Proceedings of the 6th WSEAS International Conference on Simulation, Modelling and Optimization, Lisbon, Portugal, September 22-24,

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1 Proceedigs of the 6th WSEAS Iteratioa Coferece o Simuatio Modeig ad Optimizatio isbo Portuga September O the Impemetatio of a Iterior-Poit SQP Fiter ie Search Agorithm M. FERNANDA P. COSA Departameto de Matemática para a Ciêcia e ecoogia EDIE M. G. P. FERNANDES Departameto de Produção e Sistemas Uiversidade do Miho Campus de Azurém Guimarães PORUGA Abstract: - We preset a fiter ie search sequetia quadratic programg (SQP) method based o a iteriorpoit framewor for oiear programg. Here we provide a comprehesive descriptio of the agorithm icudig the feasibiity restoratio phase for the fiter method. he proposed method has bee impemeted i Fortra 90 ad preiary umerica testig seems to idicate that the method is effective. Key-Words: - Noiear programg SQP Iterior-poit method Fiter method ie search Itroductio he proposed agorithm is a fiter ie search agorithm for sovig oiear optimizatio probems of the form F ( x ) x R s.t. b h( x) b+ r x u () where h : R R for = m ad F : R R are oiear ad twice cotiuousy differetiabe fuctios. r is the vector of rages o the costraits hx ( ) u ad are the vectors of upper ad ower bouds o the variabes x ad b is assumed to be a fiite rea vector. Eemets of the vector r ad u are rea umbers subject to the foowig imitatios: 0 r i u i for = m i =. Costraits of the form b h( x) b+ r are deoted by rage costraits. Note that equaity costraits ca sti be treated as rage costraits with r = 0. et F( x) deote the gradiet of F( x ) ad h( x) deote de Jacobia matrix of the costrait vector h( x) = [ h( x) b b+ r h( x) x u x]. A soutio of () wi be deoted by x * ad we assume that there is a fiite umber of soutios. We aso assume that the first order Kuh-ucer (K) coditios hod (with strict compemetarity) at x *. hus the costraits are verified ad there exists a * agrage mutipier vector λ 0 such that * * * * * F ( x ) = h ( x ) λ h ( x ) λ = 0. () Give a startig poit x 0 the proposed ie search agorithm geerates a sequece of improved estimates x of the soutio for the probem () usig a sequetia quadratic programg (SQP) method. At each iteratio the search directio is the soutio of a quadratic programg subprobem whose objective fuctio approximates the agragia fuctio x ( λ) = Fx ( ) λ hx ( ) ad whose costraits are iear approximatios to the costraits i (). he usua defiitio of the QP subprobem is the foowig: H + F (3) R s.t. b h + h b+ r + x u where hx ( ) deotes the Jacobia matrix of the costrait vector hx ( ) ad F h ad h deote x. he matrix the reevat quatities evauated at H is a symmetric positive defiite approximatio to the Hessia of the agragia fuctio. his probem has a soutio ad a agrage mutipier π 0 that satisfy H F h π + = π ( h + h ) = 0. Ceary the most commo approach for sovig (3) cosiders active set methods (see for exampe [6]). Sovig QP subprobems with equaity costraits is straightforward. However probems that have iequaity costraits are sigificaty more difficut to sove tha probems i which a costraits are equatios sice it is ot ow i advace which iequaity costraits are active at the soutio. I this paper we describe a ew SQP method that is based o the iterior-poit paradigm for sovig the

2 Proceedigs of the 6th WSEAS Iteratioa Coferece o Simuatio Modeig ad Optimizatio isbo Portuga September QP subprobems (3). o promote the goba covergece the fiter techique of Fetcher ad eyffer [] is used to gobaize the SQP agorithm avoidig the use of a merit fuctio ad the updatig of the peaty parameter. he uderyig cocept is that tria iterates are accepted if they improve the objective fuctio or improve the costraits vioatio istead of a combiatio of those two measures defied by a merit fuctio. he paper is orgaized as foows. Sectio describes the iterior-poit method used to sove the QP subprobems. he fiter mechaism is described i Sectio 3 ad Sectio 4 cotais the umerica resuts some cocusios ad future deveopmets. he Iterior-Poit