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The nonlinear complementarity problem with applications, part 1

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Abstract

The main result in this paper is an existence and uniqueness theorem for the following nonlinear complementarity problem: Given a mapping from then-dimensional Euclidean spaceE n into itself, find a nonnegative vector inE n whose image, under the given mapping, is also nonnegative, the two vectors being orthogonal to each other. It is shown that the above problem has a unique solution if the given mapping is continuous and strongly monotone on the nonnegative orthantE n+ ofE n. It is also shown that a sufficient condition for a differentiable mapping to be strongly monotone on an open set is that all the eigenvalues of the symmetric part of its Jacobian be bounded below by a positive constant on the given set.

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Authors and Affiliations

  1. Graduate School of Administration, University of California at Irvine, Irvine, California

    S. Karamardian (Associate Professor of Mathematics and Administration)

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  1. S. Karamardian
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This research constituted a part of the author's Ph.D. dissertation at the University of California at Berkeley, Berkeley, California. The author would like to express his appreciation to Professor G. B. Dantzig, who brought this problem to his attention and guided his research with his several suggestions and helpful criticism. Also, he thanks the referee for several important comments and recommendations.

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Karamardian, S. The nonlinear complementarity problem with applications, part 1. J Optim Theory Appl 4, 87–98 (1969). https://doi.org/10.1007/BF00927414

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  • DOI: https://doi.org/10.1007/BF00927414

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