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In this paper, we present a smoothing Newton method for solving the monotone weighted complementarity problem (WCP). In each iteration of our method, the iterative direction is achieved by solving a system of linear equations and the iterative step length is achieved by adopting a line search. A feature of the line search criteria used in this paper is that monotone and nonmonotone line search are mixed used. The proposed method is new even when the WCP reduces to the standard complementarity problem. Particularly, the proposed method is proved to possess the global convergence under a weak assumption. The preliminary experimental results show the effectiveness and robustness of the proposed method for solving the concerned WCP.

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Abstract. Based on a family of generalized merit functions, a merit function method for solving nonlinear complementarity problems was proposed by Lu, Huang and Hu [Properties of a family of merit functions and a merit function method for the NCP, Appl. Math.-J. Chinese Univ., 2010, 25: 379-390], where, the global convergence of the method was proved. However, no the result on the convergence rate of the method was reported. In this short paper, we show that the method proposed in the above paper is globally linearly convergent under suitable assumptions.

The weighted complementarity problem is an extension of the standard finite dimensional complementarity problem. It is well known that the smoothing-type algorithm is a powerful tool of solving the standard complementarity problem. In this paper, we propose a smoothing-type algorithm for solving the weighted complementarity problem with a monotone function, which needs only to solve one linear system of equations and performs one line search at each iteration. We show that the proposed method is globally convergent under the assumption that the problem is solvable. The preliminary numerical results indicate that the proposed method is effective and robust for solving the monotone weighted complementarity problem.