A NEW CLASS OF INTERIOR PROXIMAL METHODS FOR OPTIMIZATION OVER THE POSITIVE OCTANT

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1 Pesqusa Opeacoal a Socedade: Educação, Meo Ambete e Desevolvmeto a 5/09/06 Goâa, GO A NEW CLASS OF INERIOR PROXIMAL MEHODS FOR OPIMIZAION OVER HE POSIIVE OCAN Sssy da Slva Souza Uvesdade Fedeal do Ro de Jaeo Cdade Uvestáa, Ceto de ecologa, Bloco H, Ro de Jaeo, RJ sssy@cosufjb Paulo Robeto Olvea Uvesdade Fedeal do Ro de Jaeo Cdade Uvestáa, Ceto de ecologa, Bloco H, Ro de Jaeo, RJ polve@cosufjb RESUMO Neste tabalho apesetamos uma famíla de métodos teo-pomas com métca vaável paa esolve poblemas de otmzação sob estções postvdade O algotmo está defdo a foma eata e o paâmeto de egulazação é escolhdo coveetemete de foma a obte potos teoes Mostamos a boa-defção do algotmo e estabelecemos covegêca faca ao cojuto solução do poblema PALAVRAS CHAVE Algotmos de Potos Iteoes Métodos Pomas Poblemas de Otmzação Pogamação Matemátca ABSRAC I ths wo we peset a famly of vaable metc teo pomal methods fo solvg optmzato poblems ude postvty costat he algothm s defed eact fom ad the egulazato paamete s coveetly chose to guaatee teo pots We show the well defedess of the algothm ad we establsh wea covegece to the soluto set of the poblem pesets KEYWORDS Iteo pot algothms Pomal methods Optmzato poblem Mathematcal Pogammg [ 88 ]

2 Pesqusa Opeacoal a Socedade: Educação, Meo Ambete e Desevolvmeto a 5/09/06 Goâa, GO Itoducto We study the optmzato poblem defed by (OP) m f ( ) st 0 We cosde the followg assumptos: (A) f : R R { + } s a coecve cotuous pope cove fucto; (A) S (OP), the soluto set of (OP), s oempty A well ow method fo solvg poblems such as (OP) s the pomal algothm hs method seems to have bee appled the fst tme to cove poblems by Matet (970) ad the fst mpotat esults the moe geeal case of mamal mootoe opeatos ae due to Rocafella (976) May papes have geealzed the algothm peseted Rocafella (976) (see fo eample, Buach ad Iusem (998), Feea ad Olvea (00), Lemae (989)) ad some of them lead to teo pot pomal methods (see fo stace Auslede at al (999), Kapla ad chatsche (004), Olvea ad Olvea (00)) Fo solvg the poblem (OP) we popose a famly of teo pomal methods whose eels ae metcs geeated by dagoal matces, costucted upo the -powe of the teates, wth, ad the egulazato paamete s coveetly chose at each teato to guaatee the postvty of the teates We eted the techcal epot peseted by Olvea ad Olvea (00), whee the autos study the poblem (OP) by cosdeg the Assumptos (H), (H) ad that the objectve fucto s cotuously dffeetable wth lpschtza gadet, they assume ad the egulazato paametes deped of the Lpschtz costat of the gadet of f We show the same esults obtaed by Olvea ad Olvea (00) ude weae hypotheses tha those assumed by them he stuctue of the wo s smple I Secto we defe the Algothm, we peset the esults obtaed ad two umecal eamples he cocluso s gve Secto 3, ad the efeeces ae secto 4 houghout ths wo, we used the followg otato: R = {(,, ) R : 0, =,, } + ; R = {(,, ) R : > 0, =,, } + + ;, ad deotes the Eucldea e poduct ad the Eucldea om, espectvely; = (,, ) deotes the vecto whch compoets ae the -powe of R ; X deotes the dagoal mat of ode, whch elemets of the dagoal ae the compoets of ma( ) ad m( ) deote, espectvely, the lagest ad smallest compoet of, o equvaletly, the lagest ad smallest egevalue of the mat X Method VMIP ae a eal umbe ad a sequece of postve eal umbes { α } such that 0 < a α N, fo some 0 < a < Algothm VMIP Italzato: Choose ( 0 0, g ) R + + R, g 0 f ( 0 ) [ 88 ]

3 Pesqusa Opeacoal a Socedade: Educação, Meo Ambete e Desevolvmeto a 5/09/06 Goâa, GO Iteatve Step: Gve (, g ) R R, g f ( ), compute β = g (ma( )) + α () + + ad fd (, g ) R R, + g f ( + ) satsfyg + + g + β ( X ) ( ) = 0 () If g = 0 o = stop, else ae = + Well Defedess ad Popetes By usg the Assumpto (A) o f, we ca show that Poposto he sequece { } geeated by the Algothm VMIP s well defed Rema Obseve that to solve the equato () s equvalet to fd β that + = ag m{ f ( )}, whee f ( ) = f ( ) + ( X ) R such Usg the last ema, we ca pove the two et esults Poposto Let { } be the sequece geeated by the Algothm VMIP he the sequece { f ( )} s deceasg ad coveget Lemma Let { } ad β } be the sequeces geeated by the Algothm VMIP he + = 0 { + β < + K ( X ) Fom Lemma, the defto of { β } ad the lowe boudedess of the sequece { α } we obta the followg esult Coollay Let { } ad β } be the sequeces geeated by the Algothm VMIP he + = 0 + < + ( X ) { Covegece Aalyss We use the followg defto about wea covegece of a sequece { } R to a set U R Defto (Iusem (995), Defto ) A sequece { } R s sad to be wealy coveget to a U R f: () he sequece { + } s bouded; () lm ( ) = 0 ; () All cluste pots of { } belog to U + We pove that the sequece { } geeated by the Algothm VMIP coveges wealy to the soluto set S (OP) Fom the Poposto ad Assumpto (A) o f we ca show that [ 883 ]

