Local Approximation of Pareto Surface

Transcription

1 Proceedngs o the World Congress on Engneerng 007 Vol II Local Approxmaton o Pareto Surace S.V. Utyuzhnkov, J. Magnot, and M.D. Guenov Abstract In the desgn process o complex systems, the desgner s solvng an optmzaton problem, whch nvolves erent dscplnes and where all desgn crtera have to be optmzed smultaneously. Mathematcally ths problem can be reduced to a vector optmzaton problem. he soluton o ths problem s not unque and s represented by a Pareto surace n the space o the obectve unctons. Once a Pareto soluton s obtaned, t may be very useul or the decson-maker to be able to perorm a quck local approxmaton n the vcnty o ths Pareto soluton n order to explore ts senstvty. In ths paper, a method or obtanng lnear and quadratc local approxmatons o the Pareto surace s derved. he concept o a local quck Pareto analyser s proposed. hs concept s based on a local senstvty analyss, whch provdes the relaton between varatons o the erent obectve unctons under constrants. A ew examples are consdered. Index erms Pareto surace approxmaton, mult-obectve optmzaton, senstvty analyss, trade-o. I. INRODUCION In the process o desgnng complex systems, contrbutons and nteractons o multple dscplnes are taken nto account to acheve a consstent desgn. In practce, the desgn problem s made even more complcated because the decson maker (DM) has to consder many erent and oten conlctng crtera. In act, durng the optmzaton process, the DM oten has to make compromses and look or trade-o solutons rather than a global optmum, whch usually does not exst. Mult-dscplnary desgn optmzaton (MDO) has become a eld o comprehensve study or the last ew decades, especally snce the computer power has begun to satsy some mnmal requrements to tackle ths problem. MDO embodes a set o methodologes, whch provde means o coordnatng eorts and perormng the optmzaton o a complex system. wo undamental ssues assocated wth the MDO concept are the complexty o the problem (large number o varables, Manuscrpt receved March 1, 006. he research reported n ths paper has been carred out wthn the VIVACE Integrated Proect (AIP C ) whch s partly sponsored by the Sxth Framework Programme o the European Communty under prorty 4 Aeronautcs and Space. S.V. Utyuzhnkov s wth the School o Mechancal, Aerospace & Cvl Engneerng, Unversty o Manchester, Sackvlle Street, P.O. Box 88, Manchester, UK, M60 1QD (e-mal: S.Utyuzhnkov@manchester.ac.uk). J. Magnot s wth the Department o Aerospace Engneerng, School o Engneerng, Craneld Unversty, Craneld, MK4 0AL, UK (e-mal:.p..magnot.00@craneld.ac.uk). M.D. Guenov s wth the Department o Aerospace Engneerng, School o Engneerng, Craneld Unversty, Craneld, MK4 0AL, UK (e-mal: m.d.guenov@craneld.ac.uk). constrants and obectves) and the culty to explore the whole desgn space. hus, n practce the DM would benet rom the opportunty to obtan addtonal normaton about the model wthout runnng t extensvely. Fndng a soluton to an MDO problem mples solvng a vector optmzaton problem under constrants. In general, the soluton o such a problem s not unque. In ths respect, the exstence o easble solutons,.e. solutons that satsy all constrants, but cannot be optmzed urther wthout compromsng at least one o the other crtera leads to the Pareto optmal concept [1]. Each Pareto pont s a soluton o the mult-obectve optmzaton problem. he DM oten selects the nal desgn soluton among an avalable Pareto set based on addtonal requrements that are not taken nto account n the mathematcal ormulaton o the vector optmzaton problem. In spte o the exstence o many numercal methods or non-lnear vector optmzaton, there are ew methods sutable or real-desgn ndustral applcatons. In many applcatons, each desgn cycle ncludes tme-consumng and expensve computatons o each dscplne. In prelmnary desgn t s mportant to get maxmum normaton on a possble soluton at a reasonably low computatonal cost. hus, t s very desrable or the DM to be able to approxmate the Pareto surace n the vcnty o a current Pareto soluton and to provde ts senstvty normaton []. It would also