Block-Wise Alternating Direction Method of Multipliers with Gaussian Back Substitution for Multiple-Block Convex Programming

Fu, Xinghe; He, Bingsheng; Wang, Xiangfeng; Yuan, Xiaoming

doi:10.1007/978-3-030-25939-6_8

Cited by 5 publications

(9 citation statements)

References 21 publications

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“…In this section, we give the worst-case convergence rate results measured by the iteration complexity of PPRSM-S (Algorithm 1) with an accuracy of O(1/t) in both ergodic and nonergodic senses. For details of similar results, please refer to Fu et al (2014) and He et al (2014).…”

Section: Convergence Ratementioning

confidence: 83%

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An indefinite proximal Peaceman–Rachford splitting method with substitution procedure for convex programming

Deng

Liu

2019

Comp. Appl. Math.

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The strictly contractive Peaceman-Rachford splitting method (SCPRSM) has received a tremendous amount of attention for solving linearly constrained separable convex optimization problems. In this paper, we propose an indefinite proximal SCPRSM with substitution procedure (abbreviated as PPRSM-S) to improve numerical results. The prediction step takes a proximal SCPRSM cycle to update the variable blocks, then the correction step corrects the output slightly by computing a combination of the prediction step and the previous iteration. We derive the global convergence of the proposed method and analyze the convergence rate results under much mild conditions. Some experimental results on LASSO and total variation-based denoising problems demonstrate the efficiency of the substitution step and the indefinite proximal term.

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Section: Convergence Ratementioning

confidence: 83%

“…In this section, we establish global convergence of PPRSM-S (Algorithm 1) with aid of the variational inequalities framework proposed by Fu et al (2014) and He and Yuan (2015).…”

Section: Convergence Analysismentioning

confidence: 99%

An indefinite proximal Peaceman–Rachford splitting method with substitution procedure for convex programming

Deng

Liu

2019

Comp. Appl. Math.

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“…where x i ∈ R ni , θ i is convex but may be non-differentiable. Such a model finds various applications in signal/image processing, machine learning [1]- [6]. For example, in oneblock case, the model contains the Basis Pursuit model [7], augmented Basis Pursuit model [8] and Matrix Completion [9].…”

Section: Introductionmentioning

confidence: 99%

Linear Convergence Rate of Splitting Algorithms for Multi-Block Constrained Convex Minimizations

Deng¹,

Liu

Huang

2020

IEEE Access

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Multi-block linear constrained separable convex minimizations are ubiquitous and have been drawing increasing attention in recent researches. The alternating direction method of multipliers (ADMM) has been well studied and used in the literature for the two-block case. However, the direct extension of the ADMM to the multi-block case is not necessarily convergent. ADMM with Gaussian Back Substitution and ADMM with Prox-Parallel Splitting are two useful schemes to deal with the multi-block situation. Nevertheless, only a sublinear convergence rate was given in previous studies. In this paper, we prove the linear convergence rate of these two schemes under some assumptions. The proofs mainly depend on the variational inequalities. INDEX TERMS Multi-block alternating direction method of multipliers, linear convergence rate, variational inequalities, splitting algorithms.

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“…Its particular case is the Dykstra-like algorithm [6] when the problem reduces to evaluating the proximity operator of a sum of two convex functions. It is also worth mentioning the work in [7] which proposes a parallel splitting version of the Alternating Direction Method of Multipliers [8]. An appealing idea in the context of optimization is to adopt a block coordinate strategy [9], in such a way that two successive iterations deal with different blocks of variables.…”

Section: Introductionmentioning

confidence: 99%

Dual Block-Coordinate Forward-Backward Algorithm with Application to Deconvolution and Deinterlacing of Video Sequences

et al. 2017

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Optimization methods play a central role in the solution of a wide array of problems encountered in various application fields, such as signal and image processing. Especially when the problems are highly dimensional, proximal methods have shown their efficiency through their capability to deal with composite, possibly non smooth objective functions. The cornerstone of these approaches is the proximity operator, which has become a quite popular tool in optimization. In this work, we propose new dual forward-backward formulations for computing the proximity operator of a sum of convex functions involving linear operators. The proposed algorithms are accelerated thanks to the introduction of a block coordinate strategy combined with a preconditioning technique. Numerical simulations emphasize the good performance of our approach for the problem of jointly deconvoluting and deinterlacing video sequences.

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Block-Wise Alternating Direction Method of Multipliers with Gaussian Back Substitution for Multiple-Block Convex Programming

Cited by 5 publications

References 21 publications

An indefinite proximal Peaceman–Rachford splitting method with substitution procedure for convex programming

An indefinite proximal Peaceman–Rachford splitting method with substitution procedure for convex programming

Linear Convergence Rate of Splitting Algorithms for Multi-Block Constrained Convex Minimizations

Dual Block-Coordinate Forward-Backward Algorithm with Application to Deconvolution and Deinterlacing of Video Sequences

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