4 Pesqusa Opeacoal a Socedade: Educação, Meo Ambete e Desevolvmeto a 5/09/06 Goâa, GO + Poposto 3 he sequece {( +, g )} geeated by the Algothm VMIP s bouded hs last poposto mples that Coollay he sequece { β } geeated by the Algothm VMIP s bouded, ad Poposto 4 Let { } be the sequece geeated by the Algothm VMIP he + lm ( ) = 0 + Fom the chaactezato of { } peseted Rema ad some ow popetes, such as mootocty of f ad Cauchy-Schwaz equalty, we ca pove that all teates obtaed by the Algothm VMIP ae teo pots Moeove, by obsevg that g = 0 =,, X g = 0, ad by usg pevou esults we obta that all lmt pot + (, g) of {( +, g )} satsfes the complemetaty codto Smlaly to the poofs peseted Iusem (995) ad Olvea ad Olvea (00), we also show that (, g) R R hus, we coclude that all lmt pots of {( +, g )} satsfy the KK codtos of the poblem (OP), that s, Poposto 5 All lmt pots of the sequece { } geeated by the Algothm VMIP belog to S (OP) Fom Popostos 3, 4 ad 5 esults eoema he sequece { } geeated by the Algothm VMIP coveges wealy to the soluto set S (OP) We also deteme a covegece estmatve: + Poposto 6 Let {( +, g )} ad β } be the sequeces geeated by the Algothm VMIP he fo ay S (OP) the followg covegece estmate holds: f ( ) f ( { ) ma { = 0,,, g + } 3 Eamples We study thee eamples that llustat the applcato of the method VMIP We mplemeted the Algothm VMIP MatlabR Eample We cosde the poblem m Q + c + d s t 0,whee Q =, c = ( 0 ) ad d= We cosde the mplemetato of the Algothm VMIP,, α = 05 +, ad ( + ) soluto s = ( ) ( ) 0 = he eact soluto s 0 = ( ) he eact he algothm stop = 30, whe the dstace betwee successve teates s less tha the zeo of MALAB, ad ( ) s the [ 884 ]

5 Pesqusa Opeacoal a Socedade: Educação, Meo Ambete e Desevolvmeto a 5/09/06 Goâa, GO pot obtaed Eample We cosde the poblem m + e s t 0 We cosde the mplemetato of the Algothm VMIP, = 5, α = 05 +, ad 0 = (5 3) he ( + ) eact soluto s ( 0 0) he algothm stop = 8, whe the dstace betwee successve teates s less tha the zeo of MALAB, ad ( 0 004) s the pot obtaed Eample 3 We cosde the poblem m( 00) + ( 5) s t 0 We cosde the mplemetato of the Algothm VMIP, =, α = 05 +,ad 0 (5 65) ( + ) = he eact soluto s ( 00 5) he algothm stop = 0, whe the dstace betwee successve teates s less tha the zeo of MALAB, ad ( 00 50) s the pot obtaed 3 Cocluso I ths wo we have peseted a class of pomal pot method wth vaable metc fo solvg optmzato poblem o the postve octat We showed that the algothm the eact fom s well defed ad t coveges wealy to the soluto set We peset some eamples that llustate the pefomace of the Algothm VMIP A advatage of ths method s due to the fact that the egulazato paametes ae gve eplctly ad they guaatee the stct vablty of all teates We also study the eact fom of the algothm ad we guaateed the welldefedess ad the popety of wea covegece to the soluto set, but ths class of eact methods s ot dscoused hee Refeêcas Auslede, A ad eboulle, M ad Be-ba, S (999), A Logathmc-Quadatc Pomal Method fo Vaatoal Iequaltes, Computatoal Optmzato ad Applcatos,, 3-40 Buach, RS ad Iusem, AN (998), A geealzed pomal pot algothm fo the vaatoal equalty poblem a Hlbet space, SIAM Joual o Optmzato, 8, 97-6 Feea, OP ad Olvea, PR (00), Pomal Pot Algothm o Remaa Mafolds, Optmzato, 5, Iusem, AN (995), A Iteo Multplcatve Method fo Optmzato Ude Postvty Costats, Acta Applcadae Mathematcae, 38, Kapla, A ad chatsche, R (004), Iteo pomal method fo vaatoal equaltes: case of opaamootoe opeatos, Set-Valued Aalyss,, Lemae, B, he pomal pot, Iteatoal Sees of Numecal Mathematcs, JP Peot, Ed Bhause, Basel, 87, 73-87, 989 Matet, B (970) Règulasato d'équatos vaatoelles pa appomatos succesves, Revue Façase d'ifomatque et Recheche Opéatoelle,, Olvea, GL ad Olvea, PR, A New Class of Pomal Iteo-Pot Methods fo Optmzato ude Postvty Costats, ES-570, PESC/COPPE-UFRJ, 00 ( Rocafella, R (976), Mootoe Opeatos ad the pomal pot algothm, SIAM J Cotol Optm, 4, [ 885 ]