The Rate of Convergence of Augmented Lagrangian Method for Minimax Optimization Problems with Equality Constraints  

作　　者：Yu-Hong Dai Li-Wei Zhang 

机构地区：[1]LSEC,ICMSEC,AMSS,Chinese Academy of Sciences,Beijing 100190,China [2]Institute of Operations Research and Control Theory,School of Mathematical Sciences,Dalian University of Technology,Dalian 116024,China

出　　处：《Journal of the Operations Research Society of China》2024年第2期265-297,共33页中国运筹学会会刊（英文）

基　　金：the National Natural Science Foundation of China(Nos.11991020,11631013,11971372,11991021,11971089 and 11731013);the Strategic Priority Research Program of Chinese Academy of Sciences(No.XDA27000000);Dalian High-Level Talent Innovation Project(No.2020RD09)。

摘　　要：The augmented Lagrangian function and the corresponding augmented Lagrangian method are constructed for solving a class of minimax optimization problems with equality constraints.We prove that,under the linear independence constraint qualification and the second-order sufficiency optimality condition for the lower level problem and the second-order sufficiency optimality condition for the minimax problem,for a given multiplier vectorμ,the rate of convergence of the augmented Lagrangian method is linear with respect to||μu-μ^(*)||and the ratio constant is proportional to 1/c when the ratio|μ-μ^(*)||/c is small enough,where c is the penalty parameter that exceeds a threshold c_(*)>O andμ^(*)is the multiplier corresponding to a local minimizer.Moreover,we prove that the sequence of multiplier vectors generated by the augmented Lagrangian method has at least Q-linear convergence if the sequence of penalty parameters(ck)is bounded and the convergence rate is superlinear if(ck)is increasing to infinity.Finally,we use a direct way to establish the rate of convergence of the augmented Lagrangian method for the minimax problem with a quadratic objective function and linear equality constraints.

关 键 词：Minimax optimization Augmented Lagrangian method Rate of convergence Second-order sufficiency optimality 

分 类 号：O17[理学—数学]