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Inertial Proximal Alternating Linearized minimization (iPALM) for nonconvex and nonsmooth Problems

SIAM JOURNAL ON IMAGING SCIENCES 2016年第4期9卷 1756-1787页

作者： Pock, Thomas Sabach, Shoham Graz Univ Technol Inst Comp Graph & Vis A-8010 Graz Austria AIT Austrian Inst Technol GmbH Digital Safety & Secur Dept A-1220 Vienna Austria Technion Israel Inst Technol Dept Ind Engn & Management IL-3200003 Haifa Israel

In this paper we study nonconvex and nonsmooth optimization problems with semialgebraic data, where the variables vector is split into several blocks of variables. The problem consists of one smooth function of the entire variables vector and the sum of nonsmooth functions for each block separately. We analyze an inertial version of the proximal alternating linearized minimization algorithm and prove its global convergence to a critical point of the objective function at hand. We illustrate our theoretical findings by presenting numerical experiments on blind image deconvolution, on sparse nonnegative matrix factorization and on dictionary learning, which demonstrate the viability and effectiveness of the proposed method.

关键词： alternating minimization blind image deconvolution block coordinate descent heavy ball method Kurdyka-Lojasiewicz property nonconvex and nonsmooth minimization sparse nonnegative matrix factorization dictionary learning

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A generalized inertial proximal alternating linearized minimization method for nonconvex nonsmooth problems

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APPLIED NUMERICAL MATHEMATICS 2023年第1期189卷 66-87页

作者： Wang, Qingsong Han, Deren Beihang Univ Sch Math Sci LMIB Beijing 100191 Peoples R China

We propose a general inertial version of the proximal alternating linearized minimization (PALM) (denoted by NiPALM) for a class of nonconvex and nonsmooth minimization problems, whose objective function is the sum of a smooth function of the entire variables and two nonsmooth functions of each variable. NiPALM is general in the sense that it contains the popular PALM, the inertial PALM (iPALM) and Gauss-Seidel type inertial PALM (GiPALM) as special cases. Under mild assumptions, namely, the underlying functions satisfy the Kurdyka-Lojasiewicz (KL) property and some suitable conditions on the parameters, we prove that each bounded sequence generated by NiPALM globally converges to a critical point. We also apply NiPALM to nonnegative matrix factorization, sparse principal component analysis, and weighted low-rank matrix restoration problems. Comparing results with those by PALM, iPALM, and GiPALM demonstrate the robustness and effectiveness of the proposed algorithm.(c) 2023 Published by Elsevier B.V. on behalf of IMACS.

关键词： nonconvex and nonsmooth minimization Alternating minimization Inertial Proximal gradient Kurdyka-?ojasiewicz property

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Convergence of Three-Block ADMM for Weakly Convex Optimization Problems

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SIAM JOURNAL ON IMAGING SCIENCES 2025年第1期18卷 449-493页

作者： Chen, Xin Cui, Chunfeng Han, Deren Beihang Univ Sch Math Sci LMIB Minist Educ Beijing 100191 Peoples R China

This paper focuses on the convergence of the alternating direction method of multipliers (ADMM) for solving linearly constrained optimization problems whose objective function is the sum of one weakly convex and two strongly convex functions. Many applications in image denoising and machine learning fields with a sparsity-driven regularization term can be approximated unbiasedly by weakly convex functions. Our new theoretical results provide convergence guarantees of the direct extension of ADMM (E-ADMM) without Lipschitz continuity of the gradient and Kurdyka--\Lojasiewicz property, and establish the worst-case o(1/k) convergence rate in the nonergodic sense. Under further conditions such as Lipschitz continuity and nonsingularity, we derive the global linear rate of convergence. The numerical results on tensor robust principal component analysis and generalized elastic net regression illustrate that E-ADMM is efficient compared with some popular methods such as the difference-of-convex approach and accelerated proximal gradient descent.

