ANALYTIC CENTER CUTTING PLANE METHODS FOR VARIATIONAL INEQUALITIES OVER CONVEX BODIES

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1 ANALYI ENER UING PLANE MEHODS OR VARIAIONAL INEQUALIIES OVER ONVE BODIES Renin Zen School of Mathematical Sciences honqin Normal Universit honqin hina ABSRA An analtic center cuttin plane method is an iterative alorithm based on the computation of analtic centers. In this paper we propose some analtic center cuttin plane methods for solvin quasimonotone or pseudomonotone variational inequalities whose domains are bounded or unbounded conve bodies. Ke Words: Variational inequalit Quasimonotonit Pseudomonotonit Analtic center cuttin plane method onve bod MS lasses: 65K05 65K0 65K Introduction and Preliminaries Some recent developments in solvin variational inequalities are analtic center cuttin plane methods. An analtic center cuttin plane method is an interior alorithm based on the computation of analtic centers. In order to wor with analtic center cuttin plane methods some authors assume that the feasible sets of variational inequalities are poltopes e.. see [ 3 5 6] while others pa more attention to problems with infinitel man linear constraints e.. see [4 0] etc. Analtic center cuttin plane methods also can be used to other tpes of optimization problems lie Mailin address: Mathematics Department Sasatchewan Poltechnic 30 Idlwld Dr. N Sasatoon Sasatchewan anada S7L 4J7. renin.zen@saspoltech.ca

2 mathematical prorammin with equilibrium constraints [9] conve prorammin [6 3] conic prorammin [7] stochastic prorammin [8 9] and combinatorial optimization [3]. In this paper we propose some analtic center cuttin plane methods for solvin pseudomonotone or quasimonotone variational inequalities whose feasible sets are bounded or unbounded conve bodies includin the n-dimensional Euclidean space R n itself. Let be a non-empt subset of the n-dimensional Euclidean space R n and let : R n be a function. We call that a point is a solution of the variational inequalit VI if 0. he point is a solution of the dual variational inequalit VID if of VID. 0. We denote b the set of solutions of VI and b D the set of solutions Given VI[ ] VID[ ] the ap function is defined as ma f ma. We note that 0 0 and herefore we have f { 0; } ar min. ar min f { 0; } Lemma A point is a solution of VI[ ] VID[ ] if and onl if 0 f 0. A point is said to be a ε-solution of the variational inequalit if < ε.

3 3 A function : R n is said to be monotone on if 0 stronl monotone if M with a constant M > 0 quasimonotone on if > 0 0 pseudomonotone on if 0 0 strictl pseudomonotone on if > 0 0 pseudomonotone plus on if it is pseudomonotone on and if 0 0 and stronl pseudomonotone on if there eist constants M > 0 0 > α such that M 0 α. rom Auslender [] we have Lemma. Lemma if is continuous then a solution of VID is a solution of VI ; and if is continuous and strictl pseudomonotone then is a solution of VI if and onl if it is a solution of VID.. ompact onve Bodies A poltope is a set P R n which is the conve hull of a finite set.

4 A polhedron is a set n { R ; A b} R n where b R n and A is an m n matri. Ever poltope is a polhedron whereas not ever polhedron is a poltope. H. Minowsi proved the followin Lemma 3 in 896. Lemma 3 A set P R n is a poltope if and onl if it is a bounded polhedron. We mae the followin assumptions for poltopes throuhout this paper. a Interior Assumption: A poltope is alwas a full-dimensional poltope and that includes 0 e where e is a vector of all ones. We note that if a poltope has non-empt interior then a can be met b rescalin. A conve bod R n is a conve and bounded subset with non-empt interior. A rectanle B R n is defined b n B R ; a b } { n i i i where ai bi R. A rectanle can also be iven b some inequalities n B { R ; H b} where H b is a finite set of hperplanes H is an m n matri. And if we denote b V the finite set of all vertices of B then B con V. 4

5 heorem A bounded subset R n is a compact conve bod if and onl if there eists a sequence of poltopes { } which satisfies + such that U c. Proof. he sufficienc is trivial. We onl prove the necessit. Since is bounded there eists a rectanle B such that B. ae a partition P of B. hen B is divided into a set of finite subrectanles b a finite set of hperplanes. Let D U B where B are all the subrectanles which lie entirel within. Let V be the set of all vertices of B then V is a finite set. So conv is a poltope and it obviousl satisfies D. or the case of a -dimensional Euclidean space see iure. ae a finer partition P of B. Similarl we have a set D U B where B are all the subrectanles which are correspondin to P and lie entirel within ; and we have a poltope conv where V is the set of all vertices of B such that. B mathematical induction there eists a sequence of poltopes {} which satisfies +. It is eas to see that U c. 5

