A generalized alternating direction method of multipliers with semi-proximal terms for convex composite conic programming

Transform Invariant Low-Rank Textures, referred to as TILT, can accurately and robustly extract textural or geometric information in a 3D from user-specified windows in 2D in spite of significant corruptions and warping. It was discovered that the task can be characterized, both theoretically and numerically, by solving a sequence of matrix nuclear-norm and 1-norm involved convex minimization problems. For solving this problem, the direct extension of Alternating Direction Method of Multipliers (ADMM) in an usual Gauss-Seidel manner often performs numerically well in practice but there is no theoretical guarantee on its convergence. In this paper, we resolve this dilemma by using the novel symmetric Gauss-Seidel (sGS) based ADMM developed by Li, Sun & Toh (Math. Prog. 2016). The sGS-ADMM is guaranteed to converge and we shall demonstrate in this paper that it is also practically efficient than the directly extended ADMM. When the sGS technique is applied to this particular problem, we show that only one variable needs to be re-updated, and this updating hardly imposes any excessive computational cost. The sGS decomposition theorem of Li, Sun & Toh (arXiv: 1703.06629) establishes the equivalent between sGS-ADMM and the classical ADMM with an additional semiproximal term, so the convergence result is followed directly. Extensive experiments illustrate that the sGS-ADMM and its generalized variant have superior numerical efficiency over the directly extended ADMM.

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“…, (3.9).Therefore, according to Theorem B.1 in [9] and Theorem 5.1 in [23], the convergence result of Algorithm sGS-GADMM G can be listed.…”

Section: Algorithm: (Sgs-admm G)mentioning

confidence: 92%

“…Next, we quickly review another type of ADMM. In order to broadening the capability of the semi-proximal ADMM (2.6) at the special case ξ = 1, Xiao, Chen & Li [23] introduced the following generalized semi-proximal ADMM with initial…”

Section: B Classical and Generalized Semi-proximal Admmmentioning

confidence: 99%

Symmetric Gauss–Seidel Technique-Based Alternating Direction Methods of Multipliers for Transform Invariant Low-Rank Textures Problem

Ding

Xiao

2018

J Math Imaging Vis

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“…In [39], two conditions for f K   and g T I    are needed. So, the following two basic equalities are given first.…”

Section: Appendix Amentioning

confidence: 99%

“…The sparsity embodies 0 l -regularization constraint and can be relaxed to 1 l -regularization. Commonly used methods for solving 1 l -regularization problems are least angle regression (LARS) [37] and the alternating direction method of multipliers (ADMM) [38,39]. For biostatistics, ADMM has attracted a great deal of attention because it mainly deals with convex optimization problems with constraints.…”

Section: Introductionmentioning

confidence: 99%

An integrated inverse space sparse representation framework for tumor classification

Yang

Chen

et al. 2019

Pattern Recognition

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Microarray gene expression data-based tumor classification is an active and challenging issue. In this paper, an integrated tumor classification framework is presented, which aims to exploit information in existing available samples, and focuses on the small sample problem and unbalanced classification problem.Firstly, an inverse space sparse representation based classification (ISSRC) model is proposed by considering the characteristics of gene-based tumor data, such as sparsity and a small number of training samples. A decision information factors (DIF)-based gene selection method is constructed to enhance the representation ability of the ISSRC. It is worth noting that the DIF is established from reducing clinical misdiagnosis rate and dimension of small sample data. For further improving the representation ability and classification stability of the ISSRC, feature learning is conducted on the selected gene subset. The feature learning method is constructed by complementing the advantages of non-negative matrix factorization (NMF) and deep learning.Without confusion, the ISSRC combined with gene selection and feature learning is called the integrated ISSRC, whose stability, optimization and the corresponding convergence are analyzed. Extensive experiments on six public microarray gene expression datasets show the integrated ISSRC-based tumor classification framework is superior to classical and state-of-the-art methods. There are significant improvements in classification accuracy, specificity and sensitivity, whether there is a tumor in the early diagnosis, what kind of tumor, or whether metastasis occurs after tumor surgery.

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“…Another contribution of Eckstein and Bertsekas [ 4 ] is the designing of a generalized ADMM based on a generalized proximal point algorithm. Very recently, combining the idea of semi-proximal terms, Xiao et al [ 11 ] proposed a semi-proximal generalized ADMM for convex composite conic programming, and numerically illustrated that their proposed method is very promising for solving doubly nonnegative semi-positive definite programming. The method of Xiao et al [ 11 ] relaxed all the variables with a factor of , which has the potential of enhancing the performance of the classic ADMM.…”

Section: Introductionmentioning

confidence: 99%

Convergence analysis on a modified generalized alternating direction method of multipliers

Wei

2018

J Inequal Appl

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The alternating direction method of multipliers (ADMM) is one of the most powerful and successful methods for solving convex composite minimization problem. The generalized ADMM relaxes both the variables and the multipliers with a common relaxation factor in , which has the potential of enhancing the performance of the classic ADMM. Very recently, two different variants of semi-proximal generalized ADMM have been proposed. They allow the weighting matrix in the proximal terms to be positive semidefinite, which makes the subproblems relatively easy to evaluate. One of the variants of semi-proximal generalized ADMMs has been analyzed theoretically, but the convergence result of the other is not known so far. This paper aims to remedy this deficiency and establish its convergence result under some mild conditions in the sense that the relaxation factor is also restricted into .

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A generalized alternating direction method of multipliers with semi-proximal terms for convex composite conic programming

Cited by 28 publications

References 28 publications

Symmetric Gauss–Seidel Technique-Based Alternating Direction Methods of Multipliers for Transform Invariant Low-Rank Textures Problem

Symmetric Gauss–Seidel Technique-Based Alternating Direction Methods of Multipliers for Transform Invariant Low-Rank Textures Problem

An integrated inverse space sparse representation framework for tumor classification

Convergence analysis on a modified generalized alternating direction method of multipliers

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