  Minimax optimization problems are an important class of optimization problems
arising from modern machine learning and traditional research areas. While
there have been many numerical algorithms for solving smooth convex-concave
minimax problems, numerical algorithms for nonsmooth convex-concave minimax
problems are very rare. This paper aims to develop an efficient numerical
algorithm for a structured nonsmooth convex-concave minimax problem. A
majorized semi-proximal alternating coordinate method (mspACM) is proposed, in
which a majorized quadratic convex-concave function is adopted for
approximating the smooth part of the objective function and semi-proximal terms
are added in each subproblem. This construction enables the subproblems at each
iteration are solvable and even easily solved when the semiproximal terms are
cleverly chosen. We prove the global convergence of the algorithm mspACM under
mild assumptions, without requiring strong convexity-concavity condition. Under
the locally metrical subregularity of the solution mapping, we prove that the
algorithm mspACM has the linear rate of convergence. Preliminary numerical
results are reported to verify the efficiency of the algorithm mspACM.
