SEMI-PROXIMAL POINT METHOD FOR NONSMOOTH CONVEX-CONCAVE MINIMAX OPTIMIZATION  

作　　者：Yuhong Dai Jiani Wang Liwei Zhang 

机构地区：[1]LSEC,ICMSEC,AMSS,Chinese Academy of Sciences,Beijing 100190,China School of Mathematical Sciences,University of Chinese Academy of Sciences,Beijing 100049,China [2]Academy of Mathematics and Systems Science,Chinese Academy of Sciences,Beijing 100190,China [3]School of Mathematical Sciences,Dalian University of Technology,Dalian 116024,China

出　　处：《Journal of Computational Mathematics》2024年第3期617-637,共21页计算数学（英文）

基　　金：supported by the Natural Science Foundation of China(Grant Nos.11991021,11991020,12021001,11971372,11971089,11731013);by the Strategic Priority Research Program of Chinese Academy of Sciences(Grant No.XDA27000000);by the National Key R&D Program of China(Grant Nos.2021YFA1000300,2021YFA1000301).

摘　　要：Minimax optimization problems are an important class of optimization problems arising from modern machine learning and traditional research areas.While there have been many numerical algorithms for solving smooth convex-concave minimax problems,numerical algorithms for nonsmooth convex-concave minimax problems are rare.This paper aims to develop an efficient numerical algorithm for a structured nonsmooth convex-concave minimax problem.A semi-proximal point method(SPP)is proposed,in which a quadratic convex-concave function is adopted for approximating the smooth part of the objective function and semi-proximal terms are added in each subproblem.This construction enables the subproblems at each iteration are solvable and even easily solved when the semiproximal terms are cleverly chosen.We prove the global convergence of our algorithm under mild assumptions,without requiring strong convexity-concavity condition.Under the locally metrical subregularity of the solution mapping,we prove that our algorithm has the linear rate of convergence.Preliminary numerical results are reported to verify the efficiency of our algorithm.

关 键 词：Minimax optimization Convexity-concavity Global convergence Rate of con-vergence Locally metrical subregularity 

分 类 号：O17[理学—数学]