An Adaptive Surrogate-Assisted Strategy for Multi-Objective Optimization

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1 0 th World Congress on Structural and Multdscplnary Optmzaton May 9-24, 203, Orlando, Florda, USA An Adaptve Surrogate-Asssted Strategy for Mult-Objectve Optmzaton Nelen Stander Lvermore Software Technology Corporaton, Lvermore, Calforna, USA, Abstract A sequental metamodel-based optmzaton method s proposed for mult-objectve optmzaton problems. The algorthm, desgnated as Pareto Doman Reducton, s an adaptve samplng method and an extenson of the classcal Doman Reducton approach (also known as the Sequental Response Surface Method). In addton to standard benchmark examples, a Multdscplnary Desgn Optmzaton (MDO) example nvolvng a vehcle mpact s used to demonstrate that the accuracy conforms to the Drect NSGA-II exact method whle usng a small fracton of ts computatonal effort. 2. Keywords: Mult-objectve optmzaton, metamodels, surrogates, crashworthness optmzaton. 3. Introducton Mult-objectve optmzaton s typcally conducted usng a Drect Optmzaton approach such as the Non-Domnated Sortng Genetc Algorthm (NSGA-II) []. NSGA-II can be used to obtan exact results but requres a very large number of smulatons so can be expensve, especally n crashworthness optmzaton where smulatons nvolve hghly detaled Fnte Element models and costly nonlnear dynamc analyss. MOO s also possble wth metamodel-asssted methods. A naïve approach s to conduct a large number of smulatons by usng a global Space Fllng expermental desgn. Ths approach typcally reles on unform global samplng to construct the metamodel and can be faster than drect soluton. The accuracy of the approach s severely affected by scalng as the number of smulatons to obtan an accurate metamodel n a large varable space can be very large. For sngle objectve problems, ths defcency can be addressed by an teratve Doman Reducton approach (known as the Sequental Response Surface Method or SRSM) [2,3] n whch the search doman sze s gradually reduced for each teraton. Wth sutable heurstcs the SRSM approach can provde a suffcently accurate result, but so far has not been adapted to mult-objectve problems n whch multple solutons are possble. Other studes of surrogate-asssted methods can be found n the lterature. A well known method s ParEGO based on the Effcent Global Optmzaton (EGO) algorthm. In two studes [4,5], ths algorthm was found to be compettve wth NSGA-II, although the test problems were lmted to 8 varables and the results were based on a fxed, lmted number of runs rather than an attempt to converge to a fne tolerance. In ths study an adaptve doman reducton method desgnated as Pareto Doman Reducton (PDR) s ntroduced for mprovng effcency and accuracy. The method employs heurstcs whch are smlar to those of SRSM but snce multple optmal solutons are possble, t uses the rregular subregon of the Pareto Optmal Fronter (POF) as a samplng doman. The sze of ths subregon s teratvely reduced n order to ntensfy the exploraton n the predcted neghborhood of the POF. The method s conservatve n the sense that early samplng s global wth a gradual convergence to the POF. Hence t can also be vewed as an adaptve samplng approach. The proposed method has the addtonal advantage that, f a mult-objectve problem has only one optmal soluton, t degenerates to SRSM whch unfes and smplfes the methodology and user choce. Snce Radal Bass Functon Networks [6,7] are typcally used as metamodels, a Space Fllng approach s used to obtan a well spaced samplng n the reduced doman. Ths avods pont duplcaton and maxmzes the accuracy n the POF neghborhood. The remander of the paper deals wth a detaled descrpton of the methodology followed by two standard benchmark problems from the lterature and a mult-objectve mult-dscplnary (MDO) crashworthness/modal analyss example of a vehcle. The model s not very large, but suffcently realstc and representatve of a typcal model used n ndustry. The results show that the PDR method s hghly accurate and potentally an order of magntude cheaper than the drect NSGA-II approach. 4. Methodology 4. Sequental Response Surface method (SRSM) To explan the methodology, the heurstcs of a doman reducton approach for sngle-objectve optmzaton are

