Alternating Proximity Mapping Method for Strongly Convex-Strongly Concave Saddle-Point Problems

Abstract: This is a continuation of our previous work entitled \enquote{Alternating Proximity Mapping Method for Convex-Concave Saddle-Point Problems}, in which we proposed the alternating proximal mapping method and showed convergence results on the sequence of our iterates, the sequence of averages of our iterates, and the sequence of function values evaluated at the averages of the iterates for solving convex-concave saddle-point problems. In this work, we extend the application of the alternating proximal mapping method to solve strongly convex-strongly concave saddle-point problems. We demonstrate two sets of sufficient conditions and also their simplified versions, which guarantee the linear convergence of the sequence of iterates towards a desired saddle-point. Additionally, we provide two sets of sufficient conditions, along with their simplified versions, that ensure the linear convergence of the sequence of function values evaluated at the convex combinations of iteration points to the desired function value of a saddle-point.