Convergence of a smoothing-type algorithm for the monotone affine variational inequality problem

Abstract

Smoothing-type algorithms have been applied to solve various optimization problems. In the analysis on the global convergence, most existing smoothing-type algorithms need to assume that the solution set of the problem concerned is nonempty and bounded or some stronger conditions. In this paper, we investigate a smoothing-type algorithm for solving the monotone affine variational inequality problem (AVIP). Specially, we reformulate the AVIP as a system of parameterized smooth equations, and instead of solving the original AVIP, we use a Newton-type method to solve the smooth equations. We show that under mild assumptions, the iteration sequence generated by the algorithm is bounded; and the algorithm may find a maximally complementary solution to the AVIP. In our analysis on the convergence, we do not need to assume that the solution set of the AVIP is bounded.