of each objectve functon. The other s for the samplng towards the Pareto fronter. Detals about these two types of samplng gudance functons wll be gven n the descrpton of the PSP procedures. Procedures of the Pareto Set Pursung (PSP) Strategy The followng llustrates the procedures of PSP to dentfy the Pareto set ponts of an n-dmensonal MOO problem defned by Eq. (). Steps of the algorthm are llustrated n Fgure, along wth stepby-step explanatons. A well known MOO problem s taken from the lterature to facltate the explanaton of PSP []. Mnmze: F(x) = {f (x), f (x)} f ( x, x ) = ( x + x 7.5) + ( x x + ) / 4 f ( x, x ) = ( x ) / 4 + ( x 4) Subject to: g x, x ) =.5 ( x ) / x 0 (5) ( / g ( x, x ) = ( x x ) x x 0 0 x 5; 0 x Ths problem nvolves desgn varables (n = ) and objectve functons (m = ) wth constrants and varable bounds. Constrants are often handled n many dfferent ways n optmzaton, e.g., 8

9 penalty or Lagrange methods. They can also be treated as specal objectve functons as n [9]. A detaled survey of constrant handlng s n [0]. For the MOO problem defned n Eq. (), f there s any computaton-ntensve constrant, we assume that t can be ether treated as another objectve functon, or can be ntegrated to exstng objectve functons as a penalty or Lagrange term. To focus on PSP but not losng generalty, we concentrate on problems n whch all of the objectve functons are computatonally expensve black-box functons, whle constrants are nexpensve functons. Start. Intal Samplng & Functon Evaluaton Call. Ftness Computaton; Current Fronter Ponts Identfcaton. Objectve Functons Fttng; Samplng on Sngle Objectve Functons 4. Combnng the Sample Ponts from Last Step wth Current Fronter Ponts; Fronter Ponts Identfcaton from the Combned Pont Set (Expensve) Black-box Functons 5. Samplng Fronter Ponts 6. Functon Evaluaton for New Drawn Sample Ponts Call 7. Combnng New Drawn Sample Ponts wth Expensve Fronter Ponts from the Last Iteraton; Identfcaton of the New Fronter Set No 8.Convergent? Yes Stop Fgure Flowchart of the Pareto-Set-Pursung Approach Step : Intal Random Samplng and Expensve Functon Evaluaton. Intal sample ponts are used to buld an approxmaton model for each objectve functon. Ths work employs both quadratc polynomal fttng (QPF) and radal bass functon (RBF) fttng. QPF s the 9

10 most wdely used response surface model []. RBF demonstrates great promse n approxmaton []. Both QPF and RBF are very smple, ntutve, and easy to construct. RBF fttng passes all the expensve ponts, prevents unnecessary curvatures added to the unknown surface, and preserves the ( ) mnmum among the expensve ponts,.e., mn fˆ( x) = mn{ f ( x ), =,, k} because ts lnearty nature. Ths feature ensures that more expensve ponts are generated around the current mnmum of f(x), rather than beng based because of the approxmaton model f ˆ( x ). Comparatvely QPF fttng smoothes the sample data by a quadratc model; t s thus not absolutely loyal to the sample data and does ntroduce curvature to the data. These two samplng methods are automatcally alternated durng the samplng procedure based on a crteron that wll be explaned later. It s to be noted that PSP does not dctate the exclusve use of the two models. Moreover, the accuracy of the approxmaton model, as wll be dscussed later n Secton 7, s less mportant n PSP than ts common use n conventonal approxmaton-based optmzaton. An n-d quadratc polynomal model [] s usually expressed by Eq. (6). n n β x + β x + f ˆ ( x) = β + β x x (6) 0 = = < j j= n j j Where β, β, and β j represent regresson coeffcents, x,( n) = are desgn varables, and f ˆ( x ) s the response. RBF has many forms. Ths work uses ts smplest form,.e., usng k functon values f(x () ), f(x () ),, f(x (k) ) to ft a lnear splne functon k f ˆ ( ) ( x) = α x x, (7) = ˆ ) ( ) ( such that f ( x ) = f ( x ), =,,, k. Where α denotes the regresson coeffcents and () x represents the coordnates of the th sample pont. For both QPF and RBF models, the number of ntal random sample ponts s chosen as the mnmum number needed to buld a full quadratc approxmaton model, (n+)*(n+)/, where n denotes the number of varables. After random 0

