Alternating Proximal Point Algorithm with Gradient Descent and Ascent Steps for Convex-Concave Saddle-Point Problems

Inspired by the Optimistic Gradient Ascent-Proximal Point Algorithm (OGAProx) proposed by Bo{\c{t}}, Csetnek, and Sedlmayer for solving a saddle-point problem associated with a convex-concave function with a nonsmooth coupling function and one regularizing function, we introduce the Alternating Proximal Point Algorithm with Gradient Descent and Ascent Steps for solving a saddle-point problem associated with a convex-concave function constructed by a smooth coupling function and two regularizing functions. In this work, we not only provide weak and linearly convergence of the sequence of iterations and of the minimax gap function evaluated at the ergodic sequences, similarly to what Bo{\c{t}} et al.\,did, but also demonstrate the convergence and linearly convergence of function values evaluated at convex combinations of iterations under convex and strongly convex assumptions, respectively.