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Complementarity Problems and Variational Inequalities. A Unified Approach of Solvability by an Implicit Leray-Schauder Type Alternative

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Abstract

In several recent papers we obtained existence theorems for complementarity problems and variational inequalities using for each of them a particular notion of exceptional family of elements. Now, in this paper we introduce a new notion of exceptional family of elements. This notion is based on an Implicit Leray-Schauder Alternative. By this new notion we obtain a unification of the study of solvability of complementarity problems and of variational inequalities. The paper is finished with a section dedicated to variational inequalities with δ-pseudomonotone operators.

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References

  1. D. Aussel (1998) ArticleTitleSubdifferential properties of quasiconvex and pseudoconvex functions: A unified approach J. Opt. Theory Appl. 97 29–45 Occurrence Handle10.1023/A:1022618915698

    Article  Google Scholar 

  2. Baiocchi, C. and Capelo, A. (1984) Variational and Quasivariational Inequalities, Applica-tions to Free-Boundary Problems,John Wiley and Sons.

  3. H. Ben-El-Mechaiekh A. Idzik (1994) ArticleTitleA Leray-Schauder type theorem for approximable maps Proc. Amer. Math. Soc. 122 IssueID1 105–109

    Google Scholar 

  4. M. Bianchi S. Schaible (1996) ArticleTitleGeneralized monotone bifunctions and equilibrium problems J. Opt. Theory Appl. 90 31–43

    Google Scholar 

  5. M. Bianchi N. Hadjisavvas S. Schaible (1997) ArticleTitleVector equilibrium problems with generalized monotone bifunctios J. Opt. Theory Appl. 92 531–546

    Google Scholar 

  6. E Blum W. Oettli (1993) ArticleTitleForm optimization and variational inequalities to equilibrium problems The Mathematics Student 63 1–23

    Google Scholar 

  7. Bulavski, V. A., Isac, G. and Kalashnikov, V. V. (1998) Application of topological degree to complementarity problems, In: Multilevel Optimization: Algorithms and Applications (Eds. A. Migdalas, P. M. Pardalos and P. Warbrant) Kluwer Academic Publishers, 333–358.

  8. V. A. Bulavski G. Isac V. V. Kalashnikov (2001) ArticleTitleApplication of topological degree theory to semi-definite complementarity problem Optimization 49 405–423

    Google Scholar 

  9. Carbone A. and Zabreiko P.P. (2002). Some remarks on complementarity problems in a Hilbert space, (Preprint).

  10. Clarke, F. H. (1983) Optimization and Nonsmooth Analysis, Wiley-Interscience, Wiley.

  11. R. W. Cottle J. C. Yao (1992) ArticleTitlePseudo-monotone complementarity problems in Hilbert spaces J. Opt. Theory Appl. 75 IssueID2 281–295 Occurrence Handle10.1007/BF00941468

    Article  Google Scholar 

  12. J. P. Crouzeix (1997) ArticleTitlePseudomonotone variational inequality problems: existence of solutions Math. Programming 78 305–314 Occurrence Handle10.1016/S0025-5610(96)00074-3

    Article  Google Scholar 

  13. J. P. Crouzeix J. A. Ferland (1996) ArticleTitleCriteria for differentiable generalized monotone maps Math. Programming 75 399–406 Occurrence Handle10.1016/S0025-5610(96)00026-3

    Article  Google Scholar 

  14. A. Daniilidis N. Hadjisavvas (1999) ArticleTitleOn the subdifferentials of quasiconvex and pseudoconvex functions and cyclic monotonicity J. Math. Anal. Appl. 237 30–40 Occurrence Handle10.1006/jmaa.1999.6437

    Article  Google Scholar 

  15. A. Daniilidis N. Hadjisavvas (1999) ArticleTitleCharacterization of nonsmooth semistrictly quasiconvex and strictly quasiconvex functions J. Opt. Theory Appl 102 IssueID3 525–536 Occurrence Handle10.1023/A:1022693822102

    Article  Google Scholar 

  16. M.S. Gowda (1990) ArticleTitleAffine pseudomonotone mappings and the linear complementarity problems SIAM J. on Matrix Anal. Appl. 11 373–380 Occurrence Handle10.1137/0611026

    Article  Google Scholar 

  17. N. Hadjisavvas S. Schaible (1996) ArticleTitleQuasimonotone variational inequalities in Banach spaces J. Opt. Theory Appl. 90 IssueID1 95–111

    Google Scholar 

  18. N. Hadjisavvas S. Schaible (1998) ArticleTitleFrom scalar to vector equilibrium problems in the quasimonotne case J. Opt. Theory Appl. 96 IssueID2 297–309 Occurrence Handle10.1023/A:1022666014055

    Article  Google Scholar 

  19. Hadjisavvas, N. and Schaible, S. Quasimonotonicity and pseudomonotonicity in variational inequalities and equilibrium problems, In: Generalized Convexity and Monotonoicity: Recent Developments, (Eds. J. P. Crouzeix, J. E. Martinez-Legaz and M. Volle), Kluwer Academic Publishers (to appear).