2 frstly explaned. The purpose of the SRSM method s to allow convergence of the sngle-objectve soluton to a prescrbed tolerance. The SRSM method [2,3] uses a regon of nterest, a subspace of the desgn space, to determne an approxmate optmum. A range s chosen for each varable to determne ts ntal sze. A new regon of nterest centers on each successve optmum. Progress s made by movng the center of the regon of nterest as well as reducng ts sze. (0) The startng pont x wll form the center pont of the frst regon of nterest. The lower and upper bounds rl, 0 rr,0 ( x, x ) of the ntal subregon are calculated usng the specfed ntal range value r so that rl,0 (0) (0) ru,0 (0) (0) x = x 0.5r and x = x + 0.5r ; =,..., n () where n s the number of desgn varables. The modfcaton of the ranges on the varables for the next teraton depends on the oscllatory nature of the soluton and the accuracy of the current optmum. Oscllaton: A contracton parameter γ s frstly determned based on whether the current and prevous desgns (k ) ( k ) x and x are on the opposte or the same sde of the regon of nterest. Thus an oscllaton ndcator c may be determned n teraton k as where ( k ) ( k ) ( k ) (0) c = d d (2) ( k ) ( k ) ( k ) ( k ) ( k ) ( k ) ( k ) d = 2Δx / r ; Δx = x x ; d [ ;] (3) The oscllaton ndcator (purposely omttng ndces and k) s normalzed as ĉ where The contracton parameter γ s then calculated as c ˆ = c sgn( c). (4) γ = 0.5( γ ( + cˆ) + γ ( cˆ)). (5) pan The parameter γ osc s typcally representng shrnkage to dampen oscllaton, whereas γ pan represents the pure pannng case and therefore unty s typcally chosen. Accuracy: The accuracy s estmated usng the proxmty of the predcted optmum of the current teraton to the startng (prevous) desgn. The smaller the dstance between the startng and optmum desgns, the more rapdly the regon of nterest wll dmnsh n sze. If the soluton s on the bound of the regon of nterest, the optmal pont s estmated to be beyond the regon. Therefore a new subregon, whch s centered on the current pont, does not change ts sze. Ths s called pannng. If the optmum pont concdes wth the prevous one, the subregon s statonary, but reduces ts sze (zoomng). Both pannng and zoomng may occur f there s partal movement. The ( +) range r k for the new subregon n the (k + )-th teraton s then determned by: osc ( k + ) ( k ) r = λ r ; =,..., n; k = 0,..., nter (6) ( k ) where λ represents the contracton rate for each desgn varable. To determne λ, d s ncorporated by scalng accordng to a zoom parameter η that represents pure zoomng and the contracton parameter γ to yeld the contracton rate ( k ) λ = η + d ( γ η) (7) for each varable. When used n conjuncton wth neural networks or Krgng, the same heurstcs are appled as descrbed above but the nets are constructed usng all the avalable ponts, ncludng those belongng to prevous teratons. Therefore the response surfaces are progressvely updated n the regon of the optmal pont. Adaptve Contnuous Space Fllng expermental desgn s typcally used for samplng. 4.2 Space Fllng expermental desgn In metamodel-based optmzaton two mportant methods for constructng approxmatons are the approxmaton method (metamodel type) and the expermental desgn (samplng method). In mult-objectve optmzaton, sutable metamodel types are Radal Bass Functon Networks, Feedforward Neural Networks or Krgng methods. 2

