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1 SYSTEMS OPTIMIZATION LABORATORY DEPARTMENT OF OPERATIONS RESEARCH STANFORD UNIVERSITY STANFORD, CALIFORNIA NIN DTIC A Monotone Complementarity Problem in Hilbert Space by Jen-Chih Yao fl ELECTE MI TECHNICAL REPORT SOL 90-7 APR April 1990 Research and reproduction of this report were partially supported by the National Science Foundation grant DMS ; Office of Naval Research grant N J-1659, and the Department of Energy grant DE-FG03-87ER Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the author and do NOT necessarily reflect the views of the above sponsors. Reproduction in whole or in part is permitted for any purposes of the United States Government. This document has been approved for public release and sale; its distribution is unlimited. oq oo o-3

2 A Monotone Complementarity Problem.>-. in Hilbert Space Jen-Chih Yao AcSloft For April 1990 NTIS CRA & DTJC TAB 0 Department of Operations Research U0; ij f n o c ed Stanford University Stanford, CA By D..triti,, I Abstract Dist, I An existence theorem for a complementarity problem involving a weakly coercive monotone mapping over an arbitrary closed convex cone in a real Hilbert space is established Introduction Let H be a real Hilbert space with inner product (.,.) and norm ". Let K be a nonempty subset of H and f be a mapping from K into H. f is said to be weakly coercive if f is said to be monotone if (x,f(x)) --- o as Vxjl --- oo and x E K. (x - y,f(x) - f(y)) 0 for all x,y E K.

3 f is said to be strictly monotone if the above inequality is strict whenever x and y are distinct. f is said to be strongly monotone if there exists a positive number c such that (x - y,f(x) - f(y)) _ cllax - yll 2 for all x,y E K. A subset K of a real Hilbert space H is said to be a cone if Ax E K for all x E K and all A > 0. Let K be a dosed convex cone in H and dual cone K*, that is, K* ={uehi(ua' )0,Va'EK}. The complementarity problem (CP) is to find x E K such that f(x) E K" and (x,f(x)) = 0. (1) Problem (1) was formulated by Karamardian [7] and has been extensively studied in the literature. See, e.g., [2, 4, 5, 6, 7, 8] and the references therein. The purpose of this paper is to prove an existence theorem for a complementarity problem involving a weakly coercive monotone mapping over an arbitrary closed convex cone in a real Hilbert space. The main result extends some existing results; the method of the proof of this main result is to consider the family of finite-dimensional subspaces by using the known results for finite-dimensional spaces and to show that a certain net of solutions from such subspaces converges to a solution to CP. 2. The Main Result Now we prove the main result. Theorem 2.1. Let K be a closed convex cone in the real Hilbert space H. Let f be a weakly coercive monotone mapping from K into H which is continuous on K fl U for any 2

4 7 finite-dimensional subspace U of H. Then there exists x E K such that f(x) E K* and (x, f(x)) = 0. Proof. Let U be any finite-dimensional subspace of H with K fn U #. and let Pu be the orthogonal projection of H onto U. Let fu = Puf be the composition of Pu and f. Since the adjoint P of Pu is itself, we have limlul-o.oo, ueknu (u,fu(u)) = 00. Therefore fu is weakly coercive on K nl U. By 1, Corollary 2.3), there exists xu E K nl U such that (u-xu, f(xu)) O for allueknu. (2) Then by [7, Lemma 3.1], we have fu(xu) E (K n U)" and (xu,f(xu)) = 0. (3) Let A be the family of all finite-dimensional subspaces U of H with K nl U # 0 and Ku = {xv I U C V E A}. Since f is weakly coercive, it follows from (3) that there exists a constant r > 0 so that Ku C 3,. for all U E A where D, is the closure of the ball with center at 0 and radius r. For U E A, let /u' be the weak closure of Ku. Then the family I u I U E A} has the finite intersection property. Indeed, for U, V E A, let W E A be such that U U V C W. Then Ku fl Kv D Kw # 0. Since F, is weakly compact and Ku" C 3, for all U E A, it follows that nuea Ku/' # 0. Let x E n UEA U'. Suppose u E K is arbitrary and let U E A contain u. Since Ku is bounded and x E IRuW, there exists a sequence {xn} C Ku which converges to x weakly. Since f is monotone, by (2) we have (u - xz,f(u)) > 0 for all n. 3

