Brève communication. Régularisation d'inéquations variationnelles par approximations successives

Martinet, B.

doi:10.1051/m2an/197004r301541

Cited by 589 publications

(600 citation statements)

References 2 publications

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“…Theorem 3.6 shows that the sequence {x k } is bounded. The following result establishes boundedness of {y k }, and hence of {x k } in view of (6), under the condition that y k+1 is chosen to beỹ k+1 at every iteration of the A-HPE framework.…”

Section: Endmentioning

confidence: 95%

“…method, proposed by Martinet [6], and further studied by Rockafellar [16,15], is a classical iterative scheme for solving the MI problem which generates a sequence {x k } according to x k = (λ k T + I) −1 (x k−1 ), It has been used as a generic framework for the design and analysis of several implementable algorithms. The classical inexact version of the proximal point method allows for the presence of a sequence of summable errors in the above iteration, i.e.…”

mentioning

confidence: 99%

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

Monteiro¹,

Svaiter²

2013

SIAM J. Optim.

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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Section: Endmentioning

confidence: 95%

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.

113

164

View full text Add to dashboard Cite

show abstract

“…We see that each step of iterates involves only with A as the forward step and B as the backward step, but not the sum of B. This method includes, in particular, the proximal point algorithm [4,6,14,20,27] and the gradient method [3,13]. Lions-Mercier [16] introduced the following splitting iterative methods in a real Hilbert space:…”

Section: Introductionmentioning

confidence: 99%

The viscosity approximation forward-backward splitting method for solving quasi inclusion problems in Banach spaces

Zhao¹,

Yang²

2017

J. Nonlinear Sci. Appl.

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In this paper, we introduce viscosity approximation forward-backward splitting method for an accretive operator and an m-accretive operator in Banach spaces. The strong convergence of this viscosity method is proved under certain assumptions imposed on the sequence of parameters. Applications to the minimization optimization problem and the linear inverse problem are included. The results presented in the paper extend and improve some recent results announced in the current literature.

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“…4.1 A question of interest in convex optimization concerns the strong convergence of the generalized proximal point method (GPPM for short) which emerged from the works of Martinet [43], [44], Rockafellar [52] and Censor and Zenios [21]. When applied to the consistent problem ( ) described in Subsection 3.1 the GPPM produces iterates according to the rule P { } …”

Section: Regularization Of a Proximal Point Methodsmentioning

confidence: 99%

“…In Section 4 we consider the question whether or under which conditions the generalized proximal point method for optimization which emerged from the works of Martinet [43], [44], Rockafellar [52] and Censor and Zenios [21] can be forced to converge strongly in infinite dimensional Banach spaces. The origin of this question can be traced back to Rockafellar's work [52].…”

mentioning

confidence: 99%

Convergence and Stability of a Regularization Method for Maximal Monotone Inclusions and Its Applications to Convex Optimization

Alber

Butnariu

Kassay

Variational Analysis and Applications

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Abstract:In this paper we study the stability and convergence of a regularization method for solving inclusions f Ax ∈ k , where A is a maximal monotone point-to-set operator from a reflexive smooth Banach space X with the Kadec-Klee property to its dual. We assume that the data A and f involved in the inclusion are given by approximations A and k f converging to A and f, respectively, in the sense of Mosco type topologies. We prove that the sequence 1 ( )f which results from the regularization process converges weakly and, under some conditions, converges strongly to the minimum norm solution of the inclusion f Ax ∈ , provided that the inclusion is consistent.These results lead to a regularization procedure for perturbed convex optimization problems whose objective functions and feasibility sets are given by approximations. In particular, we obtain a strongly convergent version of the generalized proximal point optimization algorithm which is applicable to problems whose feasibility sets are given by Mosco approximations

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Brève communication. Régularisation d'inéquations variationnelles par approximations successives

Cited by 589 publications

References 2 publications

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

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

The viscosity approximation forward-backward splitting method for solving quasi inclusion problems in Banach spaces

Convergence and Stability of a Regularization Method for Maximal Monotone Inclusions and Its Applications to Convex Optimization

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