A UNIFIED FRAMEWORK FOR SOME INEXACT PROXIMAL POINT ALGORITHMS
            <sup>*</sup>

Solodov, M. V.; Svaiter, B. F.

doi:10.1081/nfa-100108320

Cited by 129 publications

(133 citation statements)

References 39 publications

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“…-extending convergence results obtained in [1,3,5,56,57,58,36] to richer and more flexible algorithms,…”

Section: Introductionmentioning

confidence: 83%

“…We simply need to verify that the functionf , as defined by (58), satisfies the KL inequality. Let K be a compact neighborhood ofx on which (57) holds.…”

Section: Examplesmentioning

confidence: 99%

“…From a technical viewpoint, our work blends the approach to nonconvex problems provided in [1,15,3,5] with the relative error philosophy developed in [56,57,58,36]. A valuable guideline for the error aspects is the development of an inexact proximal algorithm for equations governed by a monotone operator, and which is based on an estimation of the relative error, see [56,57,58].…”

Section: Introductionmentioning

confidence: 99%

“…A valuable guideline for the error aspects is the development of an inexact proximal algorithm for equations governed by a monotone operator, and which is based on an estimation of the relative error, see [56,57,58]. Related results without monotonicity (with a control on the lack of monotonicity) have been obtained in [36].…”

Section: Introductionmentioning

confidence: 99%

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Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods

2011

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To cite this version:Hedy Attouch, Jérôme Bolte, Benar Svaiter. Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods. Math. Program., Ser. A, 2011, 137 (1), pp.91-124. <10.1007/s10107-011-0484-9>. Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods , 15, 2010; revised: July, 21, 2011 Abstract In view of the minimization of a nonsmooth nonconvex function f , we prove an abstract convergence result for descent methods satisfying a sufficient-decrease assumption, and allowing a relative error tolerance. Our result guarantees the convergence of bounded sequences, under the assumption that the function f satisfies the Kurdyka-Lojasiewicz inequality. This assumption allows to cover a wide range of problems, including nonsmooth semi-algebraic (or more generally tame) minimization. The specialization of our result to different kinds of structured problems provides several new convergence results for inexact versions of the gradient method, the proximal method, the forward-backward splitting algorithm, the gradient projection and some proximal regularization of the Gauss-Seidel method in a nonconvex setting. Our results are illustrated through feasibility problems, or iterative thresholding procedures for compressive sensing.2010 Mathematics Subject Classification: 34G25, 47J25, 47J30, 47J35, 49M15, 49M37, 65K15, 90C25, 90C53.

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“…-extending convergence results obtained in [1,3,5,56,57,58,36] to richer and more flexible algorithms,…”

Section: Introductionmentioning

confidence: 83%

“…We simply need to verify that the functionf , as defined by (58), satisfies the KL inequality. Let K be a compact neighborhood ofx on which (57) holds.…”

Section: Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods

2011

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“…New inexact versions of the proximal point method with relative error tolerance were proposed by Solodov and Svaiter [18,19,20,21]. Iteration complexity results for one of these inexact versions of the proximal point method introduced in [18,19], namely the hybrid proximal extragradient (HPE) method, were established in [10].…”

mentioning

confidence: 99%

An Accelerated Hybrid Proximal Extragradient Method for Convex Optimization and Its Implications to Second-Order Methods

Monteiro¹,

Svaiter²

2013

SIAM J. Optim.

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113

164

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This paper presents an accelerated variant of the hybrid proximal extragradient (HPE) method for convex optimization, referred to as the accelerated HPE (A-HPE) framework. Iterationcomplexity results are established for the A-HPE framework, as well as a special version of it, where a large stepsize condition is imposed. Two specific implementations of the A-HPE framework are described in the context of a structured convex optimization problem whose objective function consists of the sum of a smooth convex function and an extended real-valued non-smooth convex function. In the first implementation, a generalization of a variant of Nesterov's method is obtained for the case where the smooth component of the objective function has Lipschitz continuous gradient. In the second implementation, an accelerated Newton proximal extragradient (A-NPE) method is obtained for the case where the smooth component of the objective function has Lipschitz continuous Hessian. It is shown that the A-NPE method has a O(1/k 7/2 ) convergence rate, which improves upon the O(1/k 3 ) convergence rate bound for another accelerated Newton-type method presented by Nesterov. Finally, while Nesterov's method is based on exact solutions of subproblems with cubic regularization terms, the A-NPE method is based on inexact solutions of subproblems with quadratic regularization terms, and hence is potentially more tractable from a computational point of view.

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A Zero-Sum Game with Two Players

Zaslavski¹

2020

Convex Optimization With Computational Errors

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A UNIFIED FRAMEWORK FOR SOME INEXACT PROXIMAL POINT ALGORITHMS ^*

Cited by 129 publications

References 39 publications

Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods

Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods

An Accelerated Hybrid Proximal Extragradient Method for Convex Optimization and Its Implications to Second-Order Methods

A Zero-Sum Game with Two Players

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A UNIFIED FRAMEWORK FOR SOME INEXACT PROXIMAL POINT ALGORITHMS *

Cited by 129 publications

References 39 publications

Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods

Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods

An Accelerated Hybrid Proximal Extragradient Method for Convex Optimization and Its Implications to Second-Order Methods

A Zero-Sum Game with Two Players

Contact Info

Product

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A UNIFIED FRAMEWORK FOR SOME INEXACT PROXIMAL POINT ALGORITHMS ^*