3 These metamodels are typcally ftted usng all avalable smulaton results, also those of prevous teratons. Because of the exstence of fxed desgn ponts from earler teratons, teratve metamodel-based methods deally requre adaptve samplng schemes. For adaptve samplng, Contnuous Space Fllng (CSF) expermental desgns [8] are hghly sutable. The key to space-fllng expermental desgns s n generatng 'good' random ponts and achevng reasonably unform coverage of sampled volume for a gven (user-specfed) number of ponts. The constraned randomzaton method termed Latn Hypercube Samplng (LHS) [9], has become a popular strategy to generate ponts on the 'box' (hypercube) desgn regon. The method mples that on each level of every desgn varable only one pont s placed, and the number of levels s the same as the number of runs. The levels are assgned to runs ether randomly or so as to optmze some crteron, e.g. so that the mnmal dstance between any two desgn ponts s maxmzed ('maxmn dstance' crteron). A probablstc search technque, adaptve smulated annealng [,2] s used to optmze the expermental desgn. 4.3 Adaptve samplng for mult-objectve optmzaton In the sngle-objectve doman reducton scheme (Secton 4.), each new doman s centered on the optmum pont of the prevous teraton. If the sub-doman concept s to be extended to mult-objectve optmzaton, the smple sub-doman can n general no longer be centered on a sngle pont snce t may exclude a large part of the POF. The POF, n ths case, s a predcted POF produced as a result of a metamodel optmzaton of the prevous teraton. Hence, t s proposed that a POF subdoman, envelopng the entre POF, be created as the unon of a sutable number of smple overlappng sub-domans of equal dmensons centered on chosen POF ponts denoted as POF kernels (Fg. 3). The number of POF kernels m depends on the current sze of the sub-doman as well as the dmensonalty of the objectve functon space. To provde a well dstrbuted set of POF kernels, these are pcked from the POF superset by maxmzng the mnmum dstance between any two kernels n the varable space (a dscrete space fllng or DSF method). A smple procedure s to select the next pont n a growng subset to be dstant from the currently selected ponts. Hence, for each teraton, the set of POF kernels s a subset of the predcted POF. As a consequence of ths strategy, t s antcpated that there wll be relatvely few, large sub-domans n the early teratons whereas the number of sub-domans wll grow durng the teratve process due to ther gradual sze reducton. The doman reducton heurstcs as descrbed n Secton 4. s appled usng the moton (proxmty and oscllaton) of the optmum of the sum of the objectves. Ths approach ensures that the method devolves to SRSM for sngle objectve optmzaton. The next step s to fll all the sub-domans wth a set of dversty bass ponts (Fg. 5). The purpose of the bass set s to serve as a pool of sutable ponts, wthn the POF regon, from whch to select desgn ponts for smulaton. A contnuous Space Fllng prncple s used [8,9]. Ths method maxmzes the mnmum dstance between any two ponts, or between any pont and a fxed pont. Begnnng wth the frst sub-doman (centered on a randomly chosen kernel), a number of ponts p s chosen wthn ths sub-doman usng CSF. The samplng process moves on to the second subdoman n whch the CSF s repeated, fllng the second sub-doman, but ths tme also consderng the prevously selected dversty ponts as part of the dstance calculaton. Ths process s contnued untl all the sub-domans are flled wth ponts. The end result s an rregular sub-doman, envelopng the POF n the desgn space and whch s space-flled wth a large set of dversty bass ponts. The fnal step before smulaton agan requres the DSF procedure to select p smulaton desgn ponts for teraton. The exstence of all prevous smulatons s consdered n the DSF executon. The set of ponts thus obtaned s well dstrbuted throughout the POF regon and ready for smulaton. 4.4 Algorthm The algorthm steps are summarzed below wth references to Fgures to 7: Intal condtons:. k := 0, choose m smulaton desgns n the full desgn space usng CSF (Fg. ). 2. Conduct the smulatons, buld the metamodel, construct an approxmate POF usng NSGA-II (Fg. 2). For each teraton k:. k : = k + 2. Select m kernels from the approxmate POF generated n teraton (k-) (Fg. 3). 3. Adapt the subregon sze based on teraton (k-). 4. Center a subregon on each kernel (Fg. 4). 5. Populate each subdoman wth a number of dversty bass ponts (Fg. 5). 6. Select p ponts for smulaton from the dversty bass ponts (Fg. 6). 7. Conduct p smulatons (Fg. 7), buld the metamodel and construct an approxmate POF usng NSGA-II 3

Fgure : Samplng ponts (X) of the frst teraton

Construct a metamodel for each desgn crteron.

4 Fgure : Samplng ponts (X) of the frst teraton accordng to the Contnuous Space Fllng expermental desgn (schematc). Fgure 2: Conduct smulatons at the X ponts. Construct a metamodel for each desgn crteron. Solve the approxmate optmzaton problem to fnd the approxmate Pareto Optmal Front ( ). Fgure 3: Usng a Dscrete Space Fllng method, select a subset of the predcted POF as POF kernels ( ). 4

5 Fgure 4: Center a sub-doman on each POF kernel ( ). Fgure 5: Fll the sub-domans wth dversty bass ponts usng a Contnuous Space Fllng procedure. Fgure 6: Sample p smulaton ponts ( ) from the dversty bass set usng the Dscrete Space Fllng approach. 5

Fgure 7: Fnal set of smulaton ponts ( ) for teraton 2. 5. Examples The method was mplemented n the LS-OPT optmzaton code [3] and appled to a number of test problems.