11 samplng (n+)*(n+)/ number of ponts, expensve black-box functons are called to evaluate these sample ponts. The number of the so-far evaluated sample ponts np at the end of the frst step thus equals to (n+)*(n+)/. For the example problem, 6 ponts for objectve functons f and f are generated n random n the desgn space ( 0 x 5 and 0 x ). The ponts are evaluated expensve ponts and are plotted n the performance space n Fgure. Step : Ftness Computaton and Current Fronter Ponts Identfcaton By usng Eq. (), the ftness value of the evaluated ntal sample ponts are computed and the current fronter ponts can be dentfed accordng to ther ftness functon values. The average ftness value of all the current fronter ponts s also computed as t s used as part of the convergence crteron. It s to be noted that the current fronter s to be updated as the procedure contnues. In Fgure P, P, and P are dentfed as the current fronter ponts. Approxmaton models for both f and f are then generated by usng Eqs. (6) or (7). Fgure Intal ponts generated for the example problem wth dentfed current fronter ponts.

12 Step : Objectve Functons Fttng and Samplng on Sngle Objectve Functons The goal of ths step s to generate a large number of cheap ponts from each approxmaton model ndependently, n preparaton for the samplng of the fronter ponts n the performance space. Frst we use the np evaluated functon values f (x () ), f (x () ),, f (x (np) ) ( =,,, m) to gan the coeffcents of the approxmaton model ˆ ( x), a n-d quadratc polynomal model Eq. (6) or a lnear splne functon f Eq. (7). We could perform random samplng to get the desred large number of cheap ponts. Alternatvely, we could put a greater emphass on ponts havng smaller functon values for each objectve n order to dentfy end ponts on the Pareto fronter. Therefore, we then construct a samplng gudance functon for the th objectve functon, zˆ ( x) = c 0 - fˆ ( x), c 0 f ˆ( x ). It s easy to see that zˆ ( x) s always postve; t s a lnear transformaton of ts approxmaton functon ˆ ( x) ; t reflects the goal to sample more ponts n regons that fˆ ( x) has smaller values; and as fˆ ( x) s bult from so-far evaluated expensve ponts, zˆ ( x) can thus adapt to ncreasngly rcher nformaton. Therefore, ˆ ( x) satsfy all of the four requrements for the samplng gudance functon; and t s the frst type of samplng gudance functon used n ths work. The other type s for the samplng of fronter set ponts whch wll be dscussed later. f z An equal number of sample ponts, e.g., 00, are drawn ndependently accordng to each objectve functon s samplng gudance functon ˆ ( x). As a result, we wll have m 00 number of cheap ponts. z It s to be noted although there are more sample ponts drawn from functon f havng small functon values of f, these ponts mght have large functon values of f j, j, and vce versa. If the pont leadng to mn(f ) s dentfed, ths pont wll be at one of the vertces of the Pareto fronter n the performance space and s usually called an extreme pont. Therefore, by ndependently pursung the mnmum of each objectve functon, we have a mxture of m 00 ponts potentally coverng all of the

13 extreme ponts on the Pareto fronter. By dong so, the dffcultes of GA-based algorthms n dentfyng extreme ponts as found n [5] can be overcome. Step 4: Combnng the Sample Ponts from Last Step wth Current Fronter Ponts; Fronter Ponts Identfcaton from the Combned Pont Set Ths step s to prepare for samplng new fronter ponts. Frst sample ponts drawn at Step are combned wth current fronter ponts for the recalculaton of the ftness value of all ponts n the combned set. It s to be noted that current fronter ponts are evaluated expensve ponts, ther real functon values are used n the ftness computaton. Sample ponts drawn from Step are not evaluated from expensve black-box functons; ther respectve fˆ ( x) functon values wll be used nstead for the ftness value computaton. Sample ponts drawn from Step are used to enrch the nformaton for the constructon of a samplng gudance functon for the next step, though ther functon values are only a predcton from the approxmaton model. Then the ftness functon values are computed by usng Eq. (). For the combned pont set, ponts havng a ftness value larger than or equal to wll be used for the samplng n the next step. Other ponts are dscarded as these ponts are lkely non-fronter ponts. Step 5: Samplng Fronter Ponts The fronter ponts obtaned from Step 4 are the best ponts among all of the exstng ponts n terms of the possblty of becomng Pareto set ponts. Fronter ponts are dfferent from Pareto set ponts as the former may turn to be non-fronter ponts f new ponts (desgns) are added as the samplng process terates. Converged fronter ponts are deemed Pareto set ponts n ths work. For the fronter ponts obtaned n Step 4 whose ftness value s larger than or equal to, defne ˆ s s s s sm sm j j j j l z( x) = G = [ max(mn( f f, f f,, f f ))]. z ˆ( x) s nonnegatve and represent the nature of the problem because t s compounded from f (x) and f ˆ ( x ). It s also used to sample more ponts n regons at whch the ftness value s hgher, because the hgher the ftness value,

