Some Saddle-function splitting methods for convex programming

Eckstein, Jonathan

doi:10.1080/10556789408805578

Cited by 130 publications

(95 citation statements)

References 14 publications

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“…It was shown by Eckstein and Bertsekas [15] that the ADM, as a special case of Douglas-Rachford splitting [20], is actually an application of the proximal point algorithm on the dual problem by means of a specially-constructed splitting operator. Based on the same argument by further applying a change of variable to the operators, Eckstein [14] presented the first proximal ADM as in (8.2) with S = µ 1 I and T = µ 2 I for positive constants µ 1 > 0 and µ 2 > 0. Later, He et al [25] further extended the idea of Eckstein [14] to monotone variational inequalities to allow λ, S, and T to be replaced by different parameters λ k , S k , and T k in each iteration.…”

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confidence: 99%

“…Based on the same argument by further applying a change of variable to the operators, Eckstein [14] presented the first proximal ADM as in (8.2) with S = µ 1 I and T = µ 2 I for positive constants µ 1 > 0 and µ 2 > 0. Later, He et al [25] further extended the idea of Eckstein [14] to monotone variational inequalities to allow λ, S, and T to be replaced by different parameters λ k , S k , and T k in each iteration. The convergence results provided in [14] and [25] for the proximal ADM both need S and T to be positive definite, which limits the applications of the method.…”

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confidence: 99%

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Hankel Matrix Rank Minimization with Applications to System Identification and Realization

Fazel

Pong

Sun

et al. 2013

SIAM J. Matrix Anal. & Appl.

485

494

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In honor of Professor Paul Tseng, who went missing while on a kayak trip on the Jinsha river, China, on August 13, 2009, for his contributions to the theory and algorithms for large-scale optimization.Abstract. In this paper, we introduce a flexible optimization framework for nuclear norm minimization of matrices with linear structure, including Hankel, Toeplitz and moment structures, and catalog applications from diverse fields under this framework. We discuss various first-order methods for solving the resulting optimization problem, including alternating direction methods, proximal point algorithm and gradient projection methods. We perform computational experiments to compare these methods on system identification problem and system realization problem. For the system identification problem, the gradient projection method (accelerated by Nesterov's extrapolation techniques) usually outperforms other first-order methods in terms of CPU time on both real and simulated data; while for the system realization problem, the alternating direction method, as applied to a certain primal reformulation, usually outperforms other first-order methods in terms of CPU time.

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confidence: 99%

mentioning

confidence: 99%

Hankel Matrix Rank Minimization with Applications to System Identification and Realization

Fazel

Pong

Sun

et al. 2013

SIAM J. Matrix Anal. & Appl.

485

494

View full text Add to dashboard Cite

show abstract

“…The corresponding extension of the (FB-M) algorithm was proposed by Chen and Teboulle [17] and a similar idea was used by Eckstein [31] to extend (ADMM) (see too [84,3]). A recent survey by Shefi and Teboulle [75] recalled the complexity issues to improve global convergence ratios, in particular for regularized versions of the splitting schemes.…”

Section: Algorithmic Enhancementsmentioning

confidence: 99%

A survey on operator splitting and decomposition of convex programs

Lenoir

Mahey

2016

RAIRO-Oper. Res.

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To cite this version:Philippe Mahey, Arnaud Lenoir. A survey on operator splitting and decomposition of convex programs. RAIRO -Operations Research, EDP Sciences, 2017, 51 (1) Abstract Many structured convex minimization problems can be modeled by the search of a zero of the sum of two monotone operators. Operator splitting methods have been designed to decompose and regularize at the same time these kind of models. We review here these models and the classical splitting methods. We focus on the numerical sensitivity of these algorithms with respect to the scaling parameters that drive the regularizing terms, in order to accelerate convergence rates for different classes of models.

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“…Many decomposition techniques (for monotone problems) are explicitly derived from the proximal point method [29,32] for maximal monotone operators, e.g., [10,41,42,44]. Sometimes the relation to the proximal iterates is less direct, e.g., the methods in [4,11,17,31,43], which were nevertheless more recently generalized and interpreted in [37,27] within the hybrid inexact proximal schemes of [39,30]. As some other decomposition methods, we might mention [22] which employs projection and cutting-plane techniques for certain structured problems, matrix splitting for complementarity problems in [6], and the applications of the latter to stochastic complementarity problems in [35].…”

Section: (1)mentioning

confidence: 99%

A class of Dantzig–Wolfe type decomposition methods for variational inequality problems

2012

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We consider a class of decomposition methods for variational inequalities, which is related to the classical Dantzig-Wolfe decomposition of linear programs. Our approach is rather general, in that it can be used with certain types of set-valued or nonmonotone operators, as well as with various kinds of approximations in the subproblems of the functions and derivatives in the single-valued case. Also, subproblems may be solved approximately. Convergence is established under reasonable assumptions. We also report numerical experiments for computing variational equilibria of the game-theoretic models of electricity markets. Our numerical results illustrate that the decomposition approach allows to solve large-scale problem instances otherwise intractable if the widely used PATH solver is applied directly, without decomposition.

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Some Saddle-function splitting methods for convex programming

Cited by 130 publications

References 14 publications

Hankel Matrix Rank Minimization with Applications to System Identification and Realization

Hankel Matrix Rank Minimization with Applications to System Identification and Realization

A survey on operator splitting and decomposition of convex programs

A class of Dantzig–Wolfe type decomposition methods for variational inequality problems

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