Inexact proximal methods for weakly convex functions

作     者：Khanh, Pham Duy Mordukhovich, Boris S. Phat, Vo Thanh Tran, Dat Ba 

作者机构：Ho Chi Minh City Univ Educ Dept Math Grp Anal & Appl Math Ho Chi Minh City Vietnam Wayne State Univ Dept Math Detroit MI 48202 USA Univ North Dakota Dept Math & Stat Grand Forks ND USA 

出 版 物：《JOURNAL OF GLOBAL OPTIMIZATION》 (J of Global Optim)

年 卷 期：2025年第91卷第3期

页      面：611-646页

核心收录：

学科分类：1201[管理学-管理科学与工程(可授管理学、工学学位)] 07[理学] 070104[理学-应用数学] 0701[理学-数学] 

基　　金：Directorate for Mathematical and Physical Sciences 

主　　题：Inexact proximal methods Weakly convex functions Forward-backward envelopes Kurdyka-& Lstrok ojasiewicz property Global convergence Linear convergence rates Proximal points 

摘      要：This paper proposes and develops inexact proximal methods for finding stationary points of the sum of a smooth function and a nonsmooth weakly convex one, where an error is present in the calculation of the proximal mapping of the nonsmooth term. A general framework for finding zeros of a continuous mapping is derived from our previous paper on this subject to establish convergence properties of the inexact proximal point method when the smooth term is vanished and of the inexact proximal gradient method when the smooth term satisfies a descent condition. The inexact proximal point method achieves global convergence with constructive convergence rates when the Moreau envelope of the objective function satisfies the Kurdyka-& Lstrok;ojasiewicz (KL) property. Meanwhile, when the smooth term is twice continuously differentiable with a Lipschitz continuous gradient and a differentiable approximation of the objective function satisfies the KL property, the inexact proximal gradient method achieves the global convergence of iterates with constructive convergence rates.