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Minimax optimization problems are an important class of optimization problemsarising from modern machine learning and traditional research areas. Whilethere have been many numerical algorithms for solving smooth convex-concaveminimax problems, numerical algorithms for nonsmooth convex-concave minimaxproblems are very rare. This paper aims to develop an efficient numericalalgorithm for a structured nonsmooth convex-concave minimax problem. Amajorized semi-proximal alternating coordinate method (mspACM) is proposed, inwhich a majorized quadratic convex-concave function is adopted forapproximating the smooth part of the objective function and semi-proximal termsare added in each subproblem. This construction enables the subproblems at eachiteration are solvable and even easily solved when the semiproximal terms arecleverly chosen. We prove the global convergence of the algorithm mspACM undermild assumptions, without requiring strong convexity-concavity condition. Underthe locally metrical subregularity of the solution mapping, we prove that thealgorithm mspACM has the linear rate of convergence. Preliminary numericalresults are reported to verify the efficiency of the algorithm mspACM.

Majorized Semi-proximal Alternating Coordinate Method for Nonsmooth Convex-Concave Minimax Optimization