be very useul or the DM to be able to carry out a local approxmaton o other optmal solutons relatvely quckly wthout addtonal ull-run calculatons. Such an approach s based on a local senstvty analyss (SA) provdng the relaton between varatons o erent obectve unctons under constrants. Currently, only a ew papers are devoted to the SA o Pareto soluton n MDO []-[6]. hey are based on the applcaton o the gradent proecton method (GPM) [7] whch was rst used n []. he SA analyss based on a local lnear approxmaton geometrcally results n ndng the hyperplane tangent to the Pareto surace at some Pareto pont. he quadratc approxmaton s based on an approxmate evaluaton o the local Hessan []. Such an approxmaton s based on the assumpton o the local avalablty o some other Pareto solutons. he generaton o such solutons can be done by the method developed n [8], [9]. However, ths assumpton may not be always vald. One o the most cult problems n the SA s related to possble non-smoothness o the Pareto surace n the obectve space and s addressed n [5], [6]. he obectve o ths work s to develop a method or local trade-o analyss and approxmaton o the Pareto surace at a erentable Pareto soluton. Lnear and quadratc analytcal local approxmatons o the Pareto ront are obtaned. It s

2 Proceedngs o the World Congress on Engneerng 007 Vol II shown that the lnear approxmaton o the Pareto surace obtaned n [] determnes n the obectve space a local hyperplane tangent to the Pareto surace only under partcular condtons. he concept o a local quck Pareto analyzer based on the local lnear and quadratc approxmatons o the Pareto surace s suggested. It enables the DM to analyze the trade-o between erent obectve unctons wthout ull tme-consumng optmzaton. Whle mprovng one obectve uncton the DM has an opportunty to determne the trade-os to be made on the others. In addton, t s possble to evaluate the gan o one obectve uncton at the expense o another one. II. MULI-OBJECIVE OPIMIZAION PROBLEM An optmzaton problem s descrbed n terms o a desgn varable vector x = (x 1,x,,x N ) n the desgn space X N. A uncton M evaluates the qualty o a soluton by assgnng t to an obectve vector y = (y 1,y,,y M ) where each obectve y = (x), : N 1, =1,, M n the obectve space Y M. hus, X s mapped onto Y by : X Y. A mult-obectve optmzaton problem can be ormulated n the ollowng orm: Subect to L nequalty constrants Mnmze [ yx ( )] (.1) g( x) 0 = 1,..., L (.) whch may also nclude equalty constrants. A easble desgn pont s a pont that does not volate any constrants. hereore the easble desgn space X * s dened as the set {x g (x) 0, =1,,L}. he easble crteron (obectve) space Y * s dened as the set {Y(x) x X * }. A desgn vector a (a X * ) s called a Pareto optmum, and only, t does not exst any b X * such that y (b) y (a), =1,, M and there exst 1 M such that: y (b) < y (a). Here and urther t s supposed that all vectors are consdered n the approprate Eucldean spaces. III. PAREO APPROXIMAION In ths secton, we assume that the Pareto surace s smooth n the vcnty o the Pareto soluton under study. A local approxmaton o the Pareto surace would allow the DM to obtan quckly both qualtatve and quanttatve normaton on the trade-o between erent local Pareto optmal solutons. A constrant s sad to be actve at a Pareto pont x * o the desgn space X a strct equalty holds at ths pont []. In ths secton, t s assumed that constrants that are actve at a partcular Pareto pont reman actve n ts vcnty. hus, the senstvty predcted at the gven Pareto pont s vald untl the set o actve constrants remans unchanged [], []. Wthout loss o generalty, let us assume that the rst I constrants are actve and the rst Q o those correspond to nequalty constrants (Q I L). Let us note the set o actve constrants (.) as G I. At the gven pont x * o the desgn easble space X * t means: * G(x ) = 0. (.1) Assume that G C 1 ( I ), then locally the constrants can be wrtten n the lnear orm: * J(x x ) = 0, (.) where J s the Jacoban o the actve constrants set at x * : J= G. I all gradents o the actve constrants are lnearly ndependent at a pont, then ths pont s