对于非凸非光滑不可分离问题的并行式惯性近端非精确ADMM算法  

Parallel Inertial Proximal Inexact ADMM Algorithm for Nonconvex Nonsmooth and Nonseparable Problems

作　　者：杨开元 党亚峥[1] 杨灿 

机构地区：[1]上海理工大学管理学院,上海

出　　处：《理论数学》2024年第12期95-111,共17页Pure Mathematics

摘　　要：交替方向乘子法(ADMM)作为一种简单有效的求解可分离问题方法,已经被广泛应用于信号处理、并行计算、机器学习等许多领域。但是当目标函数较为复杂时,ADMM算法的收敛性就会难以保证。针对一类复杂的非凸非光滑不可分离的优化问题,提出了一种非精确的并行式惯性近端交替乘子法(PIPI-ADMM),在简单的参数假设下,算法产生的迭代序列收敛到增广拉格朗日函数的稳定点。另外,当增广拉格朗日函数满足Kurdyka-Lojasiewicz性质时,算法具有强收敛性。最后,数值实验结果表明,算法的迭代策略对求解具有显著的加速作用。The Alternating Direction Method of Multipliers (ADMM) has been widely applied in various fields such as signal processing, parallel computing, and machine learning due to its simplicity and effectiveness in solving separable problems. However, the convergence of the ADMM algorithm becomes challenging to guarantee when dealing with complex objective functions. For a class of complex, non-convex, non-smooth, and non-separable optimization problems, a novel Parallel Inexact Proximal Inertial Alternating Direction Method of Multipliers (PIPI-ADMM) is proposed. Under reasonable assumptions, the iterative sequence converges to a stationary point of the augmented Lagrange function. Moreover, with the help of Kurdyka-Lojasiewicz property, the proposed algorithm is strongly convergent. The numerical results show that the iterative strategy has significant acceleration.

关 键 词：惯性技术 全局收敛 不可分离问题 交替方向乘子法 Kurdyka-Lojasiewicz性质 

分 类 号：O17[理学—数学]