Framewor his sectio describes a ifeasibe prima-dua iterior-poit method for sovig the quadratic subprobem (3). We refer to [8] for detais. Addig oegative sac variabes w p g t (3) becomes H + F (4) s.t. h w = b h h + p = b+ r h g = x + t = u x w p g t 0. he oegativity costraits are the eiated by icorporatig them i ogarithmic barrier terms i the objective fuctio trasforg (4) ito m m H + F µ ( wj) µ ( wj) j= j= µ ( gi) µ ( ti) i= i= subject to the same set of equaity costraits where µ is a positive barrier parameter. Optimaity coditios for this subprobem produce the stadard prima-dua system H + F h y z+ s = 0 y+ q v = 0 WVe = µ e PQe = µ e GZe = µ e Se = µ e h + h b w = 0 r w p = 0 (5) + x g = 0 u x t = 0 where V = diag( v j ) Q = diag( q j ) Z = diag( z i ) S = diag( s i ) W = diag( w j ) P = diag( p j ) G = diag( g i ) ad = diag( t i ) are diagoa matrices y = v q e = (...) ad e = (...) are m ad vectors respectivey. his is a oiear system of 5+5m equatios i 5+5m uows. It has a uique soutio i the strict iterior of a appropriate orthat i prima-dua space {( wgtpyzvsq): wgtpzvsq 0}. he cetra path is a arc of stricty feasibe poits. It is parameterized by the scaar µ ad each poit o the cetra path soves the prima-dua system (5). As µ teds to zero the cetra path coverges to a optima soutio to both prima ad dua probems. For a vaue of µ et ( wg q) deote the curret poit i the orthat. Our aim is to fid the directio vectors ( w q) such that the ew poit ( + w+ w q+ q) ies approximatey o the prima-dua cetra path at the poit ( µ wµ... qµ ). We see that the ew poit ( + w+ w q+ q) if it were to ie exacty o the cetra path at µ woud be defied by H + h y + z s = σ y q+ v = y+ q v β V W v+ w = µ V e w V V w γ w P Q p+ q = µ P e q P P q γ q G Z g z µ G e z G G z γz S t + s = µ e s s γs h w = w+ b h h ρ w+ p = r w p α + = (6) g = x + g υ + t = u x t τ where we have itroduced otatios σ H + F h y z+ s ad β ρ α τ υ γ w γ q γ z γ s as short-hads for the right-had side expressios. his is amost a iear system for the directio vectors ( w q). he oy oiearities appear o the right-side had of the compemetarity equatios (i.e. i γ w γ q γ z γ s the γ-vectors). he agorithm impemets a predictor-corrector [5] approach to fid a good approximatio soutio to the equatios (6). First a predictor directio ( p w p... q p ) is computed from (6) igorig the µ ad -terms of the γ-vectors. he a estimate of a appropriate target vaue for µ is made usig z g s t v w p µ = δ q m + with ( ) ( ) p p p p p p z = z+ α z g = g + α g... q = q+ α q ad p (( p ) ( 0 )) δ = α α + where p α is the ogest step egth that ca be tae aog this directio before vioatig the oegative coditios w g t p z v s q 0 with a upper boud of. he corrector step ( w q) is the obtaied by reistaig

3 Proceedigs of the 6th WSEAS Iteratioa Coferece o Simuatio Modeig ad Optimizatio isbo Portuga September the µ ad the -terms o the γ-vectors i (6). his step is used to move to a ew poit i prima-dua space. Agai we cacuate the imum step α that ca be tae aog this directio before vioatig the oegativity coditios yiedig the ew poit = + α w = w+ α w... q = q+ α q. Impemetatio detais to provide iitia vaues for a the variabes i this iterior-poit paradigm as we as to sove system (6) are described i []. A soutio of the quadratic subprobem is decared prima/dua feasibe if the reative measures of prima 4 ad dua ifeasibiities are ess tha 0. hus the QP subprobem has a soutio ( π ) with = ad π = ( vqzs ). 3 A ie Search Fiter Method i SQP After a search directio has bee computed we cosider a bactracig ie search procedure where a decreasig sequece of step sizes α (0] ( = 0...) with im α = 0 is tried uti a acceptace criterio is satisfied. he procedure that decides which tria step is accepted is a fiter method. raditioay a tria step size α is accepted if the correspodig tria poit x( α ) = x + α λ ( α ) = λ + α ξ ss( α ) = ss + α ζ provides sufficiet reductio of a merit fuctio such as the augmeted agragia fuctio [3] which has the form η ( x λ ss; η) = F ( x) λ ( h( x) ss) + ( x ss) (7) where the ifeasibiity measure ( x ss) is give by ( ) ( xss ) = h( x) ss ad η is a positive peaty parameter. Here ss is a vector of oegative sac variabes that are used oy i the ie search procedure ad at the begiig of iteratio is tae as ss = ( 0 h( x )). We treat the eemets of λ as additioa variabes so that π is used to defie a search directio ξ for the mutipier estimate λ ad the ie search is performed with respect to x λ ad ss. At iteratio a vector tripe d = ( ξ ζ) is computed to serve as directio of search for the variabes ( x λ ss). he vectors ad π are foud from the QP subprobem (3). he ξ is defied as ξ = λ π ad the vector ζ satisfies h + h = ζ + ss from which we ca see that ζ + ss is simpy the residua of the iequaity costraits from probem (3). I order to avoid the deteratio of a appropriate vaue of the peaty parameter η Fetcher ad eyffer [] proposed the cocept of a fiter method i the cotext of a trust regio SQP agorithm. he basic idea behid this approach is to iterpret the optimizatio probem () as a biobjective optimizatio probem with two goas: imizig the costraits vioatio ( x) = ( 0 h( x) ) ad imizig the objective fuctio F( x ). A certai emphasis is paced o the first measure sice a poit has to be feasibe i order to be a optima soutio of (). Foowig this paradigm we propose a approach based o the two compoets of the augmeted agragia fuctio (7): x ( λ ss) = Fx ( ) λ hx ( ) ss (8) ( ) ad ( x ss) (or equivaety ( x ss) ) rather tha o ( x) ad F( x ).(Recety i [7] a reated approach usig the agragia fuctio i a fiter trust regio based method is proposed.) he tria poit ( x ( α ) λ( α ) ss ( α )) is accepted by the fiter if it improves feasibiity i.e. if ( x( α ) ss( α )) < ( x ss) or if it improves the agragia fuctio (8) i.e. if x ( ( α ) λ ( ) ( )) ( ). α ss x ss α < λ Note that this criterio is ess demadig tha the eforcemet of decrease i the peaty fuctio (7) ad might i geera aow arger steps. 3. Sufficiet reductio ie search methods that use a merit fuctio esure sufficiet progress toward the soutio by eforcig a Armijo coditio for the augmeted agragia fuctio (7). Foowig this idea we might cosider the tria poit ( x ( α ) λ ( α ) ss ( α )) durig the bactracig ie search to be acceptabe if the ext iterate provides at east as much progress i oe of the measures or that correspods to a sma fractio of the curret costraits vioatio ( x ss) i.e if ( x ( α ) ss ( α )) ( γ ) ( x ss ) or (9a) ( x( α ) λ( α ) ss( α )) ( x λ ss)- γ ( x ss) (9b) hods for fixed costats γ γ (0). However we chage to a differet sufficiet reductio criterio wheever for the curret iterate we have

4 Proceedigs of the 6th WSEAS Iteratioa Coferece o Simuatio Modeig ad Optimizatio isbo Portuga September ( x ss) for some (0 ] foowig switchig coditios x ( λ ss) d < 0 ad ad the s s ( x ss) d > ( x ss) α λ δ (0) hod with fixed costats δ > 0 s > s > s. If ( x ss ) ad (0) is true for the curret iterate the tria poit ( x( α ) λ( α ) ss( α )) has to satisfy the Armijo coditio x ( ( α ) λ ( α ) ss( α )) x ( λ ss) + + η α x ( λ ss) d () istead of (9) i order to be acceptabe. Here η (00.5) is a costat. I accordace with the previous pubicatios o fiter methods we may ca a tria step size α for which (0) hods a -step size. Simiary if a -step size is accepted as the fia step size α i iteratio we refer to as a -type iteratio. At each iteratio the agorithm aso maitais a fiter here deoted by F { ( ) R : 0 }. Foowig the ideas i [9 0 ] the fiter here is ot defied by a ist but as a set F that cotais those combiatios of costraits vioatio vaues ad agragia fuctio vaues that are prohibited for a successfu tria poit i iteratio. So durig the ie search a tria poit ( x ( α ) λ ( α ) ss ( α )) is rejected if ( ( x ( ) ss ( )) ( x ( ) ( ) ss ( )) ) α α α λ α α F. he authors i [9 0 ] appy this simpified otatio to active set SQP ad barrier iterior-poit ie search based agorithms. At the begiig of the optimizatio the fiter is iitiaized to F = R : () 0 {( ) } for some so that the agorithm wi ever aow tria poits to be accepted that have a costrait vioatio arger tha. ater the fiter is augmeted usig the