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Accelerated primal-dual methods with adaptive parameters for composite convex optimization with linear constraints

APPLIED NUMERICAL MATHEMATICS 2024年 203卷 129-143页

作者： He, Xin Xihua Univ Sch Sci Chengdu 610039 Sichuan Peoples R China

In this paper, we introduce two accelerated primal-dual methods tailored to address linearly constrained composite convex optimization problems, where the objective function is expressed as the sum of a possibly nondifferentiable function and a differentiable function with Lipschitz continuous gradient. The first method is the accelerated linearized augmented Lagrangian method (ALALM), which permits linearization to the differentiable function;the second method is the accelerated linearized proximal point algorithm (ALPPA), which enables linearization of both the differentiable function and the augmented term. By incorporating adaptive parameters, we demonstrate that ALALM achieves the O(1/k(2)) convergence rate and the linear convergence rate under the assumption of convexity and strong convexity, respectively. Additionally, we establish that ALPPA enjoys the O(1/k) convergence rate in convex case and the O(1/k(2)) convergence rate in strongly convex case. We provide numerical results to validate the effectiveness of the proposed methods.

关键词： Nesterov acceleration Augmented Lagrangian method proximal point algorithm Convergence rate

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On the proximal Jacobian Decomposition of ALM for Multiple-Block Separable Convex Minimization Problems and Its Relationship to ADMM

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JOURNAL OF SCIENTIFIC COMPUTING 2016年第3期66卷 1204-1217页

作者： He, Bingsheng Xu, Hong-Kun Yuan, Xiaoming South Univ Sci & Technol China Dept Math Shenzhen Peoples R China Nanjing Univ Dept Math Nanjing 210093 Jiangsu Peoples R China Hangzhou Dianzi Univ Sch Sci Dept Math Hangzhou 310018 Zhejiang Peoples R China Natl Sun Yat Sen Univ Dept Appl Math Kaohsiung 80424 Taiwan Hong Kong Baptist Univ Dept Math Hong Kong Hong Kong Peoples R China

The augmented Lagrangian method (ALM) is a benchmark for solving convex minimization problems with linear constraints. When the objective function of the model under consideration is representable as the sum of some functions without coupled variables, a Jacobian or Gauss-Seidel decomposition is often implemented to decompose the ALM subproblems so that the functions' properties could be used more effectively in algorithmic design. The Gauss-Seidel decomposition of ALM has resulted in the very popular alternating direction method of multipliers (ADMM) for two-block separable convex minimization models and recently it was shown in He et al. (Optimization Online, 2013) that the Jacobian decomposition of ALM is not necessarily convergent. In this paper, we show that if each subproblem of the Jacobian decomposition of ALM is regularized by a proximal term and the proximal coefficient is sufficiently large, the resulting scheme to be called the proximal Jacobian decomposition of ALM, is convergent. We also show that an interesting application of the ADMM in Wang et al. (Pac J Optim, to appear), which reformulates a multiple-block separable convex minimization model as a two-block counterpart first and then applies the original ADMM directly, is closely related to the proximal Jacobian decomposition of ALM. Our analysis is conducted in the variational inequality context and is rooted in a good understanding of the proximal point algorithm.

关键词： Convex optimization Alternating direction method of multipliers Augmented Lagrangian method Jacobian decomposition Parallel computation proximal point algorithm Variational inequality problem

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PRODUCTS OF FINITELY MANY RESOLVENTS OF MAXIMAL MONOTONE MAPPINGS IN REFLEXIVE BANACH SPACES

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SIAM JOURNAL ON OPTIMIZATION 2011年第4期21卷 1289-1308页

作者： Sabach, Shoham Technion Israel Inst Technol Dept Math IL-32000 Haifa Israel

We propose two algorithms for finding (common) zeros of finitely many maximal monotone mappings in reflexive Banach spaces. These algorithms are based on the Bregman distance related to a well-chosen convex function and improve previous results. Finally, we mention two applications of our algorithms for solving equilibrium problems and convex feasibility problems.

