Pointwise and Ergodic Convergence Rates of a Variable Metric Proximal Alternating Direction Method of Multipliers

Gonçalves, Max L. N.; Alves, M. Marques; Melo, Jefferson G.

doi:10.1007/s10957-018-1232-6

Cited by 17 publications

(13 citation statements)

References 38 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Therefore, we only compare Algorithm 2 with other algorithms in the following experiments. In experiment 2, we compare the performance of the proposed Algorithm 2 with other state-of-the-art methods of the proximal alternating direction of the multiplier method (PADMM) [4,30,31,40] and the primal-dual Chambolle-Pock algorithm (PDCP) [19] (also known as the primal-dual hybrid gradient algorithm [20]). Note that the convex minimization problem (74) can be rewritten as follows,…”

Section: Numerical Resultsmentioning

confidence: 99%

Preconditioned Douglas-Rachford type primal-dual method for solving composite monotone inclusion problems with applications

Yang¹,

Tang²,

Wen³

et al. 2021

IPI

View full text Add to dashboard Cite

This paper is concerned with the monotone inclusion involving the sum of a finite number of maximally monotone operators and the parallel sum of two maximally monotone operators with bounded linear operators. To solve this monotone inclusion, we first transform it into the formulation of the sum of three maximally monotone operators in a proper product space. Then we derive two efficient iterative algorithms, which combine the partial inverse method with the preconditioned Douglas-Rachford splitting algorithm and the preconditioned proximal point algorithm. Furthermore, we develop an iterative algorithm, which relies on the preconditioned Douglas-Rachford splitting algorithm without using the partial inverse method. We carefully analyze the theoretical convergence of the proposed algorithms. Finally, in order to demonstrate the effectiveness and efficiency of these algorithms, we conduct numerical experiments on a novel image denoising model for salt-andpepper noise removal. Numerical results show the good performance of the proposed algorithms.

show abstract

Section: Numerical Resultsmentioning

confidence: 99%

Preconditioned Douglas-Rachford type primal-dual method for solving composite monotone inclusion problems with applications

Yang¹,

Tang²,

Wen³

et al. 2021

IPI

View full text Add to dashboard Cite

show abstract

“…which in turn, combined with (24), gives (22). The relation (23) follows immediately from (16). Now, the last statement of the lemma follows directly by (21)-(23) and definitions of T and M given in (13) and (17), respectively.…”

Section: Let Us First Introduce the Elements Required By The Setting mentioning

confidence: 81%

“…It is worth mentioning that the previous ergodic convergence results for the G-ADMM are not focused in solving (3) approximately in the sense of our paper. Iteration-complexity study of the standard ADMM and some variants in the setting of the HPE framework have been considered in [16,18,25]. Finally, convergence rates of ADMM variants using a different approach have been studied in [7,8,18,20,21,22,23,26,27], to name just a few.…”

Section: Introductionmentioning

confidence: 99%

Iteration-complexity analysis of a generalized alternating direction method of multipliers

2018

View full text Add to dashboard Cite

This paper analyzes the iteration-complexity of a generalized alternating direction method of multipliers (G-ADMM) for solving linearly constrained convex problems. This ADMM variant, which was first proposed by Bertsekas and Eckstein, introduces a relaxation parameter α ∈ (0, 2) into the second ADMM subproblem. Our approach is to show that the G-ADMM is an instance of a hybrid proximal extragradient framework with some special properties, and, as a by product, we obtain ergodic iteration-complexity for the G-ADMM with α ∈ (0, 2], improving and complementing related results in the literature. Additionally, we also present pointwise iteration-complexity for the G-ADMM.2000 Mathematics Subject Classification: 47H05, 49M27, 90C25, 90C60, 65K10.

show abstract

“…The resulting ADMM is a variable metric proximal ADMM, which is also closely related to the inexact ADMM [6]. The convergences of such methods have been studied in [12,24] but a better selection of the sequence {T k } has not been provided.…”

mentioning

confidence: 99%

Alternating direction method of multipliers with variable metric indefinite proximal terms for convex optimization

Gu¹,

Yamashita²

2020

Numerical Algebra, Control &Amp; Optimization

View full text Add to dashboard Cite

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 AD-MM 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, based on the BFGS update, we propose a new indefinite proximal term which can satisfy the conditions for the global convergence. Experiments on several datasets demonstrated that our proposed variable metric indefinite proximal ADMM outperforms most of the comparison proximal ADMMs.

show abstract

Pointwise and Ergodic Convergence Rates of a Variable Metric Proximal Alternating Direction Method of Multipliers

Cited by 17 publications

References 38 publications

Preconditioned Douglas-Rachford type primal-dual method for solving composite monotone inclusion problems with applications

Preconditioned Douglas-Rachford type primal-dual method for solving composite monotone inclusion problems with applications

Iteration-complexity analysis of a generalized alternating direction method of multipliers

Alternating direction method of multipliers with variable metric indefinite proximal terms for convex optimization

Contact Info

Product

Resources

About