Abstract
This paper introduces and analyzes generalized variational inequalities. The most general existence theory is established, traditional coercivity conditions are extended, properties of solution sets under various monotonicity conditions are investigated, and a computational scheme is considered. Similar results can be obtained for generalized complementarity and fixed-point problems.
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References
Auslender, A.,Problems de Minimax via l'Analyse Convexe et les Inegalites Variationelles: Theorie et Algorithmes, Springer-Verlag, Berlin, Germany, 1972.
Brézis, H.,Operateurs Maximaux Monotones, North Holland, Amsterdam, Holland, 1973.
Browder, F. E.,Existence and Applications of Solutions of Nonlinear Variational Inequalities, Proceeding of the National Academy of Sciences of the USA, Vol. 56, pp. 1080–1086, 1966.
Hartman, P., andStampacchia, G.,On Some Nonlinear Elliptic Differential-Functional Equations, Acta Mathematica, Vol. 115, pp. 271–310, 1966.
Lions, J. L., andStampacchia G.,Variational Inequalities, Communications of Pure and Applied Mathematics, Vol. 20, pp. 493–519, 1967.
Minty, G. J.,Monotone (Nonlinear)Operators in Hilbert Space, Duke Mathematics Journal, Vol. 29, pp. 341–346, 1962.
Minty, G. J.,On the Generalization of a Direct Method of the Calculus of Variations, Bulletin of the American Mathematical Society, Vol. 73, pp. 315–321, 1967.
Minty, G. J.,On Some Aspects of the Theory of Monotone Operators, Theory and Applications of Monotone Operators, Edited by A. Ghizzetti, Edizioni Oderisi, Gubbio, Italy, 1968.
Rockafellar, R. T.,Convex Functions, Monotone Operators, and Variational Inequalities, Theory and Applications of Monotone Operators, Edited by A. Ghizzetti, Edizioni Oderisi, Gubbio, Italy, 1968.
Stampacchia, G.,Variational Inequalities, Theory and Applications of Monotone Operators, Edited by A. Ghizzetti, Edizioni Oderisi, Gubbio, Italy, 1968.
Saigal, R.,Extensions of the Generalized Complementarity Problem, Mathematics of Operations Research, Vol. 1, pp. 260–266, 1976.
Moré, J. J.,Coercivity Conditions in Nonlinear Complementarity Problems, SIAM Review, Vol. 16, pp. 1–16, 1974.
Ortega, J. W., andRheinboldt, W. C.,Iterative Solutions of Nonlinear Equations in Several Variables, Academic Press, New York, New York, 1970.
Karamardian, S.,The Nonlinear Complementarity Problem with Applications, Part 1, Journal of Optimization Theory and Applications, Vol. 4, pp. 87–98, 1969.
Karamardian, S.,The Nonlinear Complementarity Problem with Applications, Part 2, Journal of Optimization Theory and Applications, Vol. 6, pp. 167–181, 1969.
Karamardian, S.,Generalized Complementarity Problem, Journal of Optimization Theory and Applications, Vol. 8, pp. 161–168, 1971.
Karamardian, S.,The Complementarity Problem, Mathematical Programming, Vol. 2, pp. 107–129, 1972.
Karamardian, S.,Complementarity Problems over Cones with Monotone and Pseudomonotone Maps, Journal of Optimization Theory and Applications, Vol. 18, pp. 445–454, 1976.
Moré, J. J.,Classes of Functions and Feasibility Conditions in Nonlinear Complementarity Problems, Mathematical Programming, Vol. 6, pp. 327–338, 1974.
Fang, S. C., andPeterson, E. L.,A Unification and Generalization of the Eaves and Kojima Fixed-Point Representations of the Complementarity Problem, Northwestern University, Center of Mathematical Studies for Economics and Management Sciences, Discussion Paper No. 365, 1979.
Fang, S. C.,Generalized Variational Inequality, Complementarity and Fixed-Point Problems: Theory and Applications, Northwestern University, PhD Thesis, 1979.
Merrill, O. H.,Applications and Extensions of an Algorithm that Computes Fixed Points of Certain Upper Semicontinuous Point-to-Set Mappings, University of Michigan, PhD Thesis, 1972.
Eaves, B. C., andSaigal, R.,Homotopies for Computation of Fixed Points on Unbounded Regions, Mathematical Programming, Vol. 3, pp. 225–237, 1972.
Kluge, R., andTelschow, G.,On the Convergence and Speed of Some Iteration Methods for Variational Inequalities, I, Theory of Nonlinear Operators, Edited by R. Kluge and A. Müller, Akademie-Verlag, Berlin, Germany, 1977.
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Communicated by S. Karamardian
The authors are indebted to Professor R. Saigal of Northwestern University for his continuous encouragement and helpful discussions concerning this paper.
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Fang, S.C., Peterson, E.L. Generalized variational inequalities. J Optim Theory Appl 38, 363–383 (1982). https://doi.org/10.1007/BF00935344
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DOI: https://doi.org/10.1007/BF00935344