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Convergence of multi-block bregman ADMM for nonconvex composite problems

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Science China(Information Sciences) 2018年第12期61卷 53-64页

作者： Fenghui WANG Wenfei CAO Zongben XU School of Mathematics and Statistics Xi'an Jiaotong University Department of Mathematics Luoyang Normal University School of Mathematics and Information Science Shaanxi Normal University

The alternating direction method with multipliers(ADMM) is one of the most powerful and successful methods for solving various composite problems. The convergence of the conventional ADMM(i.e.,2-block) for convex objective functions has been stated for a long time, and its convergence for nonconvex objective functions has, however, been established very recently. The multi-block ADMM, a natural extension of ADMM, is a widely used scheme and has also been found very useful in solving various nonconvex optimization problems. It is thus expected to establish the convergence of the multi-block ADMM under nonconvex frameworks. In this paper, we first justify the convergence of 3-block bregman ADMM. We next extend these results to the N-block case(N≥3), which underlines the feasibility of multi-block ADMM applications in nonconvex settings. Finally, we present a simulation study and a real-world application to support the correctness of the obtained theoretical assertions.

关键词： nonconvex regularization alternating direction method subanalytic function K-L inequality bregman distance

STRONG CONVERGENCE THEOREMS FOR EQUILIBRIUM PROBLEM AND bregman TOTALLY QUASI-ASYMPTOTICALLY NONEXPANSIVE MAPPING IN BANACH SPACES

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Acta Mathematica Scientia 2016年第5期36卷 1433-1444页

作者：朱胜黄建华 Institute of Mathematics and Computer Science Fuzhou University Fuzhou 350002 China

In this paper, we propose a new hybrid iterative scheme for finding a common solution of an equilibrium problem and fixed point of bregman totally quasi-asymptotically nonexpansive mapping in reflexive Banach spaces. Moreover, we prove some strong convergence theorems under suitable control conditions. Finally, the application to zero point problem of maximal monotone operators is given by the result.

关键词： equilibrium problem bregman totally quasi-asymptotically nonexpansive mapping bregman distance reflexive Banach space

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bregman Iterative Model Using theG-norm

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Acta Mathematicae Applicatae Sinica 2014年第1期30卷 179-186页

作者： Yu-ying SHI Xiao-zhong YANG Yong-gui ZHU Department of Mathematics and Physics North China Electric Power University School of Science Communication University of China

In this paper, we analyze the bregman iterative model using the G-norm. Firstly, we show the convergence of the iterative model. Secondly, using the source condition and the symmetric bregman distance, we consider the error estimations between the iterates and the exact image both in the case of clean and noisy data. The results show that the bregman iterative model using the G-norm has the similar good properties as the bregman iterative model using the L2-norm.

关键词： bregman distance image restoration total variation error estimation iterative regularization

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Convergence of bregman Peaceman–Rachford Splitting Method for Nonconvex Nonseparable Optimization

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Journal of the Operations Research Society of China 2023年第4期11卷 707-733页

作者： Peng-Jie Liu Jin-Bao Jian Bo He Xian-Zhen Jiang College of Mathematics and Physics Guangxi Key Laboratory of Hybrid Computation and IC Design AnalysisCenter for Applied Mathematics and Artificial IntelligenceGuangxi University for NationalitiesNanning530006GuangxiChina College of Mathematics and Information Science Guangxi UniversityNanning530004GuangxiChina School of Mathematics China University of Mining and TechnologyXuzhou221116JiangsuChina

This work is about a splitting method for solving a nonconvex nonseparable optimization problem with linear constraints,where the objective function consists of two separable functions and a coupled ***,based on the ideas from bregman distance and Peaceman–Rachford splitting method,the bregman Peaceman–Rachford splitting method with different relaxation factors for the multiplier is ***,the global and strong convergence of the proposed algorithm are proved under general conditions including the region of the two relaxation factors as well as the crucial Kurdyka–Łojasiewicz ***,when the associated Kurdyka–Łojasiewicz property function has a special structure,the sublinear and linear convergence rates of the proposed algorithm are ***,some preliminary numerical results are shown to indicate the effectiveness of the proposed algorithm.

关键词： Nonconvex nonseparable optimization Peaceman-Rachford splitting method bregman distance Kurdyka-Łojasiewicz inequality Convergence rate

The Developments of Proximal Point Algorithms

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Journal of the Operations Research Society of China 2022年第2期10卷 197-239页

作者： Xing-Ju Cai Ke Guo Fan Jiang Kai Wang Zhong-Ming Wu De-Ren Han Jiangsu Key Lab for NSLSCS School of Mathematical SciencesNanjing Normal UniversityNanjing 210023JiangsuChina School of Mathematics and Information China West Normal UniversityNanchong 637002SichuanChina 不详

The problem of finding a zero point of a maximal monotone operator plays a central role in modeling many application problems arising from various fields,and the proximal point algorithm(PPA)is among the fundamental algorithms for solving the zero-finding *** not only provides a very general framework of analyzing convergence and rate of convergence of many algorithms,but also can be very efficient in solving some structured *** this paper,we give a survey on the developments of PPA and its variants,including the recent results with linear proximal term,with the nonlinear proximal term,as well as the inexact forms with various approximate criteria.

关键词： Zero-finding problems Proximal point algorithms Variational inequality problems Optimization bregman distance Approximate criteria

STRONG CONVERGENCE THEOREMS FOR FINITE EQUILIBRIUM PROBLEMS AND bregman TOTALLY QUASI-ASYMPTOTICALLY NONEXPANSIVE MAPPING IN BANACH SPACES

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Annals of Applied Mathematics 2015年第4期31卷 372-382页

作者： Haipeng He Jianhua Huang Sheng Zhu College of Math.and Computer Science Fuzhou University

In this paper, we propose a new hybrid iterative scheme for finding a common solution to a finite of equilibrium problems and fixed point of bregman totally quasi- asymptotically nonexpansive mapping in reflexive Bana... 详细信息

关键词： equilibrium problem bregman totally quasi-asymptotically nonex-pansive mapping bregman distance reflexive Banach space

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