Semi-Proximal Point Method for Nonsmooth Convex-Concave Minimax Optimization (Journal of Computational Mathematics, 2024)
Minimax optimization problems are an important class of optimization problems arisingfrom modern machine learning and traditional research areas. While there have been manynumerical algorithms for solving smooth convex-concave minimax problems, numericalalgorithms for nonsmooth convex-concave minimax problems are rare. This paper aimsto develop an efficient numerical algorithm for a structured nonsmooth convex-concaveminimax problem. A semi-proximal point method (SPP) is proposed, in which a quadraticconvex-concave function is adopted for approximating the smooth part of the objectivefunction and semi-proximal terms are added in each subproblem. This construction enablesthe subproblems at each iteration are solvable and even easily solved when the semiproximalterms are cleverly chosen. We prove the global convergence of our algorithm under mildassumptions, without requiring strong convexity-concavity condition. Under the locallymetrical subregularity of the solution mapping, we prove that our algorithm has the linearrate of convergence. Preliminary numerical results are reported to verify the efficiency ofour algorithm.