Abstract
This paper is concerned with two well-known families of iterative methods for solving the linear and nonlinear complementarity problems. For the linear complementarity problem, we consider the class of matrix splitting methods and establish, under a finiteness assumption on the number of solutions, a necessary and sufficient condition for the convergence of the sequence of iterates produced. A rate of convergence result for this class of methods is also derived under a stability assumption on the limit solution. For the nonlinear complementarity problem, we establish the convergence of the Newton method under the assumption of a “pseudo-regular” solution which generalizes Robinson's concept of a “strongly regular” solution. In both instances, the convergence proofs rely on a common sensitivity result of the linear complementarity problem under perturbation.
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This work was based on research supported by the National Science Foundation under grant ECS-8717968.
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Pang, JS. Convergence of splitting and Newton methods for complementarity problems: An application of some sensitivity results. Mathematical Programming 58, 149–160 (1993). https://doi.org/10.1007/BF01581264
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DOI: https://doi.org/10.1007/BF01581264