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

TL;DR

This paper introduces optimality conditions and a novel numerical algorithm, PGmsAD, for solving nonsmooth minimax problems with joint linear constraints, demonstrating efficiency on large-scale applications.

Contribution

It develops a new framework for optimality conditions and proposes PGmsAD, a proximal gradient multi-step ascent-descent method with proven convergence for constrained nonsmooth minimax problems.

Findings

PGmsAD finds an $psilon$-stationary point in ${al O}(psilon^{-2}logpsilon^{-1})$ iterations.

The method is effective on large-scale problems like linear regression and generalized equations.

Numerical experiments show PGmsAD's efficiency and practical applicability.

Abstract

It is well known that there have been many numerical algorithms for solving nonsmooth minimax problems, 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 of all, we use the properties of proximal mapping and KKT system to establish optimality conditions. Secondly, we propose a framework of alternating coordinate algorithm for the minimax problem and analyze its convergence properties. Thirdly, we develop a proximal gradient multi-step ascent decent method (PGmsAD) as a numerical algorithm and demonstrate that the method can find an $\epsilon$-stationary point for this kind of nonsmooth nonconvex-nonconcave problem in ${\cal O}(\epsilon^{-2}\log\epsilon^{-1})$ iterations. Finally, we apply PGmsAD to generalized absolute value equations, generalized linear projection equations and linear regression problems and report the efficiency of PGmsAD on large-scale optimization.