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, 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 opti- mality 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 ε-stationary point for this kind of nonsmooth problem in O ε −2 log ε −1 iterations. Finally, we apply PGmsAD to generalized absolute value equa- tions, generalized linear projection equations and linear regression problems and report the efficiency of PGmsAD on large-scale optimization.

报告人简介：戴彧虹，中国科学院数学与系统科学研究院研究员、博士生导师，国家杰出青年基金获得者，中国科学院数学与系统科学研究院优化与应用中心主任，中国运筹学会理事长，亚太运筹学会联合会主席。研究方向：连续优化，整数规划，应用优化。在国际期刊Math. Prog.、SIAM J. Optim.、IEEE Trans.Signal Process.等发表论文一百余篇。主持国家自然科学基金创新研究群体项目、科技部数学与应用数学重点专项以及中国科学院先导项目。2006年获国家自然科学奖二等奖（排名第二），2015年获冯康科学计算奖，2017年获第十六届陈省身数学奖，2018年获首届萧树铁应用数学奖。应邀在2022年国际数学家大会作45分钟邀请报告和2022年第24届国际数学规划大会作一小时大会报告。