update formua F + = F ( ) R : (- γ ) ad -γ (3) { } after every iteratio i which the accepted tria step size does ot satisfy the switchig coditios (0). his esures that the iterates caot retur to the eighborhood of x. O the other had if both (0) ad () hod for the accepted step size the fiter remais uchaged. Overa this procedure esures that the agorithm caot cyce for exampe betwee two poits that aterativey decrease the costraits vioatio ad the agragia fuctio. Fiay i some cases it is ot possibe to fid a tria step size α that satisfies the above criteria. We defie a imum desired step size usig iear modes of the ivoved fuctios s γ δ[ ] γ s d d if d < 0 ad α : = γ γ α (4) γ d if d < 0 ad > γ otherwise with a safety factor γ (0]. If the bactracig ie search fids a tria step size α α < α the agorithm reverts to a feasibiity restoratio phase. Here the agorithm tries to fid a ew iterate ( x + λ+ ss+ ) which is acceptabe to the curret fiter ad for which (9) hods by reducig the costraits vioatio withi a iterative process. Our iterior-poit SQP fiter ie search agorithm for sovig iequaity costraied optimizatio probems is as foows: Agorithm Give: Startig poit ( x0 λ 0 ss ) with 0 ss0 = ( 0 h( x0 )) ; costats ( ( x0 ss0) ] > 0 ; γ (0) γ ; δ > 0 ; γα (0] ; s > ; s > s ; η η (0). Iitiaize. Iitiaize the fiter (usig ()) ad the iteratio couter 0.. Chec covergece. Stop if x is a statioary poit of the probem () i.e. if it satisfies the K m coditios () for some λ R. 3. Compute search directio. Compute the search directio ad the agrage mutipier π from the iear system (6) (usig the iterior-poit strategy preseted i Sectio ). 4. Bactracig ie search. ss = 0 h( x ) 4. Iitiaize ie search. Set ( )

5 Proceedigs of the 6th WSEAS Iteratioa Coferece o Simuatio Modeig ad Optimizatio isbo Portuga September ξ = λ π ζ = h + h ss α = Compute ew tria poit. If the tria step size becomes too sma i.e. α < α with α defied by (4) go to feasibiity restoratio phase i step 8. Otherwise compute the tria poits x ( α ) = x + α λ ( α ) = λ + α ξ ss( α ) = ss + α ζ. 4.3 Chec acceptabiity to the fiter. If ( ( x ( α ) ss ( α )) ( x ( α ) λ ( α ) ss ( α )) ) F reject the tria step size ad go to step Chec sufficiet decrease with respect to curret iterate. Case I: α is a -step size (i.e. (0) hods): If the Armijo coditio () for the fuctio hods accept the tria step ad go to step 5. Otherwise go to step 4.5. Case II: α is ot a -step size: If (9) hods accept the tria step ad go to step 5. Otherwise go to step Choose ew tria step size. Set α + α + ad go bac to step Accept tria poit. Set α α x+ x( α) ad λ+ λ( α). 6. Augmet fiter if ecessary. If is ot a -type iteratio augmet the fiter usig (3). Otherwise eave the fiter uchaged. 7. Cotiue with ext iteratio. Icrease the iteratio couter + ad go bac to step. 8. Feasibiity restoratio phase. Use a restoratio agorithm to produce a poit ( x+ λ+ ss ) that + is acceptabe to the fiter i.e. ( ( x+ ss+ ) ( x+ λ ss + + )) F. Augmet the fiter usig (3) ad cotiue with the reguar iteratio i step Feasibiity restoratio phase I this sectio we preset a restoratio agorithm. he tas of the restoratio phase is to compute a ew iterate acceptabe to the fiter by decreasig the ifeasibiity wheever the reguar bactracig ie search procedure caot mae sufficiet progress ad the step size becomes too sma. o compute a tria poit that sufficiety decreases ifeasibiity we itroduce the fuctio ( x ss) = ( h( x) ss). he restoratio agorithm herei preseted wors with the step framewor d = ( ζ ) that shoud be a descet directio for ( x ss). I fact = ( h ss) ( h ss) = < 0. d = ( h ss) h ( h ss) ζ = ( h ss) ( h ζ) Additioay we aso esure that the ew iterate x + does ot deviate too much from the curret iterate x (see step 5 i Agorithm ). Severa other restoratio agorithms are pausibe but we chose the foowig oe because it is cosistet with the step cacuatio of our iteriorpoit SQP fiter ie search method: Agorithm (restoratio agorithm) 0. Set x 0 = x λ 0 = λ ss 0 = ss = 0 ad start with step 4.. If ( x λ ss ) is acceptabe to the fiter (coditios (9)) the set x+ x λ λ + stop restoratio.. Compute ad π by sovig the QP subprobem (3) with ( x λ ) = ( x λ ). 