6 Fgure 7: Fnal set of smulaton ponts ( ) for teraton Examples The method was mplemented n the LS-OPT optmzaton code [3] and appled to a number of test problems. The doman reducton algorthm uses parameters: γ =, γ = η 0.8. pan osc = 5. Analytcal benchmark tests To valdate the method, the results of two standard benchmark examples ZDT and ZDT2 wth 30 varables (n = 30) each [] are frst presented. 25 Iteratons usng a populaton of 0 per teraton were requred resultng n a total of 2850 smulatons. 00 of these smulatons were executed to construct the fnal Pareto Optmal Fronter. The Drect GA method requred about 20,000 smulatons The Pareto solutons are shown n Fgs. 8 and 9. RBF networks were used as metamodels. It should be noted that the predcted Pareto Optmal Fronter features a few errant Pareto solutons n the range f 2 >. These occur as a result of a slght msalgnment of the fnal metamodel whch causes some solutons to be found on the f 2 axs. Mnmze f ( ) Mnmze f x) = g( x) h( f ( x), g( )) x 2( x Example ZDT: Example ZDT2: f( x) = x 9 g( x) = + n h( f, g) = f = 2 n x / g f( x) = x n 9 g( x) = + x n = 2 h( f, g) = ( f / g) 2 6

7 .2 Theory.2 Theory PDR (computed) PDR (computed) F Fgure 8: Pareto Optmal Fronter of ZDT. Comparson wth theoretcal result F Fgure 9: Pareto Optmal Fronter of ZDT2 Comparson wth theoretcal result Full vehcle MDO The crashworthness smulaton also used for optmzaton n [3] consders a model contanng approxmately 30,000 fnte elements of a Natonal Hghway Transportaton and Safety Assocaton (NHTSA) vehcle undergong a full frontal mpact. A modal analyss s performed on a body-n-whte model contanng approxmately 8,000 elements. The crash model for the full vehcle s shown n Fgure 0 for the undeformed and deformed (tme = 78ms) states, and wth only the structural components affected by the desgn varables, both n the undeformed and deformed (tme = 72ms) states, n Fgure. The NVH model s depcted n Fgure 2 n the frst torson vbratonal mode. Only body parts that are crucal to the vbratonal mode shapes are retaned n ths model. The desgn varables are all thcknesses or gages of structural components n the engne compartment of the vehcle (Fgure ), parameterzed drectly n the LS-DYNA nput fle. Twelve parts are affected, comprsng aprons, rals, shotguns, cradle rals and the cradle cross member (Fgure ). LS-DYNA v.97 [4] s used for both the crash and NVH smulatons, n explct and mplct modes respectvely. The desgn formulaton s as follows: Mnmze Mass subject to Mnmze Maxmum ntruson Maxmum ntruson(x crash ) < 55.27mm Stage pulse(x crash ) > 4.5g Stage 2 pulse(x crash ) > 7.59g Stage 3 pulse(x crash ) > 20.75g 4.38Hz < Torsonal mode frequency(x NVH ) < 42.38Hz x crash = x NVH = [ral_nner, ral_outer, cradle_rals, aprons, shotgun_nner, shotgun_outer, cradle_crossmember] T The three stage pulses are calculated from the SAE fltered (60Hz) acceleraton and dsplacement of a left rear sll node n the followng fashon: Stage pulse = k d d 2 d2 d & x dx ; k = 2 for =,.0 otherwse; wth the lmts [d ;d 2 ] = [0;84]; [84;334]; [334;Max(dsplacement)] for =,2,3 respectvely, all dsplacement unts n mm and the mnus sgn to convert acceleraton to deceleraton. The Stage pulse s represented by a trangle wth the peak value beng the value used. 7

(a) (b) Fgure 0: (a) Crash model of vehcle n the deformed state (78ms), (b) body-n-whte model of vehcle n torsonal vbraton mode (38.

varables a) Undeformed and (b) deformed (tme = 72ms) Three optmzaton strateges were used to verfy and compare the results: A drect optmzaton usng the NSGA-II algorthm.