关键词： Banach space bifunction Bregman projection convex feasibility problem equilibrium problem Legendre function monotone mapping proximal point algorithm resolvent totally convex function

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Conical averagedness and convergence analysis of fixed point algorithms

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JOURNAL OF GLOBAL OPTIMIZATION 2022年第2期82卷 351-373页

作者： Bartz, Sedi Dao, Minh N. Phan, Hung M. Univ Massachusetts Dept Math Sci Kennedy Coll Sci Lowell MA 01854 USA Federat Univ Australia Sch Engn Informat Technol & Phys Sci Ballarat Vic 3353 Australia

We study a conical extension of averaged nonexpansive operators and the role it plays in convergence analysis of fixed point algorithms. Various properties of conically averaged operators are systematically investigated, in particular, the stability under relaxations, convex combinations and compositions. We derive conical averagedness properties of resolvents of generalized monotone operators. These properties are then utilized in order to analyze the convergence of the proximal point algorithm, the forward-backward algorithm, and the adaptive Douglas-Rachford algorithm. Our study unifies, improves and casts new light on recent studies of these topics.

关键词： Adaptive Douglas-Rachford algorithm Cocoercivity Conically averaged operator Forward-backward algorithm proximal point algorithm Strong monotonicity Weak monotonicity

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Convergence of a splitting inertial proximal method for monotone operators

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2003年第2期155卷 447-454页

作者： Moudafi, A Oliny, M Univ Antilles Guyane DSI GRIMAAG Schoelcher 97200 Martinique France

A forward-backward inertial procedure for solving the problem of finding a zero of the sum of two maximal monotone operators is proposed and its convergence is established under a cocoercivity condition with respect t... 详细信息

关键词： monotone operators enlargements proximal point algorithm cocoercivity splitting algorithm projection convergence

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proximal constrained optimization approach with time penalization

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ENGINEERING OPTIMIZATION 2019年第7期51卷 1207-1228页

作者： Trejo, Kristal K. Clempner, Julio B. Poznyak, Alexander S. Natl Polytech Inst Ctr Res & Adv Studies Dept Automat Control Mexico City DF Mexico Inst Politecn Nacl Escuela Super Fis & Metemat Mexico City DF Mexico Natl Polytech Inst Sch Phys & Math Mexico City DF Mexico

This article concerns a proximal-point algorithm with time penalization. The case where the cost of moving from one position to a better one is penalized by the time taken by the agent for the decision-making is studied and the restriction employing the penalty method is incorporated. It is shown that the method converges monotonically with respect to the minimal weighted norm to a unique minimal point under mild assumptions. The gradient method is employed for solving the objective function, and its convergence is proven. The rate of convergence of the method is also estimated by computing the optimal parameters. The effectiveness of the method is illustrated by a numerical optimization example employing continuous-time Markov chains.

关键词： proximal point algorithm cost of moving time penalization convergence continuous-time Markov chains

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THE METHOD OF ALTERNATING RESOLVENTS REVISITED

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NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION 2012年第11期33卷 1280-1287页

作者： Boikanyo, Oganeditse A. Morosanu, Gheorghe Univ Botswana Dept Math Gaborone Botswana Cent European Univ Dept Math & Its Applicat Budapest Hungary

The purpose of this article is to prove a strong convergence result associated with a generalization of the method of alternating resolvents introduced by the authors in convergence of the method of alternating resolvents [4] under minimal assumptions on the control parameters involved. Thus, this article represents a significant improvement of the article mentioned above.

