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

IF 2.6 1区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Optimization Pub Date : 2024-09-03 DOI:10.1137/22m1535243

Yu-Hong Dai, Jiani Wang, Liwei Zhang

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引用次数: 0

Abstract

SIAM Journal on Optimization, Volume 34, Issue 3, Page 2883-2916, September 2024.
Abstract. 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 [math]-stationary point for this kind of nonsmooth problem in [math] 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.

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一类线性约束最小优化问题的最优条件和数值算法

SIAM 优化期刊》，第 34 卷第 3 期，第 2883-2916 页，2024 年 9 月。摘要众所周知，已有许多求解非光滑最小问题的数值算法；然而，求解有联合线性约束的非光滑最小问题的数值算法却非常罕见。本文旨在讨论具有联合线性约束的最小问题的最优性条件并开发实用的数值算法。首先，我们利用近似映射和 KKT 系统的特性来建立最优性条件。其次，我们提出了最小问题的交替坐标算法框架，并分析了其收敛特性。第三，我们开发了一种近似梯度多步上升下降法（PGmsAD）作为数值算法，并证明该方法可以在[math]迭代中找到这类非光滑问题的[math]驻点。最后，我们将 PGmsAD 应用于广义绝对值方程、广义线性投影方程和线性回归问题，并报告了 PGmsAD 在大规模优化中的效率。

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来源期刊

SIAM Journal on Optimization 数学-应用数学

CiteScore

5.30

自引率

9.70%

发文量

101

审稿时长

6-12 weeks

期刊介绍： The SIAM Journal on Optimization contains research articles on the theory and practice of optimization. The areas addressed include linear and quadratic programming, convex programming, nonlinear programming, complementarity problems, stochastic optimization, combinatorial optimization, integer programming, and convex, nonsmooth and variational analysis. Contributions may emphasize optimization theory, algorithms, software, computational practice, applications, or the links between these subjects.