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On the convergence rate of the augmented Lagrangian-based parallel splitting method

OPTIMIZATION methodS & SOFTWARE 2019年第2期34卷 278-304页

作者： Wang, Kai Desai, Jitamitra Nanyang Technol Univ Sch Mech & Aerosp Engn Singapore Singapore

The augmented Lagrangian method (ALM) is a well-regarded algorithm for solving convex optimization problems with linear constraints. Recently, in He et al. [On full Jacobian decomposition of the augmented Lagrangian method for separable convex programming, SIAM J. Optim. 25(4) (2015), pp. 2274-2312], it has been demonstrated that a straightforward Jacobian decomposition of ALM is not necessarily convergent when the objective function is the sum of functions without coupled variables. Then, Wang et al. [A note on augmented Lagrangian-based parallel splitting method, Optim. Lett. 9 (2015), pp. 1199-1212] proved the global convergence of the augmented Lagrangian-based parallel splitting method under the assumption that all objective functions are strongly convex. In this paper, we extend these results and derive the worst-case convergence rate of this method under both ergodic and non-ergodic conditions, where t represents the number of iterations. Furthermore, we show that the convergence rate can be improved from to , and finally, we also demonstrate that this method can achieve global linear convergence, when the involved functions satisfy some additional conditions.

关键词： augmented Lagrangian method separable convex programming Jacobian decomposition parallel splitting method global linear convergence convergence rate

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A proximal parallel splitting method for minimizing sum of convex functions with linear constraints

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2014年 256卷 36-51页

作者： Han, Deren He, Hongjin Xu, Lingling Nanjing Normal Univ Sch Math Sci Key Lab NSLSCS Jiangsu Prov Nanjing 210023 Jiangsu Peoples R China Hangzhou Dianzi Univ Sch Sci Hangzhou 310018 Zhejiang Peoples R China

In this paper, we propose a proximal parallel decomposition algorithm for solving the optimization problems where the objective function is the sum of m separable functions (i.e., they have no crossed variables), and the constraint set is the intersection of Cartesian products of some simple sets and a linear manifold. The m subproblems are solved simultaneously per iterations, which are sum of the decomposed subproblems of the augmented Lagrange function and a quadratic term. Hence our algorithm is named as the 'proximal parallel splitting method'. We prove the global convergence of the proposed algorithm under some mild conditions that the underlying functions are convex and the solution set is nonempty. To make the subproblems easier, some linearized versions of the proposed algorithm are also presented, together with their global convergence analysis. Finally, some preliminary numerical results are reported to support the efficiency of the new algorithms. (C) 2013 Elsevier B.V. All rights reserved.

关键词： parallel splitting method Proximal point algorithm Separable structure Linearized algorithm Global convergence

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ON THE BEHAVIOR OF THE DOUGLAS RACHFORD ALGORITHM FOR MINIMIZING A CONVEX FUNCTION SUBJECT TO A LINEAR CONSTRAINT

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SIAM JOURNAL ON OPTIMIZATION 2020年第3期30卷 2559-2576页

作者： Bauschke, Heinz H. Moursi, Walaa M. Univ British Columbia Math Kelowna BC V1V 1V7 Canada Univ Waterloo Dept Combinator & Optimizat Waterloo ON N2L 3G1 Canada

The Douglas Rachford algorithm (DRA) is a powerful optimization method for minimizing the sum of two convex (not necessarily smooth) functions. The vast majority of previous research dealt with the case when the sum has at least one minimizer. In the absence of minimizers, it was recently shown that for the case of two indicator functions, the DRA converges to a best approximation solution. In this paper, we present a new convergence result on the DRA applied to the problem of minimizing a convex function subject to a linear constraint. Indeed, a normal solution may be found even when the domain of the objective function and the linear subspace constraint have no point in common. As an important application, a new parallel splitting result is provided. We also illustrate our results through various examples.

关键词： convex optimization problem Douglas-Rachford splitting inconsistent constrained optimization least squares solution normal problem parallel splitting method projection operator proximal mapping

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A new parallel splitting augmented Lagrangian-based method for a Stackelberg game

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JOURNAL OF INEQUALITIES AND APPLICATIONS 2016年第1期2016卷 1页

作者： Yan, Xihong Wen, Ruiping Taiyuan Normal Univ Higher Educ Key Lab Engn & Sci Comp Taiyuan 030012 Shanxi Peoples R China

This paper addresses a novel solution scheme for a special class of variational inequality problems which can be applied to model a Stackelberg game with one leader and three or more followers. In the scheme, the leader makes his decision first and then the followers reveal their choices simultaneously based on the information of the leader's strategy. Under mild conditions, we theoretically prove that the scheme can obtain an equilibrium. The proposed approach is applied to solve a simple game and a traffic problem. Numerical results about the performance of the new method are reported.