6 iure conv Proposition. It is quite straihtforward to prove the followin orollar Proposition and orollar A subset R n is a compact conve bod if there eists a is uniforml bounded sequence of poltopes { } i.e. B for a iven rectanle B such that U c. Proposition Let R n be a compact conve bod and : R n a continuous function then the variational inequalit VI[ ] has solutions. Proposition Let R n be a compact conve bod and : R n a continuous and strictl psuedomonotone function then the variational inequalit VI[ ] has a unique solution. 3. Generalized Analtic enter uttin Plane Alorithms for Solvin Pseudomonotone Variational Inequalities n or an poltope { R ; A b} is associated with the potential function n { R ; A + s b s s s s ϕ ln s. n i s n i It is nown that an analtic center is the maimizer of the potential function ϕ and the unique solution of the sstem i 0} 6

7 A A Y 0 + s b s e where is a positive dual vector and Y the diaonal matri built upon. An approimate analtic center [] is the maimizer of the potential function ϕ and the unique solution of the sstem A A Z z 0 + s b s e η <. Where z is a dual vector and Z the diaonal matri built upon z. Now we modif Goffin Marcotte and Zhu s [3] Alorithm to solve VI. We propose an alorithm for solvin variational inequalities whose feasible sets are compact conve bodies. rom heorem we have a sequence of variational inequalities VI [ ] induced b the oriinal variational inequalit VI[ ] where the poltope is iven b the linear inequalities A b b R n and A is an m n matri. So we ma appl the alorithm in [3] to each VI [ ]. he followin Alorithm is usin this idea but the alorithm in [3] is applied to VI [ ] for onl certain number of iterations until we et b applin heorem of [3]. < Alorithm. Step. initialization 0 A A b n b { R ; A b }; 7

8 Step. computation of an approimate analtic center ind an approimate analtic center of ; Step 3. stop criterion ompute ma if 0 then SOP else GOO step 4; Step 4. find an ε -solution for ε ompute ma if < then increase b REURN O Step else GOO Step 5; Step 5. cut eneration Set A + A b + b n H { R ; 0} is the new cuttin plane for VI Increase b one GOO Step.. heorem Let : R n be pseudomonotone plus on a compact conve bod then Alorithm either stops with a solution of VI after a finite number of iterations or there eists a subsequence of the infinite sequence { } that converes to a point. Proof. Accordin to Alorithm and heorem of [3] for an iven after finite number of iterations such that 8

9 9 <. Since is compact there eists a subsequence { q} of { } and a point such that lim q q. p < we have p <. On the other hand due to the compactness of N > 0 such that N. Since 0 ma and 0 ma for. ] ma [ ] [ ] [ ma ma ma ma N B the continuities of and is a continuous function on. onsequentl p 0 lim lim lim q q q q q q q p p. hen we have 0. ma ma U U On the other hand U } { i such that i i lim.

10 Because U i 0 ma U we have herefore lim i 0. ma 0. i Which deduces that is a solution of VI. Alorithm usuall enerates an infinite sequence. In order to terminate at a finite number of iterations we chane the stop criterion Step 3 in Alorithm to et the followin Alorithm. Alorithm. Step Step Step 4 and Step 5 are same as those of Alorithm. Step 3. stop criterion ompute ma if < ε then SOP else GOO step 4. rom heorem we have heorem 3 Let : R n be pseudomonotone plus on a compact conve bod then Alorithm stops with a ε-solution of VI after a finite number of iterations. 0

11 4. Generalized Analtic enter uttin Plane Alorithms for Solvin Quasimonotone Variational Inequalities In this section we are oin to modif Marcotte and Zhu s [] approach to solve quasimonotone variational inequalities VI. We assume that the feasible sets are compact conve bodies. rom heorem there is a sequence of variational inequalities VI [ ] induced b the oriinal variational inequalit VI[ ]. Accordin to [] the followin are the elements required in the construction of alorithms for solvin quasimonotone variational inequalities. or an iven let the auiliar function and stron monotone in i.e. Γ : R R n n be continuous in Γ Γ β for β > 0. β is said to be the stron monotonicit constant for Γ : R R n n. he function Γ is associated with the variational inequalit AVI[Γ ] whose solution w satisfies Γ w Γ + w 0. It is nown that w are continuous [5] and that is a solution of VI[] if and onl if it is a fied point of w. Let ρ and α be positive numbers less than and β respectivel. Let l which depends on be the smallest nonneative inteer for which If Define is a solution of VI[] then l + ρ w w α w. l G + ρ w. w l 0 and G.