关键词： Alternating resolvents Maximal monotone operator Nonexpansive map proximal point algorithm Resolvent operator

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A proximal-proximal Majorization-Minimization algorithm for Nonconvex Rank Regression Problems

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2023年 71卷 3502-3517页

作者： Tang, Peipei Wang, Chengjing Jiang, Bo Hangzhou City Univ Sch Comp & Comp Sci Hangzhou 310015 Peoples R China Southwest Jiaotong Univ Sch Math Chengdu 611756 Peoples R China Zhejiang Univ Sch Comp Sci Hangzhou 310027 Zhejiang Peoples R China

In this paper, we introduce a proximal-proximal majorization-minimization (PPMM) algorithm for nonconvex rank regression problems. The basic idea of the algorithm is to apply the proximal majorization-minimization algorithm to solve the nonconvex problem with the inner subproblems solved by a sparse semismooth Newton (SSN) method based proximal point algorithm (PPA). It deserves mentioning that we adopt the sequential regularization technique and design an implementable stopping criterion to overcome the singular difficulty of the inner subproblem. Especially for the stopping criterion, it plays a very important role for the success of the algorithm. Furthermore, we also prove that the PPMM algorithm converges to a stationary point. Due to the Kurdyka-Lojasiewicz (KL) property of the problem, we present the convergence rate of the PPMM algorithm. Numerical experiments demonstrate that our proposed algorithm outperforms the existing state-of-the-art algorithms.

关键词： Nonconvex rank regression problems semismooth Newton method proximal-proximal majorization-minimization algorithm proximal point algorithm

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Solving monotone stochastic variational inequalities and complementarity problems by progressive hedging

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MATHEMATICAL PROGRAMMING 2019年第1-2期174卷 453-471页

作者： Rockafellar, R. Tyrrell Sun, Jie Univ Washington Dept Math Box 354350 Seattle WA 98195 USA Curtin Univ Dept Math & Stat Bentley WA 6102 Australia

The concept of a stochastic variational inequality has recently been articulated in a new way that is able to cover, in particular, the optimality conditions for a multistage stochastic programming problem. One of the long-standing methods for solving such an optimization problem under convexity is the progressive hedging algorithm. That approach is demonstrated here to be applicable also to solving multistage stochastic variational inequality problems under monotonicity, thus increasing the range of applications for progressive hedging. Stochastic complementarity problems as a special case are explored numerically in a linear two-stage formulation.

关键词： Progressive hedging algorithm Stochastic variational inequality problems Stochastic complementarity problems Stochastic programming problems Maximal monotone mappings proximal point algorithm Problem decomposition

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A unified hybrid iterative method for hierarchical minimization problems

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2013年 253卷 208-221页

作者： Sahu, D. R. Ansari, Q. H. Yao, J. C. Banaras Hindu Univ Dept Math Varanasi 221005 Uttar Pradesh India Aligarh Muslim Univ Dept Math Aligarh 202002 Uttar Pradesh India King Fahd Univ Petr & Minerals Dept Math & Stat Dhahran 31261 Saudi Arabia Kaohsiung Med Univ Ctr Gen Educ Kaohsiung 80708 Taiwan King Abdulaziz Univ Dept Math Jeddah 21589 Saudi Arabia

In this paper, we introduce and analyze a new unified hybrid iterative method to compute the approximate solution of the general optimization problem defined over the set D = Fix(T) boolean AND Omega vertical bar GMEP(Phi, Psi, phi)], where Fix(T) is the set of common fixed points of a family T = {T(t) : 0 <= t < infinity} of nonexpansive self-mappings on a Hilbert space H, and Omega vertical bar GMEP(Phi, Psi, phi)] is the set of solutions of the generalized mixed equilibrium problem (in short, GMEP). Such type of minimization problem is called the hierarchical minimization problem. We establish the strong convergence of the sequences generated by the proposed algorithm. Our strong convergence theorem extends, improves and unifies the previously known results in the literature. We also give a numerical example to illustrate our algorithm and results. (C) 2013 Elsevier B.V. All rights reserved.

关键词： Hierarchical minimization problems Hybrid iterative method proximal point algorithm Maximal monotone operators Metric projection mappings Resolvent operators

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