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On the linear convergence of a Proximal Gradient Method for a Class of Nonsmooth Convex Minimization Problems

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Journal of the Operations Research Society of China 2013年第2期1卷 163-186页

作者： Haibin Zhang Jiaojiao Jiang Zhi-Quan Luo College of Applied Science Beijing University of TechnologyBeijing 100124China Department of Electrical and Computer Engineering University of MinnesotaMinneapolisMN 55455USA

We consider a class of nonsmooth convex optimization problems where the objective function is the composition of a strongly convex differentiable function with a linear mapping,regularized by the sum of both l1-norm and l2-norm of the optimization *** class of problems arise naturally from applications in sparse group Lasso,which is a popular technique for variable *** effective approach to solve such problems is by the Proximal Gradient Method(PGM).In this paper we prove a local error bound around the optimal solution set for this problem and use it to establish the linear convergence of the PGM method without assuming strong convexity of the overall objective function.

关键词： Proximal gradient method Error bound linear convergence Sparse group I asso

Fully asynchronous distributed optimization with linear convergence over directed networks

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中山大学学报（自然科学版）（中英文） 2023年第5期62卷 1-23页

作者： SHA Xingyu ZHANG Jiaqi YOU Keyou Department of Automation Beijing National Research Center for Information Science and TechnologyTsinghua UniversityBeijing 100084China Beijing Sankuai Online Technology Co. Ltd.Beijing 100102China

We study distributed optimization problems over a directed network,where nodes aim to minimize the sum of local objective functions via directed communications with *** algorithms are designed to solve it for synchronized or randomly activated implementation,which may create deadlocks in *** sharp contrast,we propose a fully asynchronous push-pull gradient(APPG) algorithm,where each node updates without waiting for any other node by using possibly delayed information from ***,we construct two novel augmented networks to analyze asynchrony and delays,and quantify its convergence rate from the worst-case point of ***,all nodes of APPG converge to the same optimal solution at a linear rate of O(λ^(k)) if local functions have Lipschitz-continuous gradients and their sum satisfies the Polyak-?ojasiewicz condition(convexity is not required),where λ ∈(0,1) is explicitly given and the virtual counter k increases by one when any node ***,the advantage of APPG over the synchronous counterpart and its linear speedup efficiency are numerically validated via a logistic regression problem.

关键词： fully asynchronous distributed optimization linear convergence Polyak-Łojasiewicz condition

维普期刊数据库

Efficient algorithms for computing the largest eigenvalue of a nonnegative tensor

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Frontiers of Mathematics in China 2013年第1期8卷 155-168页

作者： Guanglu ZHOU Liqun QI Soon-Yi WU Department of Mathematics and Statistics Curtin University Perth Australia Department of Applied Mathematics The Hong Kong Polytechnic University Hong Kong China Institute of Applied Mathematics Cheng-Kung University Tainan China

Consider the problem of computing the largest eigenvalue for nonnegative tensors. In this paper, we establish the Q-linear convergence of a power type algorithm for this problem under a weak irreducibility condition. ... 详细信息

关键词： Eigenvalue nonnegative tensor power method linear convergence

linearLY CONVERGENT FIRST-ORDER ALGORITHMS FOR SEMIDEFINITE PROGRAMMING

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Journal of Computational Mathematics 2017年第4期35卷 452-468页

作者： Cong D. Dang Guanghui Lan Zaiwen Wen Department of Industrial and Systems Engineering University of Florida Gainesville FL 32611 H. Milton Stewart School of Industrial and Systems Engineering Georgia Institute of TechnologyAtlanta GA 30332 USA Beijing International Center for Mathematical Research Peking University Beijing 100871 China

In this paper, we consider two different formulations （one is smooth and the other one is nonsmooth） for solving linear matrix inequalities （LMIs）, an important class of semidefinite programming （SDP）, under a certain Slater constraint qualification assumption. We then propose two first-order methods, one based on subgradient method and the other based on Nesterov＇s optimal method, and show that they converge linearly for solving these formulations. Moreover, we introduce an accelerated prox-level method which converges linearly uniformly for both smooth and non-smooth problems without requiring the input of any problem parameters. Finally, we consider a special case of LMIs, i.e., linear system of inequalities, and show that a linearly convergent algorithm can be obtained under a much weaker assumption.

