Very Large Scale Continuous and Discrete Variable. Woptimization,

Transcription

1 Very Large Scale Contnuous and Dscrete Varable Optmzaton Garret N. Vanderplaats * Vanderplaats Research & Development, Inc S. 8 th Street Colorado Sprngs, CO An optmzaton algorthm s presented whch s capable of solvng nonlnear, constraned optmzaton tasks of well over one hundred thousand desgn varables. Ths method s an advanced exteror penalty functon approach whch uses very lttle central memory and very lttle computatonal tme wthn the algorthm. The prce pad for the large scale capablty s that ths method requres 3-5 tmes as many functon and gradent evaluatons to converge. An ad-hoc method for solvng dscrete optmzaton problems s ncluded whch relably obtans a reasonable dscrete soluton. Examples are presented to demonstrate the method. Nomenclature F(X) = objectve functon g j (X) = j-th constrant functon m = total number of constrants n = number of desgn varables P(X) = penalty to drve desgn varables to a dscrete value q = teraton number n unconstraned sub-problem p q j = ndvdual penalty parameter r p = global penalty parameter R = penalty parameter for dscrete varable optmzaton S q = search drecton at teraton q X = vector of desgn varables X = -th desgn varable L X = lower bound on the -th desgn varable U X = upper bound on the -th desgn varable Φ = pseudo-objectve functon β = conjugate drecton multpler = gradent operator I. Introducton th the ncreased use of optmzaton, the sze of the problems beng solved has grown rapdly. In structural Woptmzaton, problems wth tens of thousands of desgn varables and mllons of constrants are now beng solved. For multdscplne (MDO) problems, the number of varables and constrants can be even larger. In ths case, n the past, decomposton methods were developed due to our nablty to handle such large problems. Non gradent methods, such as genetc algorthms, have been found to be both neffcent and unrelable for more than a few varables. Common gradent based methods, such as Sequental Quadratc Programmng, can theoretcally handle large problems but two ssues quckly arse. Frst, these methods requre soluton of a large and often tme consumng sub-optmzaton task to fnd the search drecton and, second, they requre storage of large * Presdent, Fellow AIAA. 1 Amercan Insttute of Aeronautcs and Astronautcs

2 amounts of nformaton (both gradents and Lagrangan approxmaton nformaton). Ths second ssue may be handled usng spll logc, but ths can also be complcated and neffcent. There s clearly a need for methods whch wll solver very large problems wth lmted central memory and whch avod large sub-optmzaton tasks. Such a method wll be presented here. Whle t was developed prmarly for structural optmzaton, t s equally useful for MDO tasks where gradent nformaton s avalable. In ether case, t s desrable to have hgh qualty approxmatons snce ths algorthm requres more functon and gradent evaluatons than before. When desgnng composte structures or ppng systems, for example, the desgn varables must be dscrete or nteger. Genetc algorthms and branch and bound methods can be used here but are neffcent for problems of more than a few varables. The algorthm descrbed here ncludes an ad-hoc method for solvng large dscrete or mxed contnuous-dscrete varable problems. Ths method does not nsure a theoretcal optmum but does produce a reasonable dscrete soluton effcently. II. Hstorcal Background Snce the ntroducton of numercal optmzaton to structural desgn by Schmt n , the sze of problems solved by these methods has grown exponentally as shown n Fgure 1. In recent years, the focus n gradent based algorthm development has been on Sequental Quadratc Programmng and smlar methods whch possess very strong convergence characterstcs. However, as problem sze has grown, these methods have been found to have sgnfcant lmts. Frst, they requre large memory to store necessary gradent nformaton and second, they requre a sub-optmzaton task to fnd a search drecton. The memory requrement may be allevated by out of memory storage operatons but ths s complcated and neffcent. The drecton fndng sub-problem can requre sgnfcant computatonal effort whch agan leads to neffcences. In general, these modern methods can effcently solve large problems wth only a few actve constrants or small problems wth many constrants (because only a few wll be crtcal). The dffculty arses when we have many desgn varables and many actve constrants. In the 1960 s Sequental Unconstraned Mnmzaton Technques (SUMT) were popular 2 but, as noted above, where replaces by more drect methods. Recently, there has been renewed nterest n SUMT, focused prmarly on nteror pont methods 3. In developng the present capablty numerous SUMT methods were nvestgated leadng to adopton of a new exteror penalty functon approach 4, mplemented n the BIGDOT optmzaton software 5. Ths method wll be revewed here and recent enhancements wll be dentfed. General enhancements have been made, allowng for soluton of much larger problems. Also, a dscrete varable algorthm has been mplemented for soluton of dscrete or mxed contnuous-dscrete problems. NUMBER OF DESIGN VARIABLES III. The BIGDOT Algorthm The method developed here begns by solvng the contnuous varable problem and then fndng a reasonable dscrete soluton. Therefore, t s requred that the problem be solvable as a contnuous problem. Ths s consstent wth the assumptons of tradtonal branch and bound methods. A. Contnuous Varable Optmzaton The basc approach used here s to convert the orgnal constraned optmzaton problem to a sequence of unconstraned problems of the form; 2 Amercan Insttute of Aeronautcs and Astronautcs BIGDOT 500,000 VARIABLES Fgure 1. Growth n Optmzaton Problem Sze

