An indefinite-proximal-based strictly contractive Peaceman-Rachford splitting method

Gu, Yan; Jiang, Bo; Han, Deren

doi:10.48550/arxiv.1506.02221

Cited by 8 publications

(20 citation statements)

References 23 publications

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“…In addition, the convergence domain K for the stepsizes (τ, s) in (8), shown in Fig. 1, is significantly larger than the domain H given in (6) and the convergence domain in [9,15]. For example, the stepsize s can be arbitrarily close to 5/3 when the stepsize τ is close to −1/3.…”

Section: Introductionmentioning

confidence: 91%

“…What's more, the numerical performance of S-ADMM on solving the widely used basis pursuit model and the total-variational image debarring model significantly outperforms the original ADMM in both the CPU time and the number of iterations. Besides, Gu, et al [9] also studied a semiproximal-based strictly contractive Peaceman-Rachford splitting method, that is (5) with two additional proximal penalty terms for the x and y update. But their method has a nonsymmetric convergence domain of the stepsize and still focuses on the two-block case problem, which limits its applications for solving large-scale problems with multiple block variables.…”

Section: Introductionmentioning

confidence: 99%

“…But their method has a nonsymmetric convergence domain of the stepsize and still focuses on the two-block case problem, which limits its applications for solving large-scale problems with multiple block variables. Mainly motivated by the work of [14,16,9], we would like to generalize S-ADMM with more wider convergence domain of the stepsizes to tackle the multi-block separable convex programming model (1), which more frequently appears in recent applications involving big data [2,20]. Our algorithm framework can be described as follows:…”

Section: Introductionmentioning

confidence: 99%

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Generalized symmetric ADMM for separable convex optimization

Bai

et al. 2017

Comput Optim Appl

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The Alternating Direction Method of Multipliers (ADMM) has been proved to be effective for solving separable convex optimization subject to linear constraints. In this paper, we propose a Generalized Symmetric ADMM (GS-ADMM), which updates the Lagrange multiplier twice with suitable stepsizes, to solve the multi-block separable convex programming. This GS-ADMM partitions the data into two group variables so that one group consists of p block variables while the other has q block variables, where p ≥ 1 and q ≥ 1 are two integers. The two grouped variables are updated in a Gauss-Seidel scheme, while the variables within each group are updated in a Jacobi scheme, which would make it very attractive for a big data setting. By adding proper proximal terms to the subproblems, we specify the domain of the stepsizes to guarantee that GS-ADMM is globally convergent with a worst-case O(1/t) ergodic convergence rate. It turns out that our convergence domain of the stepsizes is significantly larger than other convergence domains in the literature. Hence, the GS-ADMM is more flexible and attractive on choosing and using larger stepsizes of the dual variable. Besides, two special cases of GS-ADMM, which allows using zero penalty terms, are also discussed and analyzed. Compared with several state-of-the-art methods, preliminary numerical experiments on solving a sparse matrix minimization problem in the statistical learning show that our proposed method is effective and promising.is the Lagrangian function of the problem (1) with the Lagrange multiplier λ ∈ R n . Then, the ALM procedure for solving (1) can be described as follows:However, ALM does not make full use of the separable structure of the objective function of (1) and hence, could not take advantage of the special properties of the component objective

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Section: Introductionmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Generalized symmetric ADMM for separable convex optimization

Bai

et al. 2017

Comput Optim Appl

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“…Moreover, numerical experiments show that symmetrically updating the dual variable in a more flexible way often improves the algorithm performance [10,15]. The sublinear convergence rate of GS-ADMM in the nonergodic sense and its linear convergence rate can be found in [2].…”

mentioning

confidence: 93%

“…2 ) is the stepsize for updating the dual variable λ. Inspired by enlarging stepsize region of dual variable in [14], Gu, et al [10] proposed a symmetric proximal ADMM whose dual variable is updated twice with different stepsizes. Then, He, et al [15] proposed the following symmetric ADMM (S-ADMM):…”

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confidence: 99%

Convergence on a symmetric accelerated stochastic ADMM with larger stepsizes

Bai¹,

Han²,

Sun³

et al. 2021

Preprint

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In this paper, we develop a symmetric accelerated stochastic Alternating Direction Method of Multipliers (SAS-ADMM) for solving separable convex optimization problems with linear constraints. The objective function is the sum of a possibly nonsmooth convex function and an average function of many smooth convex functions. Our proposed algorithm combines both ideas of ADMM and the techniques of accelerated stochastic gradient methods using variance reduction to solve the smooth subproblem. One main feature of SAS-ADMM is that its dual variable is symmetrically updated after each update of the separated primal variable, which would allow a more flexible and larger convergence region of the dual variable compared with that of standard deterministic or stochastic ADMM. This new stochastic optimization algorithm is shown to converge in expectation with O(1/T ) convergence rate, where T is the number of outer iterations. In addition, 3-block extensions of the algorithm and its variant of an accelerated stochastic augmented Lagrangian method are also discussed. Our preliminary numerical experiments indicate the proposed algorithm is very effective for solving separable optimization problems from big-data applications

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A Multi-Step Inertial Proximal Peaceman-Rachford Splitting Method for Separable Convex Programming

Leifu

2023

IEEE Access

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In this paper, we propose a multi-step inertial proximal Peaceman-Rachford splitting method (abbreviated as MIP-PRSM) for solving the two-block separable convex optimization problems with linear constraints, which is a unified framework for such Peaceman-Rachford splitting methods (PRSM)-based improved algorithms with inertial step. Furthermore, we establish the global convergence of the MIP-PRSM under some assumptions. Finally, some numerical experimental results on the least squares semidefinite programming (LSSP), LASSO, the convex quadratic programming problem (CQPP), total variation (TV) based denoising and medical image reconstruction problems demonstrate the efficiency of the proposed method.

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An indefinite-proximal-based strictly contractive Peaceman-Rachford splitting method

Cited by 8 publications

References 23 publications

Generalized symmetric ADMM for separable convex optimization

Generalized symmetric ADMM for separable convex optimization

Convergence on a symmetric accelerated stochastic ADMM with larger stepsizes

A Multi-Step Inertial Proximal Peaceman-Rachford Splitting Method for Separable Convex Programming

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