A stoppng crteron of 30 teratons was selected. A metamodel-asssted optmzaton usng the Pareto Doman Reducton approach. RBF networks are used as metamodels.

*Note that the number of smulatons for the metamodel asssted methods ncludes a 00 fnal verfcaton runs to construct a full trade-off curve.

8 (a) (b) Fgure 0: (a) Crash model of vehcle n the deformed state (78ms), (b) body-n-whte model of vehcle n torsonal vbraton mode (38.7Hz) Left and rght cradle rals Shotgun outer and nner Left and rght apron Inner and outer ral Front cradle upper and lower cross members (a) (b) Fgure : Structural components affected by desgn varables a) Undeformed and (b) deformed (tme = 72ms) Three optmzaton strateges were used to verfy and compare the results: A drect optmzaton usng the NSGA-II algorthm. A tolerance on the hypervolume change [5] of 0-4 was used to create a realstc benchmark result. A metamodel-asssted optmzaton usng sequental global Radal Bass Functon networks. A stoppng crteron of 30 teratons was selected. A metamodel-asssted optmzaton usng the Pareto Doman Reducton approach. RBF networks are used as metamodels. A stoppng crteron of 30 teratons was selected. The parameters for each run are shown n Table. Table : Run statstcs for Example [6]. *Note that the number of smulatons for the metamodel asssted methods ncludes a 00 fnal verfcaton runs to construct a full trade-off curve. Ths number determnes the sze and dversty of the trade-off curve. Number of teratons/ generatons Number of smulatons per teraton Total number of smulatons per case, ncludng 00 verfcaton smulatons for the last teraton Run confguraton (Number of concurrent jobs MPP cores per smulaton) Drect , NSGA-II Global * 3 4 PDR * 3 4 All runs were conducted on a 768-core Xeon-based cluster runnng up to 80 4-core MPP-DYNA smulatons n parallel. The SunGrd Engne queung system was used wth job montorng usng the LS-OPT Graphcal User Interface. Although there s no mathematcal crteron for global optmalty, the Drect GA run s consdered to be exact n 8

9 the sense that t was run to a very fne tolerance (0-4 ) of the domnated hyper-volume change. The choce of a populaton sze of 60 was the result of some expermentaton as an earler choce of 00 produced a sub-optmal result. Fg. 3 shows a comparson of the trade-off curves produced by the Drect GA optmzaton and the PDR run. The PDR results are smulaton results of the fnal trade-off curve (.e. not the approxmate results). Note that the trade-off curve s dscontnuous (two major dscontnutes) wth several concave and convex sectons. In Fg. 4, the PDR result can be seen to closely approxmate the exact result. The wall clock tme for the PDR method was approxmately 7 hours for the entre optmzaton. Fg. 4 shows a comparson of the predcted trade-off curve and the computed curve usng the PDR method whch confrms the accuracy of the metamodel and samplng strategy. Note that the metamodel s based on all 390 (3 30) smulatons, and not only the last 3. The new PDR method was also compared to the exstng global samplng approach (Fg. 5) n whch the desgn ponts for buldng the metamodel are evenly dstrbuted globally usng a Space Fllng samplng approach. The beneft of usng the PDR method s clearly vsble. Note that fner features such as dscontnutes are not properly modeled by the global samplng, obvously because of a sparsty of smulaton results near the optmum PDR (Computed) Drect GA Intruson Mass Fgure 3: Pareto Optmal Fronter: Comparson of the computed POF (00 desgns) obtaned usng the PDR approach wth the reference POF computed usng the Drect NSGA-II algorthm 0.95 PDR (predcted) PDR (computed) Intruson Mass Fgure 4: Pareto Optmal Fronter: Comparson of the PDR predcted POF and the PDR computed POF. 9