3 Mnmze m p 2 Φ ( X) = F( X) + rp qj{ MAX[0, g j( X)]} (1) j= 1 Subject to; L U X X X = 1, n (2) where X s the vector of desgn varables, F(X) s the objectve functon and g j (X) are the constrants. The subscrpt/superscrpt, p s the outer loop counter whch we wll call the cycle number. The penalty parameter, r p, s ntally set to a small value and then ncreased after each desgn cycle. The only dfference between ths formulaton and the tradtonal exteror penalty functon s the addton of ndvdual penalty parameters, q, on each constrant. These multplers are smlar to the Lagrange multplers used n the Augmented Lagrange Multpler Method 6, but are calculated by a propretary formula. Equaton 2 mposes lmts on the desgn varables (sde constrants) whch are handled drectly. If equalty constrants are consdered, they can just be converted to two equal and opposte nequalty constrants. The unconstraned penalzed functon defned by Eq. 1 s solved by the Fletcher-Reeves conjugate drecton method 7, whch requres vrtually no memory. The gradent of Φ(X) s requred durng the optmzaton process. m p p j j j j= 1 Φ ( X) = F( X) + 2 r q { MAX[0, g ( X) g ( X)]} (3) p j Here, the choce of the exteror penalty functon becomes apparent because only gradents of volated constrants are requred. Furthermore, t s not necessary to store all gradents at once. Notng that Eq. 3 s a smple addton of gradent vectors so, n the lmt, we can calculate only one gradent (objectve or constrant) at a tme. As an ndcaton of computer memory requred by varous methods, the proposed method s compared wth the three methods used by the DOT program 8. Ths s presented n Table 1, where MMFD s the Modfed Method of Feasble Drectons, SLP s the Sequental Lnear Programmng Method and SQP s the Sequental Quadratc Programmng Method. The memory requrements for the DOT methods are the Table 1. Storage Requrements Number of Desgn Varables Method 100 1,000 10,000 MMFD 53,000 5X10 6 5X10 8 SLP 113,000 11X X10 8 SQP 119, X X10 8 BIGDOT 1,400 to 11,000 number of words for storage of all nternal arrays. For the BIGDOT method, the two memory requrements are the mnmum, where only 1 gradent vector s calculated at a tme, and the maxmum, were all possble gradents are stored n memory. The number of constrants equals the number of desgn varables. As can be seen, as the problem sze grows, storage requrements for the earler methods grows exponentally. However, for the present method, storage s much less and the requrement grows only lnearly wth problem sze. If there are many more constrants than desgn varables, the requested storage for the earler methods grows even more rapdly. As noted above, the unconstraned mnmzaton sub-problem s solved by the Fletcher-Reeves algorthm. Here, at teraton q, the search drecton s found as; If q = 1 14,000 to 10X ,000 to 10X Amercan Insttute of Aeronautcs and Astronautcs

4 q S 0 = Φ ( X ) (4) If q > 1 q 1 1 ( q S X ) β S q = Φ + (5) where q 1 Φ( X ) β = (6) Φ q 2 ( X ) Once the search drecton s calculated, the one-dmensonal search s calculated by polynomal nterpolaton. It can be argued that more modern quas-newton methods are a better choce for solvng the sub-problem. However, these methods requre much more storage. Also, computatonal experence has shown that the Fletcher- Reeves algorthm s comparable n effcency and relablty f carefully programmed. Durng the unconstraned mnmzaton sub-problem, almost no effort s requred to calculate the search drecton, so the computatonal tme spent n the optmzer tself s neglgble. The cost of optmzaton s almost completely the cost of analyss and gradent computatons. Thus, t s desrable (but not essental) that hgh qualty approxmatons are avalable, as s the case n modern structural optmzaton. Verson 1 of BIGDOT requred that gradent nformaton be provded drectly and, f only a few gradents could be stored n memory, repeated returns would be made to the user to calculated the needed set. Verson 2 allows the user to provde gradents n compacted form va a bnary fle. If convenent, more than the requested number of gradents can be provded. Ths has the advantage that, for strctly lnear problems, gradents need only be calculated and stored once. Also, for problems wth a nonlnear objectve functon but lnear constrants, the objectve functon gradent s calculated as needed but the constrant gradents need to be calculated only once. B. Dscrete Varable Optmzaton Tradtonal branch and bound methods or genetc algorthms become hopelessly neffcent when used for optmzaton problems of more than a few varables (say 10 or 20). For large scale optmzaton, no theoretcally correct method s avalable for nonlnear constraned optmzaton problems. Therefore, an ad-hoc method s used here wth the goal of fndng a reasonable dscrete soluton whch s feasble. One approach to ths s offered n ref. 9, where a sn or cosne shaped penalty functon was created to drve the desgn to a nearby dscrete soluton. Unlke ref. 9, the sn shape was found here to be most relable. Also, t has been modfed slghtly from the reference to be of the form; n L U 1 X 0.25( X + 3 X ) PX ( ) = R 1= sn2π U L = 1 2 X X (7) where X s the next lower dscrete value and L X U s the next hgher dscrete value. Ths, by ts nature, creates a multtude of relatve mnma. Therefore, f these penaltes are mposed from the begnnng, there s a hgh probablty of fndng a soluton that s not near the true optmum. Thus, t s desrable to frst solve the contnuous varable problem to provde a good startng pont. Even dong ths, the approach s very senstve to penalty parameter values and may produce ether a non-dscrete soluton or an nfeasble one. 4 Amercan Insttute of Aeronautcs and Astronautcs

