  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.
