A proximal point algorithm for the monotone second‐order cone complementarity problem

This paper is devoted to the study of the proximal point algorithm for solving monotone second‐order cone complementarity problems. The proximal point algorithm is to generate a sequence by solving subproblems that are regularizations of the original problem. After given an appropriate criterion for approximate solutions of subproblems by adopting a merit function, the proximal point algorithm is verified to have global and superlinear convergence properties. For the purpose of solving the subproblems efficiently, we introduce a generalized Newton method and show that only one Newton step is eventually needed to obtain a desired approximate solution that approximately satisfies the appropriate criterion under mild conditions. Numerical comparisons are also made with the derivative‐free descent method used by Pan and Chen (2010), which confirm the theoretical results and the effectiveness of the algorithm.