10 0.95 Unform Global Samplng (computed) PDR (computed) Drect GA Intruson Mass Fgure 5: Pareto Optmal Front: Comparson of the smulaton results for unform global samplng ( ) and PDR samplng (+) wth the reference soluton obtaned usng the Drect NSGA-II algorthm ( ). 6. Conclusons Ths study presents an adaptve metamodel-based technque for mult-objectve optmzaton. Although the set of test examples s sparse, the results seem to ndcate that the method holds promse for beng effcent as well as accurate. For a 7-varable problem only about 4% of the effort requred by a drect method could provde a result of smlar accuracy. For a 30-varable problem ths number rose to about 5%. It s apparent that the larger the problem (n terms of the number of varables), the less effcent the PDR method becomes. Industral crash problems typcally requre desgn varables and n ths range the PDR method should stll be compettve. A sgnfcant advantage s that PDR devolves to SRSM for a sngle-objectve optmzaton problem. It should be noted that a large number of smulatons per teraton was requred for the analytcal problems whereas the much smaller (n terms of the varables) crash applcaton requred sgnfcantly less. As a rule t s recommended to use.5n smulatons per teraton (as was done for the crash example), but that number was obvously too lmted for the 30-varable test problems. It s dffcult to recommend a specfc number of smulaton ponts per teraton. Ths dlemma s not unque to PDR, but also appears n drect soluton methods. For example the populaton sze of a Genetc Algorthm such as NSGA-II s also typcally problem dependent. As noted above, the author had to experment wth populaton sze when applyng the drect GA solver to the crash problem to produce an exact result. It s therefore recommended that a large array of test problems be tested as future research to determne a robust populaton sze for PDR. 7. References [] Deb, K. Mult-Objectve Optmzaton usng Evolutonary Algorthms. Wley [2] Stander, N., Crag, K.J. On the robustness of a smple doman reducton scheme for smulaton-based optmzaton, Engneerng Computatons, 9(4), pp , [3] Stander, N., Roux, W.J., Basudhar, A., Eggleston, T., Goel, T., Crag, K.J. LS-OPT User s Manual Verson 5.0, Aprl 203. [4] Knowles, J. ParEGO: A hybrd algorthm wth on-lne landscape approxmaton for expensve multobjectve optmzaton problems. Techncal report TR-COMPSYSBIO , September [5] Fu, G., Khu, S.-T., Butler, D. Multobjectve optmzaton of urban wastewater systems usng ParEGO: a comparson wth NSGA II, Proceedngs of the th Conference on Urban Dranage, Ednburgh, Scotland, UK, [6] Hartman, E.J., Keeler, J.D., Kowalsk, J.M. Layered neural networks wth Gaussan hdden unts as unversal approxmatons. Neural Computaton, 2(2), pp , 990. [7] Park, J., Sandberg, I.W. Approxmaton and radal bass functon networks. Neural Computaton, 5(2), pp , 993. [8] Morrs, M. D. and Mtchell, T. J. Exploratory Desgns for Computer Experments. J. Statst. Plann. Inference 43, ,

11 [9] Johnson, M.E., Moore, L.M., Ylvsaker, D. Mnmax and Maxmn dstance desgns. J. Statst. Plann. Inference, 26, 3-48, 990. [0] McKay, M.D., Conover, W.J., Beckman, R.J. A comparson of three methods for selectng values of nput varables n the analyss of output from a computer code. Technometrcs, pp , 979. [] Ingber, L. Very fast smulated re-annealng, Mathematcal Computer Modelng, 2, pp ,989. [2] Ingber, L., Adaptve smulated annealng (ASA), [ftp.alumn.caltech.edu: /pub/ngber/asa.tar.gz], Lester Ingber Research, McLean VA, 993. [3] Crag K.J., Stander, N., Dooge, D., Varadappa, S. MDO of automotve vehcle for crashworthness and NVH usng response surface methods. Paper AIAA2002_5607, 9th AIAA/ISSMO Symposum on Multdscplnary Analyss and Optmzaton, 4-6 Sept 2002, Atlanta, GA. [4] Hallqust, J.O. LS-DYNA User s Manual, Lvermore Software Technology Corporaton. [5] Whle, L., Bradstreet, L. Barone, L., Hngston, P. Heurstcs for Optmzng the calculaton of hypervolume for mult-objectve optmzaton problems IEEE Congress on Evolutonary Computaton, CEC 2005, Vol. 3, , IEEE, September [6] Stander, N. An Effcent New Sequental Strategy for Mult-Objectve Optmzaton usng LS-OPT. Proceedngs of the 2 th Internatonal LS-DYNA User s Conference, Detrot, Mchgan, June 3-5, 202.

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