5 Durng ths phase of the optmzaton, t s mportant to nclude the orgnal penalty functon as well to mantan feasblty wth respect to the general constrants. Equaton 7 attempts to drve the varables to a dscrete value. However, ths penalty functon creates numerous relatve mnma and has lttle assurance of nsurng feasblty wth respect to the orgnal constrants or of actually drvng all dscrete varables to a dscrete value. Therefore, after several cycles, progress s evaluated. If all dscrete varables are not drven to an allowed value, we ask how to change each varable such that t wll move to a dscrete value wth mnmum effect on the objectve functon and at the same tme mantanng feasblty wth respect to the orgnal constrants. To acheve ths, we frst get the gradent of the objectve functon and the penalty term of the pseudo-objectve functon, wth the penalty multplers set to unty (get the sum of volated constrant gradents). We then seek to drve the soluton to a dscrete value wth mnmum ncrease n the objectve functon whle remanng feasble wth respect to the constrants. The general algorthm for ths s; 1. Includng only dscrete varables, and bypassng varables that are already at a dscrete value, search for P/ X F/ X (8) 2. Calculate the changes n X that wll move X to ts next larger and smaller dscrete value. 3. For each such δx estmate the maxmum constrant value based on a lnear approxmaton. 4. Move X to the dscrete value that mantans feasblty. 5. Repeat from step 1 untl all dscrete varables are at a dscrete value. Ths algorthm drves the desgn varables to a dscrete value whle stll ncludng the orgnal constrants va the orgnal SUMT algorthm. Ths has the lmtaton that varables can move only one dscrete value up or down from the contnuous soluton durng each cycle. The fnal algorthm fnds a feasble, dscrete soluton effcently and seldom fals. However, t must be remembered that ths s not guaranteed to be a theoretcal optmum, only a good dscrete soluton. Ths algorthm has the advantage that t s a straghtforward addton to the contnuous varable algorthm and that t requres very lttle memory and computatonal effort. IV. Examples Examples are presented here to demonstrate the algorthm descrbed above. A. Cantlevered Beam The cantlevered beam shown n Fgure 2 s desgned for mnmum materal volume. Here, fve segments are shown. In general, N segments wll be consdered. The desgn varables are the wdth, b, and heght, h of each of the N segments. The beam s subject to stress lmts, σ, at the left end of each segment and the geometrc requrement that the heght of any segment does not exceed twenty tmes the wdth. There are a total of 2N desgn varables and 2N constrants, as well as lower bounds on the varables. Therefore, at the contnuous optmum, t s expected that the desgn wll be fully constraned (as many actve constrants as desgn varables) because the structure s statcally determnate. If the beam was allowed to be completely contnuous wthout lower bounds on the desgn varables, the theoretcal optmum s 53,714. Y l 1 l 2 l 3 l 4 l 5 b h 4 P = 50,000 N 5 E = 2.0x10 N/cm L = 500 cm σ = 14,000 N/cm Fgure 2. Cantlevered Beam P X 5 Amercan Insttute of Aeronautcs and Astronautcs