14 the more lkely Pareto set ponts exst. Thus ths z ˆ( x) functon reflects our samplng goal. Snce t s bult on f (x) and f ˆ ( x ), t adapts to ncreasngly rcher nformaton as more sample ponts are evaluated. Therefore z ˆ( x) satsfes all four requrements and t s n fact the second type of samplng gudance functon n ths work. Assumng L fronter ponts are obtaned from Step 4, a random sample x (np+), x (np+),, etc. from the L fronter ponts are drawn, guded by the z ˆ( x) functon. To determne the number of to-be-drawn ponts, a smple heurstcs s used n ths work. If the rato of L and the number of the current fronter ponts n the last teraton s less than, L ponts are drawn. If the rato s between and 4, the same number of the current fronter ponts s drawn. Otherwse, a double number of the fronter ponts n the last teraton are drawn. All the new sample ponts form a new sample set. If the drawn ponts have been evaluated before or repettve, they are dscarded from the new sample set to avod re-evaluaton. Step 6: Functon Evaluaton for New Sample Ponts The ponts n the new sample set obtaned from Step 5 are evaluated by callng expensve black-box functons. Fgure 4 Sample ponts at the end of the st teraton for the example problem. 4

15 For the example problem, n total 00 cheap ponts are generated ndependently from each approxmaton model at Step. These 00 ponts are also evaluated by all the approxmated objectve models. They are then combned wth ponts P, P, and P to calculate ftness functon values. Among all these ponts, ponts havng ftness values larger than or equal to are chosen as a new pont set. New samples are drawn from ths pont set as shown n Fgure 4 by performng Step 5. The + symbol ndcates locatons of the new sample ponts PN, PN and PN, after expensve functon evaluatons. Step 7: Combnng New Sample Ponts wth Expensve Fronter Ponts from the Last Iteraton and Identfcaton of the New Fronter Set At ths step, the newly evaluated sample ponts PN, PN and PN are then combned wth formerly evaluated expensve fronter ponts P, P, and P shown n Fgure. Ftness values of ths combned pont set are then calculated and the fronter ponts are dentfed as the fnal fronter ponts of ths teraton. It s easy to see that PN, PN and PN wll replace the prevous P, P, and P as the new fronter ponts. All of the fnal fronter ponts are evaluated expensve ponts. All ponts evaluated by expensve functon evaluatons, n ths case PN, PN, PN and the ntal 6 ponts, wll partcpate the modelng of each objectve functon at the nd teraton. Step 8: Checkng Convergence If the convergence crtera are met, the procedure termnates; otherwse, back to Step. Fgure 5 shows all of the evaluated sample ponts and converged fronter ponts wth respect to the feasble performance space for the example problem. As one can see that the converged fronter ponts overrde the real Pareto fronter, even at the sngular pont at the left upper corner. These ponts also dstrbute closely and evenly across the entre curve. The rato of Pareto ponts obtaned over the total number of sample ponts s hgh. Snce the proposed method calls random processes, 0 ndependent runs have 5