关键词： weakly convex robust principal component analysis alternating direction method of multipliers nonconvex and nonsmooth minimization global linear rate of convergence

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nonconvex Lagrangian-Based Optimization: Monitoring Schemes and Global Convergence

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MATHEMATICS OF OPERATIONS RESEARCH 2018年第4期43卷 1210-1232页

作者： Bolte, Jerome Sabach, Shoham Teboulle, Marc Univ Toulouse Capitole Toulouse Sch Econ F-31015 Toulouse France Technion Israel Inst Technol Fac Ind Engn & Management IL-3200003 Haifa Israel Tel Aviv Univ Sch Math Sci IL-69978 Ramat Aviv Israel

We introduce a novel approach addressing global analysis of a difficult class of nonconvex-nonsmooth optimization problems within the important framework of Lagrangian-based methods. This genuine nonlinear class captures many problems in modern disparate fields of applications. It features complex geometries, qualification conditions, and other regularity properties do not hold everywhere. To address these issues, we work along several research lines to develop an original general Lagrangian methodology, which can deal, all at once, with the above obstacles. A first innovative feature of our approach is to introduce the concept of Lagrangian sequences for a broad class of algorithms. Central to this methodology is the idea of turning an arbitrary descent method into a multiplier method. Second, we provide these methods with a transitional regime allowing us to identify in finitely many steps a zone where we can tune the step-sizes of the algorithm for the final converging regime. Then, despite the min-max nature of Lagrangian methods, using an original Lyapunov method we prove that each bounded sequence generated by the resulting monitoring schemes are globally convergent to a critical point for some fundamental Lagrangian-based methods in the broad semialgebraic setting, which to the best of our knowledge, are the first of this kind.

关键词： nonlinear composite minimization nonconvex and nonsmooth minimization Lagrangian-based methods proximal method of multipliers semialgebraic optimization nonsmooth Kurdyka-Lojasiewicz property global convergence

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Bregman reweighted alternating minimization and its application to image deblurring

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INFORMATION SCIENCES 2019年 503卷 401-416页

作者： Sun, Tao Qiao, Linbo Li, Dongsheng Natl Univ Def Technol Coll Comp Changsha 410073 Hunan Peoples R China

In this paper, we consider a class of nonconvex problems with linear constraints appearing frequently in the area of image processing. We solve this problem by the penalty method and propose the Bregman reweighted alternating minimization algorithm. The penalty parameter is set to increase in the iterations to speed up the algorithm. A convergence result is proved for the algorithm. If another smooth assumption is promised, with the semi-algebraic property, we can prove the algorithm globally converges to a critical point of the penalty objective function. Numerical results demonstrate the efficiency of the proposed algorithm. (C) 2019 Published by Elsevier Inc.

关键词： Bregman distance Reweighted alternating minimization Semi-algebraic Convergence analysis nonconvex and nonsmooth minimization Deblurring

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A variational proximal alternating linearized minimization in a given metric for limited-angle CT image reconstruction

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APPLIED MATHEMATICAL MODELLING 2019年 67卷 315-336页

作者： Wang, Chengxiang Luo, Xiaoqiang Yu, Wei Guo, Yumeng Zhang, LingLi Univ Elect Sci & Technol China Sch Math Sci Chengdu 611731 Sichuan Peoples R China Chongqing Univ Educ Minist China Engn Res Ctr Ind Computed Tomog Nondestruct Testi Chongqing 400044 Peoples R China Sichuan Univ Arts & Sci Dept Math Dazhou 635000 Peoples R China Hubei Univ Sci & Technol Sch Biomed Engn Xianning 437100 Peoples R China Chongqing Univ Coll Math & Stat Chongqing 401331 Peoples R China

Due to the restriction of computed tomography (CT) scanning environment, the acquired projection data may be incomplete for exact CT reconstruction. Though some convex optimization methods, such as total variation minimization based method, can be used for incomplete data reconstruction, the edge of reconstruction image may be partly distorted for limited-angle CT reconstruction. To promote the quality of reconstruction image for limited-angle CT imaging, in this paper, a nonconvex and nonsmooth optimization model was investigated. To solve the model, a variational proximal alternating linearized minimization (VPALM) method based on proximal mapping in a given metric was proposed. The proposed method can avoid computing the inverse of a huge system matrix thus can be used to deal with the larger-scale inverse problems. What's more, we show that each bounded sequence generated by VPALM globally converges to a critical point based on the Kurdyka-Lojasiewicz property. Real data experiments are used to demonstrate the viability and effectiveness of VPALM method, and the results show that the proposed method outperforms two classical CT reconstruction methods. (C) 2018 Elsevier Inc. All rights reserved.

