A hybrid inexact Logarithmic–Quadratic Proximal method for nonlinear complementarity problems

Abstract

Inspired by the Logarithmic–Quadratic Proximal method [A. Auslender, M. Teboulle, S. Ben-Tiba, A logarithmic–quadratic proximal method for variational inequalities, Comput. Optim. Appl. 12 (1999) 31–40], we present a new prediction–correction method for solving the nonlinear complementarity problems. In our method, an intermediate point is produced by approximately solving a nonlinear equation system based on the Logarithmic–Quadratic Proximal method; and the new iterate is obtained by convex combination of the previous point and the one generated by the improved extragradient method at each iteration. The proposed method allows for constant relative errors and this yields a more practical Logarithmic–Quadratic Proximal type method. The global convergence is established under mild conditions. Preliminary numerical results indicate that the method is effective for large-scale nonlinear complementarity problems.