16 been carred out. Smlar accuracy and effcency have been acheved n all of the 0 runs. Detaled results on ths test functon wll be reported n Secton 5. Fgure 5 Performance space, evaluated ponts, and Pareto set ponts for the example problem. Convergence crtera Because no property of a black-box functon s avalable, t s unlkely to develop a rgorous convergence crteron such as Kuhn-Tucker condtons. Many drect search algorthms use the maxmum number of teratons or allowable budget as the crteron, e.g., []. Ths work apples two convergence crtera. The frst can be thought of as a crteron along the vertcal drecton, whch measures the progress of teratons. It s smlar to the one used n [5],.e., the dfference between fronter ponts after two consecutve teratons s suffcently small. Assumng (PS) k s the fronter pont set at the k-th teraton, n(ps) k,k+ s the total number of ponts n (PS) k that also exst n (PS) k+, and n(ps) k+ s the total number of ponts n (PS) k+. Then ρ n(ps) k, k+ = n( PS) s used as the frst convergence crteron, whch means most fronter ponts dentfed n the last teraton are also n the current fronter and therefore only few new fronter ponts are found. The second crteron can be k + 6

17 consdered as a convergence crteron n the horzontal drecton, whch measures the closeness and dstrbuton of Pareto ponts on the fronter at the last teraton. Referrng to the ftness functon defned n Eq. () and ts propertes, when Pareto ponts are closely and evenly dstrbuted, the ftness value of all Pareto set ponts tends to be. Therefore, n ths work we set the average ftness value of all the current fronter ponts, G = L L G =, close to as an addtonal convergence crteron, where L s the number of fronter ponts n the last PSP teraton. In practce, we use G.0(or.00). Ths crteron ensures the Pareto ponts spread over the fronter, a desred property supported by many researchers [, 5, 4]. Approxmaton Model Alternaton At Step, QPF and RBF models are automatcally alternated to take the advantages of both and make the algorthm more effcent. There are two crtera for alternaton. In general QPF s used n the earler stages because QPF emphaszes more on predctng the overall trend than lnear RBF. RBF s used n the later stages because t s more loyal to sample ponts. The earler and later stages are marked by evaluatng the average ftness value of current fronter ponts, G k, as compared aganst the desred convergence crteron for G d, e. g..00, plus a constant devaton δ (=0.05 n ths work). If G k G > d + δ, whch ndcates that the current fronter s far from convergence so that t s regarded as an earler stage, QPF s used. Otherwse, RBF s used. The second crteron s based on the number of new sample ponts. Models are alternated to speed up f the number of new sample ponts generated after an teraton s less than a gven small constant. 7

18 5. Numercal Studes The proposed PSP method s tested wth a number of well-known MOO problems taken from the lterature. Three test problems are chosen. The frst s a convex, b-objectve MOO problem. The second, as descrbed by Eq. (5), has a non-smooth Pareto fronter featured by a sngular Pareto pont. The thrd s a hghly non-lnear problem wth objectves. PSP s then appled to a real engneerng desgn problem. The formulas of the test problems are as below. Problem []. Mnmze: f ( x, x ) = ( x ) + ( x ) f ( x, x ) = x + ( x 6) g ( x, x ) = x.6 ( x, x ) = 0.4 x 0 g ( x, x ) = x 5 0 g 4 ( x, x ) = x 0 Subject to : 0 g (8) Problem s the example problem defned by Eq. (5). Problem [4]: Mnmze F x) = { f ( x), f ( x), f ( )} ( x ( x) = x x x x 5,,,. Subject to: g 0 (9) 0 = where the objectve functons are gven by f = 5 ( x + x ( + x + x ) + x + x ) /0 f f = 5 ( x = 50 ( x + x + x + x + x ( + x + x ( + x + x ) + x ) /0 + x )) /0 Snce random processes are used n PSP, 0 runs have been carred out for each problem. The test results are presented below. For each test problem, the number of teratons, total number of evaluated ponts (expensve), number of converged fronter ponts, as well as the rato of the number of converged fronter ponts to the total number of evaluated ponts are recorded for 0 dfferent runs. The medan and varaton range for all nteger numbers are recorded. For the pont rato, the mean value and ts standard devaton (S.D.) are used nstead. The results are lsted n Table, along wth 8