12 Alorithm 3. Step. initialization Let β > 0 be the stron monotonicit constant for Γ : R R n n with respect to and let α 0 β. 0 A A b n b { R ; A b }; Step. computation of an approimate analtic center ind an approimate analtic center of ; Step 3. stop criterion ompute ma if 0 then SOP else GOO step 4; Step 4. find an ε -solution for ε ompute ma if < then increase b REURN O Step else GOO Step 5; Step 5. auiliar variational inequalit Let w satisfies the variational inequalit Let + Γ w Γ w 0.

13 3 l w + ρ and G where l is the smallest inteer which satisfies l w w w + α ρ Step 6. cuttin plane eneration Set + G A A + G b b 0} ; { n G H R is the new cuttin plane for VI. Increase b one GOO Step. B heorem of [] similar to the proof of heorem we have heorem 4 Let : R n be Lipschitz continuous i.e. there eists a constant L > 0 such that L on a compact conve bod and D be nonempt. hen Alorithm 3 either stops with a solution of VI after a finite number of iterations or there eists a subsequence of the infinite sequence { } that converes to a point. Alorithm 4. Step Step Step 4 Step 5 and Step 6 are same as those in Alorithm 3. Step 3. stop criterion ompute ma if ε < then SOP else GOO step 4.

14 B heorem 4 we have and heorem 5 Let : R n be Lipschitz continuous on a compact conve bod D be nonempt. hen Alorithm 4 stops with a ε-solution of VI after a finite number of iterations. 5. Generalized Analtic enter uttin Plane Alorithms for Variational Inequalities with Unbounded Domains his section presents an analtic center cuttin plane alorithms for solvin a stronl pseudomonotone variational inequalit VI[ ] whose domain is an unbounded conve bod includin n-dimensional Euclidean space of R n itself. B use of Proposition and due to the pseudomonotonicit VI[ ] has an unique solution over. Let { } be a sequence of poltopes that satisfies then VI[ ] have an unique solution c + and U over. We can alwas assume that contains all boundar points of if there are an. Since the solution of VI[ ] is a fied point lies in if is lare enouh sa > therefore b Lemma >. he followin Alorithm 5 is proposed here to find. Alorithm 5. Step. initialization 0 A A b n b { R ; A b }; 4

15 Step. find an ε -solution for ε ind an approimate analtic center of ompute ma if < then increase b REURN O Step else GOO Step 3; Step 3. cut eneration Set A + A b + b n H { R ; 0} is the new cuttin plane for VI Increase b one GOO Step.. heorem 6 Let : R n be stronl monotone on then Alorithm 5 either stops with a solution of VI after a finite number of iterations or there eists a subsequence of the infinite sequence { } that converes to a point. that [8] Proof. is stronl monotone on implies that there eists a constant N > 0 such Let ma N. be the unique solution of VI[ ] over. Suppose all boundar points of if there are an are in. hen if is lare enouh sa > b Lemma we have >. If Alorithm 5 does not stop after finite number of iterations then eists an infinite sequence { } with such that 5

16 Hence N N <. ma < > which implies that { } is a bounded sequence. herefore subsequence of { } which is converent to in. Similar to the proof of heorem is a solution for VI[ ] and so. We notice that in the proof of heorem 6 the e is that { subsequence. herefore similarl have the followin heorem 7. } is a bounded hreorem 7 Let : R n be stronl pseudomonotone on then Alorithm 5 either stops with a solution of VI after a finite number of iterations or there eists a subsequence of the infinite sequence in that converes to a point. heorem 6 and 7 state that Alorithm 5 can alwas stop and output an approimate solution after finite number of iterations. 6. onclusion Remar his paper wors with variational inequalities whose feasible sets are bounded or unbounded conve bodies. We present some analtic center cuttin plane alorithms that etend the alorithms proposed in [ 3 6]. We should mention that our approach can be used to etend man interior methods which are associated with polhedral feasible reions e.. the alorithms iven b [5 6]. We can also etend some other alorithms for variational inequalities over polhedral feasible sets [7 7 8]. 6

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