关键词： Inverse problem CT reconstruction Alternating minimization nonconvex and nonsmooth minimization Limited-angle Cr

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Iteratively Linearized Reweighted Alternating Direction Method of Multipliers for a Class of nonconvex Problems

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2018年第20期66卷 5380-5391页

作者： Sun, Tao Jiang, Hao Cheng, Lizhi Zhu, Wei Natl Univ Def Technol Dept Math Changsha 410073 Hunan Peoples R China Natl Univ Def Technol Coll Comp Changsha 410073 Hunan Peoples R China Natl Univ Def Technol State Key Lab High Performance Computat Changsha 410073 Hunan Peoples R China Xiangtan Univ Sch Math & Computat Sci Hunan Key Lab Computat & Simulat Sci & Engn Xiangtan 411105 Peoples R China

In this paper, we consider solving a class of nonconvex and nonsmooth problems frequently appearing in signal processing andmachine learning research. The traditional alternating directionmethod of multipliers encounters troubles in both mathematics and computations in solving the nonconvex and nonsmooth subproblem. In view of this, we propose a reweighted alternating direction method of multipliers. In this algorithm, all subproblems are convex and easy to solve. We also provide several guarantees for the convergence and prove that the algorithm globally converges to a critical point of an auxiliary function with the help of the Kurdyka-Lojasiewicz property. Several numerical results are presented to demonstrate the efficiency of the proposed algorithm.

关键词： Alternating direction method of multipliers Iteratively reweighted algorithm nonconvex and nonsmooth minimization Kurdyka-Lojasiewicz property Semi-algebraic functions

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Convergent Nested Alternating minimization Algorithms for nonconvex Optimization Problems

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MATHEMATICS OF OPERATIONS RESEARCH 2023年第1期48卷 53-77页

作者： Gur, Eyal Sabach, Shoham Shtern, Shimrit Technion Israel Inst Technol Fac Ind Engn & Management IL-3200003 Haifa Israel

We introduce a new algorithmic framework for solving nonconvex optimization problems, that is called nested alternating minimization, which aims at combining the classical alternating minimization technique with inner iterations of any optimization method. We provide a global convergence analysis of the new algorithmic framework to critical points of the problem at hand, which to the best of our knowledge, is the first of this kind for nested methods in the nonconvex setting. Central to our global convergence analysis is a new extension of classical proof techniques in the nonconvex setting that allows for errors in the conditions. The power of our framework is illustrated with some numerical experiments that show the superiority of this algorithmic framework over existing methods.

关键词： nonconvex and nonsmooth minimization nested algorithms nondescent methods nonsmooth Kurdyka-Lojasiewicz property global convergence

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Canonical dual methods for general box constrained nonconvex and nonsmooth optimization problems

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JOURNAL OF COMPUTATIONAL METHODS IN SCIENCES AND ENGINEERING 2019年第3期19卷 765-777页

作者： Liu, Huicheng C. Gao, Jianwei Liu, Jing North China Elect Power Univ Sch Econ & Management Beijing 102206 Peoples R China North China Elect Power Univ Beijing Key Lab New Energy & Low Carbon Dev Beijing Peoples R China Wuyi Univ Dept Math & Comp Sci Jiangmen Guangdong Peoples R China

This paper presents a canonical duality theory for general box constrained nonconvex and nonsmooth optimization problems. The theory is illustrated perfectly by showing mainly a canonical dual transformation and a triality theory. It is proved that the general box constrained nonconvex and nonsmooth optimization problems can be changed into the perfect dual problems, which can be solved easily. The perfect dual transformation is developed for eliminating the duality gap,while the triality theorem studies global and local extreme conditions. Applications and examples are illustrated.

关键词： Canonical (perfect) dual transformation triality theory box constrains nonconvex and nonsmooth minimization global and local optimization

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A Dynamic Alternating Direction of Multipliers for nonconvex minimization with Nonlinear Functional Equality Constraints

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2022年第1-3期193卷 324-353页

作者： Cohen, Eyal Hallak, Nadav Teboulle, Marc Tel Aviv Univ Sch Math Sci Tel Aviv Israel Technion Fac Ind Engn & Management Haifa Israel

This paper studies the minimization of a broad class of nonsmooth nonconvex objective functions subject to nonlinear functional equality constraints, where the gradients of the differentiable parts in the objective and the constraints are only locally Lipschitz continuous. We propose a specific proximal linearized alternating direction method of multipliers in which the proximal parameter is generated dynamically, and we design an explicit and tractable backtracking procedure to generate it. We prove subsequent convergence of the method to a critical point of the problem, and global convergence when the problem's data are semialgebraic. These results are obtained with no dependency on the explicit manner in which the proximal parameter is generated. As a byproduct of our analysis, we also obtain global convergence guarantees for the proximal gradient method with a dynamic proximal parameter under local Lipschitz continuity of the gradient of the smooth part of the nonlinear sum composite minimization model.

关键词： Augmented Lagrangian-based methods nonconvex and nonsmooth minimization Proximal gradient method Kurdyka-Lojasiewicz property Global convergence

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