3. Compute ss ( ) d ( ζ ) ξ ζ ad defie the vector = which is used as directio of search for the variabes ( x ss ). 4. Set α =. 5. If ( x ( ) ss ( ) ) ( x ss )+ d α α α η ad x ( α ) x ε ( + x ( α ) ) the set x + = x ( α) ss + = ss ( α) λ + = λ ( α) = + ad retur to step. Otherwise α α ad repeat step 5. 4 Resuts ad Cocusios o test this SQP framewor based o the iteriorpoit strategy with a fiter ie search method we seected 3 sma iequaity costraied probems from the Hoc ad Schittowsi (HS) coectio [4]. he tests were doe i doube precisio arithmetic with a Petium 4 ad Fortra 90. For the successfu teratio of the agorithm the iterative sequece of x-vaues must coverge ad the fia poit must satisfy the first-order Kuh-ucer coditios (see ()) with a 0-4 toerace. he chose vaues for the

6 Proceedigs of the 6th WSEAS Iteratioa Coferece o Simuatio Modeig ad Optimizatio isbo Portuga September costats are simiar to the oes used i []: 4 4 = 0 { ( x ss )} = 0 { ( x ss )} γ 5 = 0 s =.3 η = 0 δ = γ α = 0.05 s =. 4 4 = 0 η = 0 ad ε = 0. where γ x 0 is the startig poit ad ss0 h x0 = (0 ( )). Our umerica resuts are reported i the first part of the abe. Each set of two coums cotais the umber of QP subprobems soved (N QP ) ad the umber of fuctio evauatios (N fe ). For comparative purposes we icude i the secod part of the tabe the resuts obtaied whe a ie search method based o the merit fuctio (7) is used as i [] istead of the herei proposed fiter method. I most probems the resuts are simiar whie i 9 of the 3 probems the fiter method requires ess fuctio evauatios tha the merit fuctio based ie search. Sighty better resuts were obtaied with the merit fuctio i 3 probems. abe : Comparative resuts Probem Fiter Method Merit Fuctio SNOP N QP N fe N QP N fe N QP HS HS HS3 HS4 3 0 HS HS HS HS HS HS HS HS 3 3 HS HS HS30 5 HS HS HS HS HS HS HS HS HS4 3 3 HS HS HS HS55 3 HS HS HS HS We aso icude the resuts obtaied by the sover SNOP a specific SQP impemetatio of a activeset method based o a smooth augmeted agragia merit fuctio which is avaiabe i the NEOS Server ( he compariso with SNOP is ot meat to be a rigorous assessmet of the performace of our agorithm sice the teratio criteria are ot comparabe. However the umerica resuts show that our iterior-poit SQP fiter ie search method is effective o sma dimesioa probems. arge scaig probems testig wi foow i the ear future. Despite some simiarities our method has basicay two differeces from the fiter ie search SQP method proposed ad aayzed i [9 0]: defiitio (4) ad the feasibiity restoratio phase. Future deveopmets wi focus o the goba covergece aaysis of the proposed method. Refereces: [] M.F.P. Costa ad E.M.G.P. Ferades A SQP based o a iterior-poit strategy for oiear iequaity costraied optimizatio probems Proceedigs of the VII Cogreso Gaego de Estatística e Ivestigació de Operaciós ISBN (CD-Rom) 005. [] R. Fetcher ad S. eyffer Noiear programg without a peaty fuctio Mathematica Programg Vo.9 00 pp [3] P.E. Gi W. Murray ad M. Sauders Some theoretica properties of a augmeted agragia merit fuctio echica Report SO 86-6R 986. [4] W. Hoc ad K. Schittowsi est Exampes for Noiear Programg Spriger-Verag 98. [5] S. Mehrotra O the impemetatio of a (primadua) iterior poit method SIAM Joura o Optimizatio Vo. 99 pp [6] J. Noceda ad S.J. Wright Numerica Optimizatio Spriger-Verag 00. [7] S. UBrich O the superiear oca covergece of a fiter-sqp method Mathematica Programg. Vo pp [8] R.J. Vaderbei OQO: A iterior-poit code for quadratic programg echica Report SOR [9] A. Wachter ad.. Bieger ie search fiter methods for oiear programg: motivatio ad goba covergece SIAM Joura o Optimizatio Vo.6 () 005 pp. -3. [0] A. Wachter ad.. Bieger ie search fiter methods for oiear programg: oca covergece SIAM Joura o Optimizatio Vo.6 () 005 pp [] A. Wachter ad.. Bieger O the impemetatio of a iterior-poit fiter ie search agorithm for arge-scae oiear programg. Mathematica Programg. Vo pp

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