Fixing and extending some recent results on the ADMM algorithm

Banert, Sebastian; Boţ, Radu Ioan; Csetnek, Ernö Robert

doi:10.48550/arxiv.1612.05057

Cited by 11 publications

(29 citation statements)

References 13 publications

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“…Let us show now that an appropriate choice of M 2 leads (both in continuous and discrete case) to an implementable proximal step in the second inclusion. This is crucial for numerical results in applications, see also [12] and [10]. For every t ∈ [0, +∞), we define…”

Section: Solution Concept Discretizations Examplementioning

confidence: 99%

A Dynamical Approach to Two-Block Separable Convex Optimization Problems with Linear Constraints

Bitterlich

Csetnek

Wanka

2021

Numerical Functional Analysis and Optimization

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The aim of this manuscript is to approach by means of first order differential equations/inclusions convex programming problems with two-block separable linear constraints and objectives, whereby (at least) one of the components of the latter is assumed to be strongly convex. Each block of the objective contains a further smooth convex function. We investigate the dynamical system proposed and prove that its trajectories converge weakly to a saddle point of the Lagrangian of the convex optimization problem. The dynamical system provides through time discretization the alternating minimization algorithm AMA and also its proximal variant recently introduced in the literature.

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Section: Solution Concept Discretizations Examplementioning

confidence: 99%

A Dynamical Approach to Two-Block Separable Convex Optimization Problems with Linear Constraints

Bitterlich

Csetnek

Wanka

2021

Numerical Functional Analysis and Optimization

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“…VMSP-ADMM is studied in [14] where the T k is assumed to be positive definite. The convergence and complexity results have been studied in [1,12,18]. Moreover it is also closely related to the inexact ADMM, where the subproblems in (1.4) or (1.5) to be solved approximately with certain implementable criteria [3,5,7,8,14,22].…”

Section: Introductionmentioning

confidence: 99%

An Alternating Direction Method of Multipliers with the BFGS Update for Structured Convex Quadratic Optimization

Yamashita

2019

Preprint

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The alternating direction method of multipliers (ADMM) is an effective method for solving wide fields of convex problems. At each iteration, the classical ADMM solves two subproblems exactly. However, in many applications, it is expensive or impossible to obtain the exact solutions of the subproblems. To overcome the difficulty, some proximal terms are added to the subproblems. This class of methods normally solves the original subproblem approximately, and thus takes more iterations. This fact urges us to consider that a special proximal term can lead to a better result as the classical ADMM. In this paper, we propose a proximal ADMM whose regularized matrix in the proximal term is generated by the BFGS update (or limited memory BFGS) at every iteration. These types of matrices use second-order information of the objective function. The convergence of the proposed method is proved under certain assumptions. Numerical results are presented to show the effectiveness of the proposed proximal ADMM.

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“…The resulting ADMM is a variable metric proximal ADMM, which is also closely related to the inexact ADMM [6,13,15,16,27,44]. The convergences of such methods have been studied in [2,23,36] but a better selection of the sequence {T k } has not been provided.…”

Section: Introductionmentioning

confidence: 99%

Alternating Direction Method of Multipliers with Variable Metric Indefinite Proximal Terms for Convex Optimization

Yamashita

2019

Preprint

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This paper studies a proximal alternating direction method of multipliers (ADMM) with variable metric indefinite proximal terms for linearly constrained convex optimization problems. The proximal ADMM plays an important role in many application areas, since the subproblems of the method are easy to solve. Recently, it is reported that the proximal ADMM with a certain fixed indefinite proximal term is faster than that with a positive semidefinite term, and still has the global convergence property. On the other hand, Gu and Yamashita studied a variable metric semidefinite proximal ADMM whose proximal term is generated by the BFGS update. They reported that a slightly indefinite matrix also makes the algorithm work well in their numerical experiments. Motivated by this fact, we consider a variable metric indefinite proximal ADMM, and give sufficient conditions on the proximal terms for the global convergence. Moreover, we propose a new indefinite proximal term based on the BFGS update which can satisfy the conditions for the global convergence.

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Fixing and extending some recent results on the ADMM algorithm

Cited by 11 publications

References 13 publications

A Dynamical Approach to Two-Block Separable Convex Optimization Problems with Linear Constraints

A Dynamical Approach to Two-Block Separable Convex Optimization Problems with Linear Constraints

An Alternating Direction Method of Multipliers with the BFGS Update for Structured Convex Quadratic Optimization

Alternating Direction Method of Multipliers with Variable Metric Indefinite Proximal Terms for Convex Optimization

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