19 the result for the panel desgn problem whch wll be descrbed n the next secton. The total number of expensve functon evaluatons for each problem equals to the number of evaluated ponts tmes the number of objectve functons, m. Table Results of testng and applcaton of PSP. # of teratons # of evaluated ponts # of converged fronter ponts # of converged fronter ponts # of evaluated ponts Medan Range Medan Range Medan Range Mean S.D. Problem 5 [4 5] 7 [60 86] 64 [5 80] Problem.5 [9 0] 8.5 [ 55] 4.5 [9 0] Problem.5 [6 48] 99.5 [7 46] 04.5 [9 49] Panel Desgn [9 45] 86.5 [50 ] 7 [ 4] As one can see from the table, the total number of expensve ponts s small for all problems. Consderng the small number of teratons, parallel computaton could be potentally mplemented. Among the modest number of sample ponts, the rato of converged fronter ponts to the total number of sample ponts are more than 60% for all the test problems. The low rato of the panel desgn problem wll be dscussed later. PSP s also found robust wth small varatons on the pont rato between dfferent runs. To better observe the accuracy of converged fronter ponts, all of the evaluated ponts are plotted together wth the feasble performance space, as shown n Fgures 5~7. It s clear from the plots that the converged fronter ponts overrde exactly or are very close to the real Pareto fronter. 6. Mult-objectve Optmzaton for Fuel Cell Component Desgn The multple functon panel s a key component of a patented Proton Exchange Membrane (PEM) fuel stack [6]. The panel conssts of a copper fn sandwched between two square flat copper sheets. Fgure 8 llustrates a secton of the panel. 9

20 Fgure 6 Performance space, evaluated ponts, and Pareto set ponts for Problem. Fgure 7 Performance space, evaluated ponts, and Pareto set ponts for Problem. 0

21 h t w Fgure 8 A secton of the multple functon panel and desgn varables. The panel has three functons: ) to functon as a dstrbuted sprng system to compensate the enormous amount of hydro and thermal expansons durng the fuel cell operaton, ) to functon as a radator to get rd of the heat generated durng the operaton, and ) to functon as a conductor to collect electrons for a stack of many fuel cells. The panel s to be made of copper coated wth graphte to satsfy the thrd functon requrement, and n the mean tme to avod contamnaton of metal partcles to PEM fuel cells. Desgn of the panel for the rest two functons s formulated as a MOO problem as follows, wth h, w, and t as the desgn varables. Mnmze: ( h, w, t) = h ( u) w 0.04 / f sn hθ 4 Maxmze: f ( h, w, t) = fndht( h, w, ) (0) Subject to: t f ( h, w, t) 40 ; 4 h 9 ; 4 w 9 ;. 06 t 0. 6 Where h tg = w Pwh u =. θ, Et sn θ f denotes the rato of the dfference between the panel s real deformaton and the deal deformaton over the deal deformaton under a gven pressure, P. E s the Young s modulus of copper. f s the ext ar temperature from the panel, whch s to be maxmze for the radaton purpose. fndht s a program whch fnds the ext ar temperature, whch can be

22 consdered as a black-box functon. Detals of the program can be found n Ref. [6]. Results obtaned by the Pareto-Set-Pursung method are lsted n Table. Fgure 9 Performance space, evaluated ponts, Pareto set ponts for the panel desgn. Fgure 9 llustrates the results. The unt for f s Kelvn. Values of f are lsted as negatve as the optmzaton process mnmzes the negatve of f for maxmzaton. One can see from Table that the rato of Pareto set ponts s not very hgh for ths problem. It s because the performance space s dscontnuous wth long narrow tals. It took extra expenses to generate sample ponts n the upper left regon of the Pareto set. 7. Dscussons Features of PSP One can possbly observe from the PSP procedure that PSP has followng two propertes: ) as PSP always samples n the entre desgn space, every Pareto set pont has a probablty to be drawn and ths probablty s much hgher than that of non-pareto set ponts, accordng to the samplng gudance functon; ) as the teraton contnues, ponts n (PS) k wll be closer to the real Pareto fronter than those

23 n (PS) k-, and thus PSP converges to the real Pareto fronter f samplng contnues to nfnty. From test results and the desgn applcaton, the proposed PSP method demonstrates great effcency for MOO problems. By reflectng the PSP process and especally ts samplng strategy, t s found that a reducton of search space s nherent by utlzng the nature of the Pareto set. Let s stll use the example n Eq. (5) for explanaton. Assume at Iteraton we randomly sample two desgn ponts a and b n the desgn space, as shown n Fgure 0. Ponts a and b are then evaluated by callng the expensve black-box functons and thus ther objectve functon values are obtaned. As a result, ponts A and B n the performance space correspondng respectvely to ponts a and b are shown n Fgure. x b j a x Fgure 0 Sample ponts n the desgn space at the st teraton. Assume the coordnates of A and B n the performance space are (f A, f A ) and (f B, f B ). Currently these two samplng ponts A and B are fronter ponts n the performance space. Accordng to the defnton of Pareto set ponts, any pont I falls n the regon defned by f I >f A and f I >f A s domnated by Pont A. Smlarly, any pont J falls n the regon defned by f J >f B, f J >f B s domnated by Pont B. Only ponts n the shaded area n Fgure have potental to domnate Pont A or B. Ponts and j correspondng to I and J respectvely are also plotted n the desgn space, as shown n Fgure 0. As