关键词： Stackelberg game variational inequality separable structure convergence parallel splitting method

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A hybrid splitting method for smoothing Tikhonov regularization problem

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JOURNAL OF INEQUALITIES AND APPLICATIONS 2016年第1期2016卷 1-13页

作者： Zeng, Yu-Hua Peng, Zheng Yang, Yu-Fei Hunan Univ Coll Math & Econometr Changsha 410082 Hunan Peoples R China Hunan First Normal Univ Coll Math & Computat Sci Changsha 410205 Hunan Peoples R China Fuzhou Univ Coll Math & Comp Sci Fuzhou 350108 Peoples R China

In this paper, a hybrid splitting method is proposed for solving a smoothing Tikhonov regularization problem. At each iteration, the proposed method solves three subproblems. First of all, two subproblems are solved in a parallel fashion, and the multiplier associated to these two block variables is updated in a rapid sequence. Then the third subproblem is solved in the sense of an alternative fashion with the former two subproblems. Finally, the multiplier associated to the last two block variables is updated. Global convergence of the proposed method is proven under some suitable conditions. Some numerical experiments on the discrete ill-posed problems (DIPPs) show the validity and efficiency of the proposed hybrid splitting method.

关键词： Tikhonov regularization augmented Lagrangian method parallel splitting method alternative direction method of multipliers

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A partial parallel splitting augmented Lagrangian method for solving constrained matrix optimization problems

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2010年第6期60卷 1515-1524页

作者： Peng, Zheng Wu, Donghua Fuzhou Univ Dept Math Fuzhou 350108 Peoples R China Shanghai Univ Dept Math Shanghai 200444 Peoples R China Nanjing Univ Dept Math Nanjing 210093 Peoples R China

Alternating directions methods (ADMs) are very effective for solving convex optimization problems with separable structure. However, when these methods are applied to solve convex optimization problems with three separable operators, their convergence results have not been established as yet. In this paper, we consider a class of constrained matrix optimization problems. The problem is first reformulated into a convex optimization problem with three separable operators, then it is solved by a proposed partial parallel splitting method. The proposed method combines the parallel splitting (augmented Lagrangian) method (PSALM) and the alternating directions method (ADM), and it is referred to as PADALM in short. The main difference between PADALM and PSALM is that in PADALM, two operators are handled first by a parallel method, then the third operator and the former two are dealt with by an alternating method. Finally, the convergence result for PADALM is established and numerical results are provided to show the efficacy of PADALM and its superiority over PSALM. Crown Copyright (C) 2010 Published by Elsevier Ltd. All rights reserved.

关键词： parallel splitting method Alternating directions method Separable convex optimization Constrained matrix optimization

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Global extrapolation with a parallel splitting method

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Numerical Algorithms 1992年第1期3卷 427-440页

作者： Tai, Xue-Cheng Department of Mathematics University of Jyväskylä Jyväskylä SF-40351 P.O. Box 35 Finland

Extrapolation with a parallel splitting method is discussed. The parallel splitting method reduces a multidimensional problem into independent one-dimensional problems and can improve the convergence order of space variables to an order as high as the regularity of the solution permits. Therefore, in order to match the convergence order of the space variables, a high order method should also be used for the time integration. Second and third order extrapolation methods are used to improve the time convergence and it was found that the higher order extrapolation method can produce a more accurate solution than the lower order extrapolation method, but the convergence order of high order extrapolation may be less than the actual order of the extrapolation. We also try to show a fact that has not been studied in the literature, i.e. when the extrapolation is used, it may decrease the convergence of the space variables. The higher the order of the extrapolation method, the more it decreases the convergence of the space variables. The global extrapolation method also improves the parallel degree of the parallel splitting method. Numerical tests in the paper are done in a domain of a unit circle and a unit square. © 1992, J.C. Baltzer A.G. Scientific Publishing Company. All rights reserved.

关键词： 65B05 65N30 global extrapolation parabolic problem parallel LOD method parallel splitting method

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