  20. P.T. Harker J.S. Pang (1990) ArticleTitleFinite–dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applica-tions Math. Programming 48 161–220 Occurrence Handle10.1007/BF01582255

    Article  Google Scholar 

  21. Z. Huang (2001) Generalization of an existence theorem for variational inequality problems Academia SINICA Beijing, China

    Google Scholar 

  22. N. J. Huang C. J. Gao X. P. Huang (2001) Exceptional family of elements and feasibility for nonlinear complementarity problems Preprint, Academia SINICA Beijing China

    Google Scholar 

  23. D. H. Hyers G. Isac Th. M. Rassias (1997) Topics in Nonlinear Analysis and Applications World Scientific Singapore, New Jersey, London, Hong Kong

    Google Scholar 

  24. Isac G. (1992). Complementarity Problems, Lecture Notes in Math., Springer-Verlag, Nr 1528.

  25. Isac, G. (2000) Topological Methods in Complementarity Theory, Kluwer academic Publishers.

  26. G. Isac (1999) ArticleTitlea generalization of Karamardian’s condition in complementarity theory Nonlinear Analysis Forum 4 49–63

    Google Scholar 

  27. Isac, G. (2000) Exceptional family of elements, feasibility, solvability and continuous path of ε-solutions for nonlinear complementarity problems,In: Approximation and Complexity in Numerical Optimization: Continuous and Discrete Problems, (Ed, Panos M. Pardalos), Kluwer Academic Publishers, 323–337.

  28. Isac, G. On the solvability of multivalued complementarity problem: A Topological method, Fourth European Workshop on Fuzzy Decision Analysis and Recognition Technology, EFDAN”99, Dorthmund, Germany, June 14–15, (Ed. R. Felix), 51–66.

  29. G. Isac (2001) ArticleTitleLeray-Schauder type alternatives and the solvability of complementarity problems Topological Meth. Nonlinear Analysis 18 191–204

    Google Scholar 

  30. G. Isac (2000) ArticleTitleOn the solvability of complementarity problems by Leray-Schauder type alternatives Libertas Mathematica 20 15–22

    Google Scholar 

  31. G. Isac V. A. Bulavski V. V. Kalashnikov (1997) ArticleTitleExceptional families, topological degree and complementarity problems J. Global Opt. 10 207–225 Occurrence Handle10.1023/A:1008284330108

    Article  Google Scholar 

  32. Isac, G., Bulavski, V. A. and Kalashnikov, V. V. Complementarity, Equilibrium, Efficiency and Economics, Kluwer Academic Publishers (Forthcoming).

  33. G. Isac A. Carbone (1999) ArticleTitleExceptional families of elements for continuous functions: Some applications to complementarity theory J. Global Opt. 15 181–196 Occurrence Handle10.1023/A:1008376709933

    Article  Google Scholar 

  34. Isac, G. and Cojocaru, M. G. Functions without exceptional family of elements and the solvability of variational inequalities on unbounded sets, (Forthcoming, Topological Meth. Nonlinear Analysis).

  35. G. Isac V. V. Kalashnikov (2001) ArticleTitleExceptional families of elements, Leray-Schauder alternative, pseudomonotone operators and complementarity J. Opt. Theory Appl. 109 IssueID1 69–83 Occurrence Handle10.1023/A:1017509704362

    Article  Google Scholar 

  36. Isac, G. and Kalashnikov, V. V. Exceptional families of elements, Leray-Schauder alternative and the solvability of implicit complementarity problems, Annals of Oper. Res. (Forthcoming).

  37. Isac, G., Kostreva, M. M. and Polyashuk, M. (2001) Relational complementarity problem, From local to Global Optimization, A. Migdalas et al (eds.), Kluwer Academic Publishers, 327–339.

  38. G. Isac W. T. Obuchowska (1998) ArticleTitleFunctions without exceptional families of elements and complementarity problems J. Opt. Theory Appl. 99 147–163 Occurrence Handle10.1023/A:1021704311867

    Article  Google Scholar 

  39. G. Isac Y. B. Zhao (2000) ArticleTitleExceptional families of elements and the solvability of variational inequalities for unbounded sets in infinite dimensional Hilbert spaces J. Math. Anal. Appl. 246 544–556 Occurrence Handle10.1006/jmaa.2000.6817

    Article  Google Scholar 

  40. V. V. Kalashnikov (1995) Complementarity Problem and the Generalized Oligopoly Model CEMI Moscow

    Google Scholar 

  41. S. Karamardian (1976) ArticleTitleComplementarity problems over cones with monotone and pseudomonotone maps J. Opt. Theory Appl 18 445–454 Occurrence Handle10.1007/BF00932654

    Article  Google Scholar 

  42. S. Karamardian S Schaible (1990) ArticleTitleSeven kinds of monotone maps J. Opt. Theory Appl. 66 37–46 Occurrence Handle10.1007/BF00940531

    Article  Google Scholar 

  43. S. Karamardian S. Schaible J. P. Crouzeix (1993) ArticleTitleCharacterizations of genera-lized monotone maps J. Opt. Theory Appl., 76 399–413

    Google Scholar 

  44. D. Kinderlehrer G. Stampacchia (1980) An Introduction to Variational Inequalities and their Applications Academic Press New York, London

    Google Scholar 

  45. Konnov, I. (2001) Combined Relaxation Methods for variational Inequalities, Lecture Notes in Economics and Mathematical Systems, Springer-Verlag, Nr. 495.