24 PSP only draws sample ponts from current fronter ponts (though they are a mx of real ponts and predcted ponts), t mples that roughly only the shaded regons n Fgure are explored further n the entre performance space. The rest of the regons are excluded. In ths way the search space from the perspectve of performance shrnks a great deal by usng only two ponts. By contnung samplng n the shaded area n Fgure, new fronter ponts can be dentfed and the shaded area wll be further shrunk. Therefore, n bref, because of the characterstcs of the Pareto set, the PSP samplng dscards domnated regons n the performance space and thus converges rapdly. f I A J B Fgure Sample ponts n the performance space at the st teraton In the process of PSP development, an alternatve method to generate ntal sample ponts s explored. That s, we run optmzaton on each sngle objectve functon regardless of other objectves. The optmum of each objectve functon s used as ntal sample ponts for PSP. In theory these optma are n the Pareto set. By searchng for these optma, t s expected that PSP can be sped up. It s found through testng that there s no obvous mprovement on the convergence speed of PSP. Furthermore, the optmzaton processes used to fnd ntal ponts consume many expensve functon evaluatons. Overall, ths dea requres more functon evaluatons wth no apparent benefts. f 4

25 Comparson of PSP wth Other Approxmaton-based Methods Current state-of-art approxmaton-based methods strve to buld an accurate approxmaton model [, 5], on whch the Pareto fronter s searched. Usng the method n Ref. [] as an example of these methods, ts procedure s llustrated n Fgure a, whch s adapted from Ref. []. In contrast, the procedure of PSP s smply llustrated n Fgure b. As one can see that PSP ) does not need to valdate the approxmaton model as the model s only used to gude the samplng; ) no thrd-party optmzaton algorthm such as GA s called n PSP; and ) no verfcaton of the fronter ponts s requred snce all the dentfed fronter ponts n PSP have been evaluated n the process. The buldaccurate-model-frst approach as n Ref. [] could be effcent when the black-box functon s smple and of low dmenson. However, the cost to dentfy and valdate the model could be hgh, especally when the model s complex. Ref. [] stated that t was almost mpossble to nexpensvely construct an accurate approxmaton model for hgh-dmensonal problems, thus the relance on the accuracy of approxmaton model rases concerns. PSP provdes an alternatve n whch the approxmaton accuracy s not crtcal but the Pareto fronter ponts can stll be effcently obtaned. Sample Desgn Space Buld Approxmaton Model Valdate the Model Sample Desgn Space Buld Approxmaton Model Sample towards Pareto Fronter Identfy Pareto Fronter Verfy Pareto Fronter (a) Fgure Comparson of PSP wth the method n Wlson et al. []: (a) the procedure of the method n Ref. []; (b) a smplfed llustraton of PSP procedure. (b) 5