called a regular pont [1]. hus, we say that a pont x * X * s regular rank(j) = I. Let us urther assume that n the obectve space Y the Pareto surace s gven by: S ( y ) = 0 (.) and at the Pareto pont y * = (x * ) uncton S C ( 1 ). he values o the gradent o any erentable uncton F at pont x * under constrants are dened by the reduced gradent ormula (see, e.g., [10]): F S = P F (.4) l where S l s the hyperplane tangent to the easble space X * : S = { x J( x-x ) = 0} (.5) l and P s proecton matrx onto ths hyperplane : * 1 P = IJ ( JJ ) J. (.6) Drectonal dervatves on correspondng to (.4) n the obectve space are represented by: df df dx = S l. dx (.7) he rst element o the product corresponds to the reduced gradent. In the second element, dx represents the nntesmal change n the desgn vector x requred to accommodate the nntesmal sht n the obectve vector tangent to the Pareto surace. he last dervatve n (.7) can be represented va the gradents n the desgn space X as ollows. Assume that matrx P, ( = ( 1,,, M ) ) has n < M lnearly ndependent columns. It s to be noted that n M. Indeed, snce = ( P ) dx, (.8) n = M would mean that or any, n partcular one where all obectves are mproved together, there would exst a dx so that the set o actve constrants remans unchanged. hs contradcts the act that the pont under study s a Pareto soluton. Indeed, n vew o (.) we have M S = 0 (.9) = 1 S

3 Proceedngs o the World Congress on Engneerng 007 Vol II and t s easy to see that to move locally on the Pareto surace, ( = 1,,M) cannot be chosen ndependently. Wthout loss o generalty, let us assume urther that the rst n components o P are lnearly ndependent and represented by: hereore, (.8) reduces to P ( P,..., P ). (.10) 1 n = ( P ) dx. (.11) Now, let us wrte dx n the ollowng orm: dx= A. (.1) hen, havng multpled both sdes o (.1) by ( P ) and takng nto account (.11) we obtan that Hence, ( P ) A =. (.1) 1 [( ) ]. (.14) A = P P P hus, matrx A s the rght-hand generalzed nverse matrx to ( P ). It s possble to prove that the nverse matrx 1 [( P ) P ] s always non-sngular because all the vectors P, ( = 1,, n ) are lnearly ndependent. From the denton o matrx A t ollows that ( P ) A = I and AP = δ, where I s the unt matrx and δ s the Kronecker symbol. Hence, the system o vectors { A } ( = 1,..., n ) creates the bass recprocal to the bass o vectors { P } ( = 1,..., n ). From (.14) t ollows that PA = A and dx n (.1) belongs to the tangent plane S l at the Pareto pont. hus, dx 1 = A P [( P ) P ] (.15) and or any n : dx = A, (.16) where A = ( A1, A,..., An ). hen, rom (.4), (.7) and (.15) t ollows that or any n : df = ( P F) A = A P F = A F (.17) I F =, (n < M), then we can obtan the senstvty o an obectve along the easble descent drecton o an obectve. hus, = A (0 n, n < M). (.18) It s mportant to note that ths ormula concdes wth the ormula: ( P, P ) (, P ) = (.19) ( P, P ) (, P ) obtaned n [] and only ether the vectors P create an orthogonal bass or n = 1. In ths case, the matrx ( P ) P s dagonal. In partcular, these ormulas always concde n the case o two-obectve optmzaton snce n = 1. On the Pareto surace n the obectve space the operator o the rst dervatve can be dened by: d = A. (.0) By applyng ths operator to the rst order dervatve ound prevously, one can obtan the reduced Hessan as ollows: d F ( ) F F n = A A A A (0, ). (.1) hus, the Pareto surace can be locally represented as a lnear hyperplane: n ds Δ = 0 (.) or a quadratc surace: n where = (x * ). = 1 n k 1, k 1 k ds 1 d S Δ + Δ Δ = 0 (.) Approxmatons (.) and (.) can be rewrtten wth respect to the trade-o relatons between the obectve unctons as ollows: n * p p p = 1 = + Δ ( p = n + 1,..., M), (.4) n n * p 1 ( p) p p k k = 1, k= 1 = + Δ + H Δ Δ ( p = n + 1,..., M), where H d ( p) p k k =. (.5)

