Convergence of Bregman Alternating Direction Method of Multipliers for Nonseparable Nonconvex Objective with Linear Constraints  

作　　者：Xiaotong Zeng Junping Yao Haoming Xia Xiaotong Zeng;Junping Yao;Haoming Xia(Key Laboratory of Optimization Theory and Applications, School of Mathematics and Information, China West Normal University, Nanchong, China)

机构地区：[1]Key Laboratory of Optimization Theory and Applications, School of Mathematics and Information, China West Normal University, Nanchong, China

出　　处：《Journal of Applied Mathematics and Physics》2024年第2期639-660,共22页应用数学与应用物理（英文）

摘　　要：In this paper, our focus lies on addressing a two-block linearly constrained nonseparable nonconvex optimization problem with coupling terms. The most classical algorithm, the alternating direction method of multipliers (ADMM), is employed to solve such problems typically, which still requires the assumption of the gradient Lipschitz continuity condition on the objective function to ensure overall convergence from the current knowledge. However, many practical applications do not adhere to the conditions of smoothness. In this study, we justify the convergence of variant Bregman ADMM for the problem with coupling terms to circumvent the issue of the global Lipschitz continuity of the gradient. We demonstrate that the iterative sequence generated by our approach converges to a critical point of the issue when the corresponding function fulfills the Kurdyka-Lojasiewicz inequality and certain assumptions apply. In addition, we illustrate the convergence rate of the algorithm.In this paper, our focus lies on addressing a two-block linearly constrained nonseparable nonconvex optimization problem with coupling terms. The most classical algorithm, the alternating direction method of multipliers (ADMM), is employed to solve such problems typically, which still requires the assumption of the gradient Lipschitz continuity condition on the objective function to ensure overall convergence from the current knowledge. However, many practical applications do not adhere to the conditions of smoothness. In this study, we justify the convergence of variant Bregman ADMM for the problem with coupling terms to circumvent the issue of the global Lipschitz continuity of the gradient. We demonstrate that the iterative sequence generated by our approach converges to a critical point of the issue when the corresponding function fulfills the Kurdyka-Lojasiewicz inequality and certain assumptions apply. In addition, we illustrate the convergence rate of the algorithm.

关 键 词：Nonseparable Nonconvex Optimization Bregman ADMM Kurdyka-Lojasiewicz Inequality 

分 类 号：O17[理学—数学]