Abstract
Given a continuous mapF:R n→R n and a lower semicontinuous positively homogeneous convex functionh:R n→R, the nonlinear complementarity problem considered here is to findxεR n+ andy∈∂h(x), the subdifferential ofh atx, such thatF(x)+y≥0 andx T(F(x)+y)=0. Some existence theorems for the above problem are given under certain conditions on the mapF. An application to quasidifferentiable convex programming is also shown.
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References
Cottle, R. W.,Nonlinear Programs with Positively Bounded Jacobians, SIAM Journal on Applied Mathematics, Vol. 14, pp. 147–158, 1966.
Karamardian, S.,The Complementarity Problem, Mathematical Programming, Vol. 2, pp. 107–129, 1972.
Kojima, M.,A Unification of the Existence Theorems of the Nonlinear Complementarity Problem, Mathematical Programming, Vol. 9, pp. 257–277, 1975.
Mangasarian, O. L.,Locally Unique Solutions of Quadratic Programs, Linear and Nonlinear Complementarity Problems, Mathematical Programming, Vol. 19, pp. 200–212, 1980.
McLinden, L.,The Complementarity Problem for Maximal Monotone Multifunctions, Variational Inequalities and Complementarity Problems, Edited by R. W. Cottle, F. Giannessi, and J. L. Lions, John Wiley and Sons, Chichester, England, pp. 251–270, 1980.
Moré, J. J.,Coercivity Conditions in Nonlinear Complementarity Problems, SIAM Review, Vol. 16, pp. 1–16, 1974.
Parida, J., andRoy, K. L.,An Existence Theorem for the Nonlinear Complementarity Problem, Indian Journal of Pure and Applied Mathematics, Vol. 13, pp. 615–619, 1982.
Saigal, R.,Extension of the Generalized Complementarity Problem, Mathematics of Operations Research, Vol. 1, pp 260–266, 1976.
Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.
Tanimoto, S.,Nondifferentiable Mathematical Programming and Convex-Concave Functions, Journal of Optimization Theory and Applications, Vol. 31, pp. 331–342, 1980.
Clarke, F. H.,A New Approach to Lagrange Multipliers, Mathematics of Operations Research, Vol. 1, pp. 165–174, 1976.
Mangasarian, O. L.,Nonlinear Programming, McGraw-Hill Book Company, New York, New York, 1969.
Kakutani, S.,A Generalization of Brouwer's Fixed-Point Theorem, Duke Mathematical Journal, Vol. 8, pp. 457–459, 1941.
Mangasarian, O. L., andMcLinden, L.,Simple Bounds for Solutions of Monotone Complementarity Problems and Convex Programs, Mathematical Programming, Vol. 32, pp. 32–40, 1985.
Parida, J., andSen, A.,Duality and Existence Theory for Nondifferentiable Programming, Journal of Optimization Theory and Applications Vol. 48, pp. 451–458, 1986.
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Communicated by O. L. Mangasarian
The authors are grateful to Professor O. L. Mangasarian and the referee for their substantive suggestions.
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Parida, J., Sen, A. A class of nonlinear complementarity problems for multifunctions. J Optim Theory Appl 53, 105–113 (1987). https://doi.org/10.1007/BF00938819
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DOI: https://doi.org/10.1007/BF00938819