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

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 -stationary point for this kind of nonsmooth nonconvex-nonconcave problem in O −2 log −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.

A first-order augmented Lagrangian method for constrained minimax optimization

“…where L k is given in (10). Consequently, it follows from Theorem 2 that Algorithm 3 can be suitably applied to problem (9) for finding an ǫ k -stationary point (x k+1 , y k+1 ) of it.…”

“…(iii) In view of Theorem 2 (see Appendix A), one can see that an ǫ k -stationary point of (9) can be successfully found in step 3 of Algorithm 1 by applying Algorithm 3 to problem (9) and thus Algorithm 1 is well-defined.…”

“…Finally, we introduce an (approximate) stationary point (e.g., see [9,10,21]) for a general minimax problem min…”

“…In addition, a multiplier gradient descent method was proposed in [44] for solving (1) with c(x) ≡ 0, d(x, y) being an affine mapping, and p and q being the indicator function of a simple compact convex set. Also, a proximal gradient multi-step ascent decent method was developed in [9] for (1) with c(x) ≡ 0, d(x, y) being an affine mapping, and f (x, y) = g(x) + x T Ay − h(y), under the assumption that f (x, y) − q(y) is strongly concave in y. Besides, primal dual alternating proximal gradient methods were proposed in [53] for (1) with c(x) ≡ 0, d(x, y) being an affine mapping, and {f (x, y) being strongly concave in y or [q(y) ≡ 0 and f (x, y) being a linear function in y]}.…”

“…Besides, primal dual alternating proximal gradient methods were proposed in [53] for (1) with c(x) ≡ 0, d(x, y) being an affine mapping, and {f (x, y) being strongly concave in y or [q(y) ≡ 0 and f (x, y) being a linear function in y]}. For these methods, an iteration complexity for finding an approximate stationary point of the aforementioned special minimax problem was established in [9,14,53], respectively. Yet, their operation complexity, measured by the amount of fundamental operations such as evaluations of gradient of f and proximal operator of p and q, was not studied in these works.…”