26 Comparson of PSP wth other Drect Search Methods for MOO Problems The PSP method can also be used as an ndependent drect search method for all MOO problems. It would be deal f PSP can be compared wth other drect search methods such as GA-based methods by examnng ther converged Pareto sets. There are a number of dffcultes, however. Frst, the obtaned Pareto sets are usually not publshed. If those data are avalable, we can see f two sets domnate each other. If all the ponts n one set are domnated by the ponts n the other, we can safely conclude that the latter s a better soluton set. But the more lkely s that only some ponts n one set are domnated by ponts n the other. In ths case, t s almost mpossble to establsh an objectve crteron to compare the qualty of the two [7]. The same s true f both sets are real Pareto sets and therefore no pont s domnated by any other pont n ether set. Gven such dffcultes, our result s thus only compared vsually wth the real fronter generated by evaluatng a fne grd of ponts as shown n Fgures 5-7, Concluson Ths work presents a new mult-objectve optmzaton (MOO) method, the Pareto Set Pursung (PSP) method. PSP s especally sutable for desgn problems nvolvng expensve black-box functons, though t could also be used as a general-purpose MOO method. Ths approach provdes decson makers wth a set of Pareto set for choces wthout any a pror knowledge of the objectve functons or preferences. Solutons obtaned by ths method can reflect the entre Pareto optmal fronter, even when the fronter surface s hghly non-lnear, e.g., non-convex or concave. Ths approach also automatcally captures the Pareto optmal fronter wthout callng any formal optmzaton process. It uses approxmaton to gude the samplng process only and does not demand an accurate approxmaton model. Through tests and applcatons, PSP s found robust and effcent, and the Pareto set ponts found by PSP are real or close-to-real Pareto set ponts and spread closely and evenly over the entre Pareto optmal fronter. The defned samplng gudance functon may be useful for developng other dscrmnatve samplng schemes for varous purposes. 6

27 Future research wll examne closely the applcablty of PSP to hgh-dmensonal problems wth many desgn objectves, how to compare PSP wth other drect search MOO methods, and how PSP can be mproved by replacng the random samplng wth more controlled samplng methods such as Latn Hypercube samplng, and so on. Acknowledgements Fundng from Natural Scence and Engneerng Research Councl (NSERC) of Canada s gratefully apprecated. The authors would also lke to thank anonymous revewers for ther valuable comments that resulted n a much mproved manuscrpt. References. Keeney, R.L. and Rafa, H. (976). Decsons wth multple objectve: preferences and value tradeoff. John Wley and Sons, New York.. Marler, R. T. and Arora, J. S. (004) Survey of mult-objectve optmzaton methods for engneerng. Structural and Multdscplnary Optmzaton, 6: pp Chen, W., Wecek, M.M. and Zhang, J. (999). Qualty utlty -- a compromse programmng approach to robust desgn. Journal of Mechancal Desgn, Transactons of the ASME, : pp Messac, A. (996). Physcal programmng: effectve optmzaton for computatonal desgn. AIAA Journal, 4(): pp Tappeta, R.V. and Renaud, J.E. (999). Interactve multobjectve optmzaton procedure. AIAA Journal, 7(7): pp Tappeta, R.V., Renaud, J.E., Messac, A. and Sundararaj, G. (000). Interactve physcal programmng: tradeoff analyss and decson makng n multcrtera optmzaton. AIAA Journal, 8(5): pp Schaumann, E.J., Ballng, R.J. and Day, K. (998). Genetc algorthms wth multple objectves. 7th AIAA/USAF/NASA/ISSMO Symposum on Multdscplnary Analyss and Optmzaton, St. Lous, MO, AIAA Vol., Sept. -4, 998, pp. 4-, Paper No. AIAA Deb, K. (999). Evolutonary algorthms for mult-crteron optmzaton n engneerng desgn. Proceedngs of evolutonary algorthms n engneerng & computer scence, Eurogen Deb, K., Mohan, M. and Mshra, S. (00). A fast mult-objectve evolutonary algorthm for fndng well-spread pareto-optmal solutons. Report No. 0000, Indan Insttute of Technology Kanpur, Kanpur. 0. Srnvas, N. and Deb, K. (995). Multobjectve optmzaton usng nondomnated sortng n genetc algorthms. Journal of Evolutonary Computaton, (): pp