5 Since (u -., f(u)) is wealdy continuous, we have (u -x,f(u)) > 0 for all u E K. For any u E K and any 0 < t < 1, let ut = tu + (1 - t)x. By substituting ut into (2), we have (ut- X,f(ut)) _ 0 for allo<t< 1. (4) Letting t approach 0 in (4), we get By (5) and [7, Lemma 3.1], it follows that (u -X, f(w)) _ 0 for all u E K. (5) f(x) E K* and (x,f(z)) = 0. The next corollary follows from Theorem 2.1 directly. Corollary 2.2. Let K be a closed convex cone in the real Hilbert space H. Let f be a mapping from K into H which is continuous on K fl U for any finite-dimensional subspace U of H. Then there exists a unique x E K such that f(z) E K* and (z,f(x)) = 0 under each of the following conditions: I. f is strictly monotone and weakly coercive, 2. f is strongly monotone. We note that Corollary extends a result of Nanda and Nanda [8, Theorem] where f is assumed to be strongly monotone and Lipschitzian. 4

6 References 1. M. Aganagi6, "Variational inequalities and generalized complementarity problems," Technical Report SOL 78-11, System Optimization Laboratory, Department of Operations Research, Stanford University, Stanford, California, [See also "Contributions to Complementarity Theory," Ph.D. thesis, Department of Operations Research, Stanford University, Stanford, California, July 1978.] 2. M. S. Bazaraa, J. J. Goode and M. Z. Nashed, A nonlinear complementarity problem in mathematical programming in Banach space, Proc. Amer. Math. Soc. 35 (1972), K. Deimling, "Nonlinear Functional Analysis," Springer-Verlag, Berlin, G. Isac and M. Th6ra, Complementarity problem and the existence of the post-critical equilibrium state of a thin elastic plate, J. Optim. Theory Appl. 58 (1988), G. Isac, Nonlinear complementarity problem and Galerkin method, J. Math. Anal. Appl. 108 (1985), S. Karamardian, Complementarity problems over cones with monotone and pseudomonotone maps, J. Optim. Theory Appl. 18 (1976), S. Karamardian, Generalized complementarity problem, J. Optim. Theory Appl. 8 (1971), S. Nanda and S. Nanda, A nonlinear complementarity problem in mathematical programming in Hilbert space, Bull. Austral. Math. Soc. 20 (1979),

7 UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (Man Da0nIW4 REPORT DOCMkENTA.TION PAGE RFR OPZDGFR I. REORT NMBERGOVT ACCESSION NO4. RECIPIENT'lS CATALOG NUMBER SOL r*-s. TYPEf OF REPORT G PERIOD COVERED A Monotone Complementarity Problem in Hilbert Space Technical Rteport 6. PERPO@ORG 1. REPORT NummaR 7. AUTHORWa 0. CONTRACT Oft GRANT NUMUEW@') Jen-Chih Yao NOOOl PERFORMING ORGANIZATION NAKE AND ADDRESS I. PROGRAM ELEMENT PROJECT, TASK AREAS 6ORK UNIT NUMBER$ Department of Operations Research - SOL Stanford University 1111 MA Stanford, CA It. CONTROLLING OFFICE 1NMEANO ADDRESS $11- REPORT DATE Office of Naval Research - Dept. of the Navy April N. Quincy Street IS. NUMBER OFPAGEts Arlington, VA Up IS. SECURITY CLASS. (of enie repen) UNCLASSIFIED I"a DECJ.ASMJICATIOM/DOWNGRACING 10. DISTRIBUTION STATEMENT (of ilif Rep.i) This document has been approved for public release and sale; its distribution is unlimited. 17. DISTRIBUTION STATEMENT (0900 abel 00m6di 81. "*., HAD@ df* S RSmt) IS. SUPPLEMENTARY NOTES IS. KEY WORDS (CmI*.u. as etvrn. aid. 110"00 a..m me Of 6F*S 410-*--&-)~ Hilbert space; complementarity problem; monotone mapping; weakly coercive mapping. 20. LUST RACT (Cendvee uowe side 41 aeee0ewr -and~& 6 W-* -0n000 An existence theorem for a complementarity problem involving a weakly coercive monotone mapping over an arbitrary closed convex cone in a real Hilbert space is established. DD AN EDITION or i Novsoi *mg-jg nes SECURITY CLASSIICATON OP "WIS PAGE (Vb WS ~

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