6 The desgn task s now defned as; Mnmze V N = bhl (9) = 1 Subject to; Here, N=25000, 50000, and was used to create problems of 50000, , and desgn varables, respectvely. For the dscrete soluton, each varable was chosen from sets of dscrete values n ncrements of 0.1. The results are presented n Table 2. The nformaton n σ 1 0 1, N σ = (10) h 20b 0 = 1, N (11) b 1.0 = 1, N (12) h 5.0 = 1, N (13) Table 2. Optmzaton Results Contnuous Optmum Dscrete Optmum Number of Desgn Varables 50, , , ,000 53,744 53,720 53,755 53,730 (243/46) (209/38) (262/49) (266/50) [49,979/46] [99,927/150] [249,919/211] [499,700/1732] 54,864 (92/38) 54,848 (96/25) 54,887 (143/24) 54,821 Infeasble g max = parentheses (../..) are the number of functon evaluatons over the number of gradent evaluatons. The numbers n brackets [../..] are the number of actve constrants over the number of actve sde constrants at the optmum, where a constrant s consdered actve f ts value s greater than and no more postve than Except for the 500,000 varable problem, all constrants were satsfed at the optmum. The 500,000 varable problem faled to acheve a feasble dscrete soluton, even though t generated 500,000 dscrete values. Ths problem ended wth 79 volated constrants havng a maxmum volaton of just over one percent. In each case, the optmum acheved was about the same and the dscrete soluton was only slghtly larger, ndcatng a good dscrete soluton. It s partcularly noteworthy that the number of functon and gradent evaluatons s nearly constant. Ths suggests that the algorthm s very scalable regardless of problem sze. B. Car Body Renforcement The BIGDOT optmzer has been added to the GENESIS structural optmzaton program 10 to perform large scale structural optmzaton. To date, topology optmzaton problems n excess of 2,000,000 varables and member szng problems n excess of 450,000 varables have been solved. Fgure 3 shows a car body model whch we wsh to renforce to ncrease the bendng and/or torson frequency. The approach used here was to allow every element n the model was optmzed for thckness (wth a lower bound of the orgnal desgn) wth the constrant that only a specfed fracton of the materal may be used. Here, 34,560 6 Amercan Insttute of Aeronautcs and Astronautcs

7 szng varables were used. Whle somewhat dffcult to see n Fgure 3 (unless vewed n color), renforcement was added n the areas of the frewall, rocker panels and rear fender areas. Table 3 gves the ncrease n bendng or torson frequency for dfferent values of added mass. Table 3. Frequency Increases Added Mass (Kg) Increased Frequency (Hz) Maxmze Frst Torson Frequency Maxmze Frst Bendng Frequency Fgure 3. Car Body Renforcement C. Topology Optmzaton of a Support Fgure 4 s a topology optmzaton example where just over one mllon desgn varables were used. Ths structure was optmzed to mnmze stran energy under the appled load. Intal Desgn Fnal Desgn Fgure 14. Skeletal Support V. Summary An algorthm has been developed for solvng very large optmzaton tasks, whch uses lmted central memory and whch avods soluton of a large sub-optmzaton task. The memory requrements grow only lnearly wth problem sze. The algorthm s based on a modern exteror penalty functon method whch exhbts approxmately constant effcency, ndependent of problem sze. Once a contnuous optmum has been acheved, a dscrete soluton s found usng an ad-hoc algorthm whch drves the optmum to a feasble dscrete soluton wth mnmal ncrease n the objectve functon. Experence has shown that the optmzaton problem sze s no longer lmted by the optmzer, but nstead by the ablty of the analyss and senstvty software to provde the needed nformaton. VI. References 1 Schmt, L.A., Structural Desgn by Systematc Synthess, Proceedngs, 2nd Conference on Electronc Computaton, ASCE, New York, 1960, pp Facco, A. V., and G. P. McCormck: Nonlnear Programmng: Sequental Unconstraned Mnmzaton Technques, John Wley and Sons, New York, Hager, W.W., D. W. Hearn and P. M. Pardalos: Large Scale Optmzaton; State of the Art, Kluwer Academc Publshers, 1994, pp Very Large Scale Optmzaton. NASA Langley Research Center Phase I SBIR Contract NAS and contnued under a Phase II Contract NAS BIGDOT User s Manual, Verson 2.0: Vanderplaats Research & Development, Inc., Colorado Sprngs, CO, Rockafellar, R. T.: The Multpler Method of Hestenes and Powell Appled to Convex Programmng, J. Optmzaton Theory Applcatons, vol. 12, no. 6, pp , Amercan Insttute of Aeronautcs and Astronautcs

8 7 Fletcher, R. and C. M. Reeves: Functon Mnmzaton by Conjugate Gradents, Br. Computer J., vol. 7, no. 2, pp , DOT User s Manual, Verson 5.0: Vanderplaats Research & Development, Inc., Colorado Sprngs, CO, Shn, D. K., Gurdal, Z. and Grffn, O. H., Jr., A Penalty Approach for Nonlnear Optmzaton wth Dscrete Desgn Varables, Eng. Opt., Vol. 16, pp , GENESIS User s Manual, Verson 7.5: Vanderplaats Research & Development, Inc., Colorado Sprngs, CO, Amercan Insttute of Aeronautcs and Astronautcs