4 Proceedngs o the World Congress on Engneerng 007 Vol II Quadratc approxmaton (.5) wth n = M - 1 s used n [] where the reduced Hessan matrx H s evaluated wth a least-squared mnmzaton usng the Pareto set generated around the orgnal Pareto pont. In an ndustral stuaton, such evaluaton can be unsutable because t would requre generatng more Pareto ponts n the vcnty o the pont under study. Instead, the local determnaton o the reduced Hessan usng (.1) s more accurate and s entrely based on the value o the obectve and constrant gradents wth respect to the ndependent desgn varables. hese gradents are calculated and used durng the optmzaton procedure; thereore the local approxmatons can be obtaned at no extra computatonal cost. It s mportant to note that n contrast to [] the developed approxmatons precsely correspond to the rst three terms o the aylor expanson n the general case. IV. LOCAL QUICK PAREO ANALYSIS he rst order dervatves p provde us wth rst order senstvty o an obectve p along the easble descent drecton o an obectve when all other obectves are kept constant. It s to be noted here that all the dervatves p are non-postve (1 n and n < p M). Otherwse, two obectves could be locally mproved whch would contradct the Pareto-soluton assumpton. he local approxmatons o the Pareto surace can be used to study the local adaptablty o a Pareto soluton. Snce n a real-le problem t can be very computatonally expensve to obtan even a sngle Pareto soluton, local approxmate solutons around a Pareto pont can be obtaned usng ether (.4) or (.5). As dscussed above, n the prelmnary desgn t can be very benecal to the DM s/he s able to perorm quck SA o the soluton obtaned. Usng the local approxmaton o the Pareto surace the DM has an opportunty to perorm the SA wthout addtonal ull-run computatons. It s also easy to obtan the normaton on trade-o between erent obectves. Usually, the number o obectves consdered n an ndustral case s larger than two. In ths case, the change o one obectve does not ully determne the changes o the others. I the DM reezes all obectves apart rom two or three, t s then possble to obtan normaton whch s useul or understandng the trade-o between the selected obectves. he analyss o solutons around a Pareto pont allows the DM to correct locally the soluton wth respect to addtonal preerences. Furthermore, the DM s able to analyze possble volatons o the constrants as part o the trade-o analyss. In the desgn practce, the opportunty o urther mprovement o some obectves at the expense o local degradaton o some other obectves can also be mportant. Representatons (.4) and (.5) are only local approxmatons and there s a queston on the range o Δx where the approxmaton s vald. In the ramework o a local analyss, gvng a strct answer to ths queston s not possble. Nevertheless, t s possble to evaluate qualtatvely the relable range o the varaton o Δx by comparng the solutons obtaned by the lnear and quadratc approxmatons. It s reasonable to expect that the approxmatons are sutable as long as the erence between the two approxmatons s small. As a qualtatve example, let us consder the case o an optmzaton problem wth three obectves and assume that lnear and quadratc approxmatons or obectve are avalable. Assume that the DM compromses obectve and mproves obectve 1. he local approxmatons (.4) and (.5) provde the DM wth the normaton on how obectve s aected. I the dscrepances between the lnear and quadratc approxmaton reman relatvely small n some norm, the local approxmaton may be consdered as relable, as shown n Fgure Lnear Quadratc Fgure 1: «Relable» local Pareto approxmaton Otherwse, the local approxmaton s not relable or the chosen range o varaton o the desgn varables, as llustrated n Fgure Lnear Quadratc Fgure : «Non-relable» local Pareto approxmaton In general, the Pareto surace can be non-smooth. I the desgned Pareto soluton appears at a pont o lack o smoothness, the approxmatons derved above are not ormally vald. In such a case a substantal dscrepancy can appear between the rst and second order approxmatons n the vcnty o the pont. In the SA, due to a perturbaton δ and the approprate dsplacement δx some constrants, whch are nactve at pont x *, can become ether volated or actve. he exact vercaton o the constrants valdaton may be tme consumng. In [6], t s suggested to obtan a local lnear approxmaton o the nactve constrants at x * to study the degree o constrant volaton. he approach developed n secton three above can be used to obtan the approprate lnear and quadratc approxmatons or non-actve nequalty constrants as ollows:

