Optimality Conditions and Numerical Algorithms for a Class of Linearly Constrained Minimax Optimization Problems

It is well known that there have been many numerical algorithms for solving nonsmooth minimax problems; however, numerical algorithms for nonsmooth minimax problems with joint linear constraints are very rare. This paper aims to discuss optimality conditions and develop practical numerical algorithms for minimax problems with joint linear constraints. First, we use the properties of proximal mapping and the KKT system to establish optimality conditions. Second, we propose a framework of an alternating coordinate algorithm for the minimax problem and analyze its convergence properties. Third, we develop a proximal gradient multistep ascent descent method (PGmsAD) as a numerical algorithm and demonstrate that the method can find an [Formula: see text]-stationary point for this kind of nonsmooth problem in [Formula: see text] iterations. Finally, we apply PGmsAD to generalized absolute value equations, generalized linear projection equations, and linear regression problems, and we report the efficiency of PGmsAD on large-scale optimization.