28 . Nan, P.K.S. and Deb, K. (00). A computatonally effectve mult-objectve search and optmzaton technque usng coarse-to-fne gran modelng. Report No , Indan Insttute of Technology Kanpur, Kanpur.. Luh, G.-C., Chueh, C.-H. and Lu, W.-W. (00). MOIA: mult-objectve mmune algorthm. Journal of Engneerng Optmzaton, 5(): pp Wlson, B., Cappeller, D.J., Smpson, T.W. and Frecker, M.I. (00). Effcent Pareto fronter exploraton usng surrogate approxmatons. Optmzaton and Engneerng, : pp L, Y., Fadel, G.M. and Wecek, M.M. (998). Approxmatng Pareto curves usng the hyperellpse. 7th AIAA/USAF/NASA/ISSMO Symposum on Multdscplnary Analyss and Optmzaton, St. Lous, Paper No. AIAA Yang, B. S., Yeun, Y.-S., and Ruy, W.-S. (00). Managng Approxmaton Models n Multobjectve Optmzaton. Structural and Multdscplnary Optmzaton, 4: pp Ross, S. M. (00). Smulaton, rd edton, Academc Press, San Dego, CA. 7. Fu, J.C. and Wang, L. (00). A random-dscretzaton based Monte Carlo samplng method and ts applcatons. Methodology and Computng n Appled Probablty, 4: pp Wang, L., Shan, S. and Wang, G.G. (004). Mode-Pursung samplng method for global optmzaton on expensve black-box functons. Journal of Engneerng Optmzaton, 6(4): pp Audet, C. and Denns, J. E. (004). A Pattern Search Flter Method for Nonlnear Programmng Wthout Dervatves. SIMA Journal on Optmzaton, 4(4): pp Mchalewcs, Z. (995). A Survey of Constrant Handlng Technques n Evolutonary Computaton Methods. Proceedngs of the Fourth Annual Conference on Evolutonary Programmng, J. McDonnell, R. Reynolds, and D. Fogel (ed.), MIT Press, Cambrdge, MA: pp Myers, R. H. and Montgomery, D. C. (995). Response surface methodology, process and product optmzaton usng desgned experments. John Wley & Sons, Inc.. Jn, R., Chen, W. and Smpson, T. W. (00). Comparatve Studes of Metamodelng Technques under Multple Modelng Crtera. Journal of Structural and Multdscplnary Optmzaton, (): pp. -.. Ong, Y. S., Nar, P. B., and Keane, A. J. (00). Evolutonary Optmzaton of Computatonally Expensve Problems va Surrogate Modelng. AIAA Journal, 4(4): pp Ray, T., and Tsa, H. M. (004). Swarm Algorthm for Sngle- and Multobjectve Arfol Desgn Optmzaton. AIAA Journal, 4(): pp Dong, Z. (999). Fuel cell stack assembly, Patent Cooperaton Treaty Patent WO, AU, CA, CN, IN, JP, KR, SG, US, European patent (AT, BE, CH, CY, DE, DK, ES, FI, FR, GB, GR, IE, IT, LU, MC, NL, PT, SE). Patent No. 99/5778, November, Wang, G.G. (999). A quanttatve concurrent engneerng desgn method usng vrtual prototypng-based global optmzaton and ts applcaton n transportaton fuel cells. Ph.D. Dssertaton Thess, Unversty of Vctora, Vctora, Canada. 7. Ztzler, E., Thele, L., Laumanns, M., Fonseca, C. M., and da Fonseca, V. G. (00). Performance Assessment of Multobjectve Optmzers: An Analyss and Revew. IEEE Transactons on Evolutonary Computaton, 7(): pp

29 Nomenclature c 0 A constant np Number of so-far evaluated sample ponts l The exponent for ftness value computaton L Number of ponts havng ftness value larger than or equal to m Number of desgn objectves n Number of desgn varables k Number of ponts; an ndex of teraton p Number of nequalty constrants P Pressure on the mult-functon panel wthn a fuel cell stack q Number of equalty constrants t Thckness of the fuel cell panel X The desgn varable vector, x = [x, x,, x n ] x () The -th desgn pont x * A Pareto optmum l x r Lower bound of the r th varable u x r Upper bound of the r th varable h(x) Equalty constrant functons g(x) Inequalty constrant functons G Pareto ftness value of the th desgn G Average ftness value of fronter ponts f Scaled k th objectve functon value of the th desgn sk rawf, Un-scaled value of the frst objectve for the th desgn rawf,mn Mnmum un-scaled value of the frst objectve among all desgns rawf,max Maxmum un-scaled value of the frst objectve among all desgns ẑ Samplng gudance functon f(x) A general objectve functon f ˆ( x ) The approxmaton model of f(x) F(x) Vector of all of the objectves f (x) The -th objectve functon, =,,m fˆ ( x) Approxmaton model of the -th objectve functon f (x) (PS) k The fronter pont set at the k-th teraton α Regresson coeffcents of the RBF model β Regresson coeffcent of the quadratc polynomal functon model δ ρ A small constant for model alternaton A rato between number of ponts for PSP convergence 9