5 Proceedngs o the World Congress on Engneerng 007 Vol II dg g ( x) = g ( x ) + Δ, ( I < k L), (4.1) * k k k * k 1 k k = k + dg d g Δ + Δ g ( x) g ( x ) ( ). (4.) Note that (5.5) are erent rom the exact analytcal rst order dervatves (5.4). hey result n the approxmaton gven n Fgure. hese equatons can be used to very that nactve constrants reman nactve at a new approxmate Pareto pont. Such vercaton s necessary to ensure that the assumpton that the set o actve constrants remans unchanged s vald and thereore that the approxmaton s legtmate. V. EXAMPLE o compare wth the approach descrbed n [], let us consder the ollowng mult-obectve problem: Mnmse: Subect to: = (x, y, z). (5.1) g( x) = 1x y z 0, x > 0, y > 0, z > 0. (5.) he desgn space and obectve space concde n ths example. It s easy to see that the Pareto surace corresponds to the part o the unt sphere n the rst quadrant and s represented by the ollowng ormulas: Fgure : Lnear approxmaton [] (unt sphere) Accordng to the method developed n ths paper, we obtan: 1 0 A = 0 1. (5.6) x y z z Usng (.18) and (.1), one can easly ensure that we obtan the exact rst order and second order dervatves. he resultng lnear and quadratc approxmatons are shown n Fgure 4 and Fgure 5 respectvely. z = 1x y, x > 0, (5.) y > 0. he analytcal rst order dervatves can be easly derved: Fgure 4: New lnear approxmaton (unt sphere) dz x 1 dx 1 x y dz y dy 1 x y,. (5.4) Let us derve the rst order approxmaton usng approach [] and the method descrbed n ths paper. Usng (.19) and (.6) one can obtan the rst order dervatves as n []: Fgure 5: New quadratc approxmaton (unt sphere) d -xz ( 1-x -y )(-x) d 1 [] y + z 1-x 1-x -y d yz ( ) ( 1x y )(-y) d [] x + z 1y 1x y ( ),. (5.5) he relatve error o the predcton n obectve s gven n Fgure 6 n uncton o the local changes n obectve 1 and.

6 Proceedngs o the World Congress on Engneerng 007 Vol II Fgure 6: Relatve error n predctng (unt sphere) VI. CONCLUSION A method or local approxmaton o the Pareto ronter s presented n ths paper. he exact general ormulas or the rst and second order approxmatons are derved. An approach s suggested to evaluate the vcnty o the Pareto soluton where the local analyss s vald. he developed concept o the local Pareto analyser allows the decson maker to perorm a local analyss o the Pareto solutons and trade-os between erent obectves. Future work wll concentrate on testng and applcaton o the method to complex MDO ndustral test cases. REFERENCES [1] K.M., Mettnen, Nonlnear Multobectve Optmzaton, Kluwer Academc, Boston, [] S., Hernandez, A general senstvty analyss or unconstraned and constraned Pareto optma n Multobectve Optmzaton, AIAA-188-CP, Proceedngs o the 6 th AIAA/ASME Structures Dynamcs and Materals Conerence, [] R.V., appeta, and J.E., Renaud, Interactve MultObectve Optmzaton Procedure, AIAA Paper , Aprl [4] R.V., appeta, J.E., Renaud, A., Messac, and G.J., Sundarara, Interactve physcal programmng: traeo analyss and decson makng n multdscplnary optmzaton, AIAA Journal, 8, Vol.5, 000, pp [5] W.H., Zhang, Pareto senstvty analyss n multcrtera optmzaton, Fnte Elements n Analyss and Desgn, 9, 00, pp [6] W.H., Zhang, On the Pareto optmum senstvty analyss n multcrtera optmzaton, Internatonal Journal or Numercal Methods n Engneerng, 58, 00, pp [7] J.B., Rosen, he gradent proecton method or nonlnear programmng. Part I. Lnear constrants, Journal o the Socety or Industral and Appled Mathematcs, 1, Vol.8, 1958, pp [8] S.V., Utyuzhnkov, M.D., Guenov, and P., Fantn, Numercal method or generatng the entre Pareto ronter n multobectve optmzaton, Proceedngs o Eurogen 005, Munch, September 1-14, 005. [9] M.D., Guenov, S.V., Utyuzhnkov, and P., Fantn, Applcaton o the moed physcal programmng method to generatng the entre Pareto ronter n multobectve optmzaton, Proceedngs o Eurogen 005, Munch, September 1-14, 005. [10] R., Fletcher, Practcal methods o optmzaton, John Wley & Sons, 1989.