关键词： Semi-definite Programming linear Matrix Inequalities Error Bounds linear convergence

A Method with Parameter for Solving the Spectral Radius of Nonnegative Tensor

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Journal of the Operations Research Society of China 2017年第1期5卷 3-25页

作者： Yi-Yong Li Qing-Zhi Yang Xi He School of Mathematical Sciences and Lab of Pure Mathematics and Combinatic Nankai UniversityTianjin 300071China

In this paper,a method with parameter is proposed for finding the spectral radius of weakly irreducible nonnegative *** is more,we prove this method has an explicit linear convergence rate for indirectly positive ***,the algorithm is exactly the NQZ method(proposed by Ng,Qi and Zhou in Finding the largest eigenvalue of a non-negative tensor SIAM J Matrix Anal Appl 31:1090–1099,2009)by taking a specific ***,we give a modified NQZ method,which has an explicit linear convergence rate for nonnegative tensors and has an error bound for nonnegative tensors with a positive Perron ***,we promote an inexact power-type ***,some numerical results are reported.

关键词： Nonnegative tensor Indirectly positive tensors linear convergence Perturbation Complexity

Gradient Descent for Symmetric Tensor Decomposition

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Annals of Applied Mathematics 2022年第4期38卷 385-413页

作者： Jian-Feng Cai Haixia Liu Yang Wang Department of Mathematics The Hong Kong University of Science and TechnologyClear Water BayKowloonHong KongChina School of Mathematics and Statistics&Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and TechnologyWuhanHubei 430074China

Symmetric tensor decomposition is of great importance in *** studies have employed a greedy approach,where the main idea is to first find a best rank-one approximation of a given tensor,and then repeat the process to the residual tensor by subtracting the rank-one *** this paper,we focus on finding a best rank-one approximation of a given orthogonally order-3 symmetric *** give a geometric landscape analysis of a nonconvex optimization for the best rank-one approximation of orthogonally symmetric *** show that any local minimizer must be a factor in this orthogonally symmetric tensor decomposition,and any other critical points are linear combinations of the ***,we propose a gradient descent algorithm with a carefully designed initialization to solve this nonconvex optimization problem,and we prove that the algorithm converges to the global minimum with high probability for orthogonal decomposable *** result,combined with the landscape analysis,reveals that the greedy algorithm will get the tensor CP low-rank *** results are provided to verify our theoretical results.

关键词： Gradient descent random initialization symmetric tensor decomposition CP decomposition linear convergence

维普期刊数据库

Distributed Augmented Primal-Dual Algorithm for Multi-Agent Constrained Optimization

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Distributed Augmented Primal-Dual Algorithm for Multi-Agent ...

基于新的随机选择方式的随机Kaczmarz算法

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作者： Kaixin Du Min Meng Shanghai Research Institute for Intelligent Autonomous Systems Tongji University Department of Control Science and Engineering College of Electronics and Information Engineering Tongji University

This paper deals with the distributed optimization problem while respecting coupled affine equality constraints. Resorting to the augmented Lagrangian function, a novel distributed augmented primal-dual algorithm is developed by virtue of the gradient tracking and dynamic consensus mechanisms. Based on the carefully constructed Lyapunov function, it is proved that the proposed algorithm can achieve linear convergence and an exact upper bound on the stepsize is given. Finally, a numerical example is presented to corroborate the effectiveness of the designed algorithm.

关键词： Distributed optimization Coupled affine equality constraints Augmented Lagrangian method linear convergence

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高等学校计算数学学报 2018年第1期40卷 65-75页

作者：郭俊含李维国中国石油大学(华东)理学院

1引言Kaczmarz算法[8]是求解超定相容线性方程组的最受欢迎的方法之一,Ax=b,（1.1）其中矩阵A=(a;)∈C;（m≥n）,b=（b;,b;,…,b;）;∈C;,a;（i=1,2,…,m）表示矩阵A的行.由于其简单且收敛速度快,Kaczmarz算法的应用范围非常广泛,最早是被

关键词： overdetermined consistent linear system Randomized Kaczmarz method the distance linear convergence

Accelerated Design for Distributed Asynchronous Optimization

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Accelerated Design for Distributed Asynchronous Optimization