  46. J. Leray J. Schauder (1934) ArticleTitleTopologie et equations fonctionnelles Ann. Sci. Ecole Norm. Sup. 51 45–78

    Google Scholar 

  47. W. T. Obuchowska (2001) ArticleTitleExceptional families and existence results for nonlinear complementarity problem J. Global Opt. 19 183–198 Occurrence Handle10.1023/A:1008352130456

    Article  Google Scholar 

  48. A. J. B. Potter (1972) ArticleTitleAn elementary version of the Leray-Schauder theorem J. London Math. Soc. (2) 5 414–416

    Google Scholar 

  49. J. Ch. Yao (1994) ArticleTitleMulti-valued variational inequalities with k-pseudomonotone operator J. Opt. Theory Appl. 83 IssueID2 391–403

    Google Scholar 

  50. Zhao, Y. B. (1998) Existence Theory and Algorithms for Finite-Dimensional Variational Inequality and Complementarity Problems, Ph.D. Dissertation Institute of Applied Mathematics, Academia SINICA, Beijing, (China).

  51. Y. B. Zhao (1999) ArticleTitleD-orientation sequences for continuous and nonlinear complementarity problems Applied Math. Computation 106 221–235 Occurrence Handle10.1016/S0096-3003(98)10125-X

    Article  Google Scholar 

  52. Y. B. Zhao J. Y. Han (1999) ArticleTitleExceptional family of elements for variational inequality problem and its applications J. Global Opt. 14 313–330 Occurrence Handle10.1023/A:1008202323884

    Article  Google Scholar 

  53. Y.B. Zhao J.Y. Han H.D. Qi (1999) ArticleTitleExceptional families and existence theorems for variational inequality problems J. Opt. Theory Appl. 101 IssueID2 475–495 Occurrence Handle10.1023/A:1021701913337

    Article  Google Scholar 

  54. Y. B. Zhao G. Isac (2000) ArticleTitleQuasi-P* and P(t, α, β)-maps, exceptional family of elements and complementarity problems J. Opt. Theory Appl. 105 IssueID1 213–231 Occurrence Handle10.1023/A:1004674330768

    Article  Google Scholar 

  55. Y. B. Zhao G. Isac (2000) ArticleTitleProperties of multivalued mapping associated with some nonmonotone complementarity problems SIAM J. Control Optim. 39 IssueID2 571–593 Occurrence Handle10.1137/S0363012998345196

    Article  Google Scholar 

  56. Y. B. Zhao D. Li (2000) ArticleTitleStrict feasibility conditions in nonlinear complementarity problems J. Opt. Theory Appl. 107 IssueID2 641–664 Occurrence Handle10.1023/A:1026459501988

    Article  Google Scholar 

  57. Y. B. Zhao D. Li (2001) Characterizations of a homotopy solution mapping for quasi-and semi-monotone complementarity problems Academia SINCA Beijing, China

    Google Scholar 

  58. Y.B. Zhao D. Li (2001) ArticleTitleOn a new homotopy continuation trajectory for nonlinear complementarity problems Math. Oper. Res. 26 IssueID1 119–146 Occurrence Handle10.1287/moor.26.1.119.10594

    Article  Google Scholar 

  59. Y. B. Zhao D. Sun (2001) ArticleTitleAlternative theorems for nonlinear projection equations and applications Nonlinear Anal. 46 853–868 Occurrence Handle10.1016/S0362-546X(00)00154-1

    Article  Google Scholar 

  60. Y.B. Zhao J.Y. Yuan (2000) ArticleTitleAn alternative theorem for generalized variational inequalities and solvability of non-linear quasi- PM*-complementarity problems Appl. Math. Computation 109 167–182 Occurrence Handle10.1016/S0096-3003(99)00019-3

    Article  Google Scholar 

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  1. Department of Mathematics, Royal Military College of Canada, P. O. Box 17000 STN Forces, Kingston, Ontario, Canada, K7K 7B4

    G. Isac

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Isac, G. Complementarity Problems and Variational Inequalities. A Unified Approach of Solvability by an Implicit Leray-Schauder Type Alternative. J Glob Optim 31, 405–420 (2005). https://doi.org/10.1007/s10898-003-5611-6

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