Department of Applied Mathematics & Physics, Kyoto University

Technical Report 2001-004 (April 10, 2001)

A Superlinearly Convergent Algorithm for the Monotone Nonlinear Complementarity Problem without Uniqueness and Nondegeneracy Conditions
by Hiroshige Dan, Nobuo Yamashita and Masao Fukushima

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The purpose of this paper is to present an algorithm for solving the monotone nonlinear complementarity problem (NCP) that enjoys superlinear convergence in a genuine sense without the uniqueness and nondegeneracy conditions. Recently, Yamashita and Fukushima proposed a method based on the proximal point algorithm (PPA) for monotone NCP. The method has the favorable property that a generated sequence converges to the solution set of NCP superlinearly. However, when a generated sequence converges to a degenerate solution, the subproblems may become computationally expensive and the method does not have genuine superlinear convergence. More recently, Yamashita, Dan and Fukushima presented a technique to identify whether a solution is degenerate or not. Using this technique, we construct a differentiable system of nonlinear equations whose solution is a solution of the original NCP. Moreover, we propose a hybrid algorithm which is based on the PPA and uses this system. We show that the proposed algorithm has a genuine quadratic or superlinear rate of convergence even if it converges to a solution that is neither unique nor nondegenerate.