Abstract
In this paper we establish several sufficient conditions for the existence of a solution to the linear and some classes of nonlinear complementarity problems. These conditions involve a notion of the ``exceptional family of elements'' introduced by Smith [19] and Isac, Bulavski and Kalashnikov [4], where the authors have shown that the nonexistence of the ``exceptional family of elements'' implies solvability of the complementarity problem. In particular, we establish several sufficient conditions for the nonexistence as well as for the existence of the exceptional family of elements.
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References
Cottle, R.W., Pang, J.S., and Stone, R.E. (1992), The Linear Complementarity Problem, Academic Press, New York.
Eaves, B.C. (1971), On the Basic Theorem of Complementarity, Mathematical Programming 1: 68–75.
Habetler, G.J., and Price, A.L. (1971), Existence Theorem for Generalized Nonlinear Complementarity Problems, Journal of Optimization Theory and Applications 4: 223–239.
Isac, G., Bulavski V. and Kalashnikov, V. (1997), Exceptional Families, Topological Degree and Complementarity Problems, Journal of Global Optimization 10: 207–225.
Isac, G. (1992), Complementarity Problems, Lecture Notes in Mathematics, Springer-Verlag, No. 1528.
G. Isac, G., Kostreva, M.M. and Wiecek, M.M. (1995), Multiple-Objective Approximation of Feasible but Unsolvable Linear Complementarity Problems, Journal of Optimization Theory and Applications 86: 389–405.
Isac, G., and Obuchowska, W.T. (1998), Functions Without Exceptional Family of Elements and Complementarity Problems, Journal of Optimization Theory and Applications 99: 147–163.
Karamardian, S. (1976), An Existence Theorem for the Complementarity Problem, Journal of Optimization Theory and Applications 19: 227–232.
Karamardian, S. (1971), Generalized Complementarity Problem, Journal of Optimization Theory and Applications 8: 161–168.
Kojima, M., Mizuno, S., and Noma, T. (1989), A New Continuation Method for Complementarity Problems with Uniform P-functions, Mathematical Programming 43: 107–113.
Kojima, M. (1975), A Unification of the Existence Theorems of the Nonlinear Complementarity Problem, Mathematical Programming 9: 257–277.
Moré, J.J. (1974), Coercivity Conditions in Nonlinear Complementarity Problem, SIAM Reviews 16: 1–16.
Moré, J.J., and Rheinboldt, W. (1973), On P-and S-functions and Related Classes of n-dimensional Nonlinear Mappings, Linear Algebra and its Applications 6: 45–68.
Moré, J.J. (1996), Global Methods for Nonlinear Complementarity Problems, Math. of Oper. Research 21(3): 589–614.
Murty, K.G. (1987), Linear Complementarity, Linear and Nonlinear Programming, Heldermann Verlag, West Berlin.
Pang, J.S. (1991), Iterative Descent Algorithms for a Row Sufficient Linear Complementarity Problem, SIAM Journal on Matrix Analysis and Applications 12: 611–624.
Pang, J.S., and Yao, J.C. (1995), On Generalization of a Normal Map and Equation, SIAM J. Control Optimization 33: 168–184.
Rockafellar, R.T. (1970), Convex Analysis, Princeton University Press, Princeton, New Jersey.
Rockafellar, R.T. (1970), Some Convex Programs Where Duals Are Linearly Constrained, in: J.B. Rosen, O.L. Mangasarian and K. Ritter (eds), pp. 293–322, Nonlinear Programming, Academic Press, New York, New York.
Smith, T.E. (1984), A Solution Condition for Complementarity Problems with an Application to Special Price Equilibrium, Applied Mathematics and Computation 15: 61–69.
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Obuchowska, W. Exceptional Families and Existence Results for Nonlinear Complementarity Problem. Journal of Global Optimization 19, 183–198 (2001). https://doi.org/10.1023/A:1008352130456
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DOI: https://doi.org/10.1023/A:1008352130456