A Logarithmic-Quadratic Proximal Method for Variational Inequalities

Auslender, Alfred; Teboulle, Marc; Ben-Tiba, Sami

doi:10.1007/978-1-4615-5197-3_3

Cited by 50 publications

(119 citation statements)

References 11 publications

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“…This kind of discrepancies is actually currently used in the convex optimization literature and may be seen a regularized log-proximal method (see for instance Ausslender et al [1]). The resulting optimization algorithm leads to efficient tractable interior point solutions even when the number of constraint is large.…”

Section: Quasi-kullback or Log-proximal Divergencementioning

confidence: 99%

“…In Hjort et al [24], convergence of empirical likelihood is investigated when q is allowed to increase with n. They show that convergence to a χ 2 distribution still holds when q = O(n 1 3 ) as n tends to infinity.…”

Section: Remarkmentioning

confidence: 99%

“…Then we minimize the dual expression of this divergence under the constraints of the model. It can be seen as a quasi-empirical likelihood or a regularized log-proximal empirical likelihood (see for instance Ausslender et al [1], for the use of such divergences in the convex literature). The domain of the corresponding divergence is the whole real line making the algorithmic aspects of the problem much more tractable than for empirical likelihood when the number of constraints is large.…”

Section: Introductionmentioning

confidence: 99%

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Empirical Phi-discrepancies and quasi-empirical likelihood: exponential bounds

2015

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Abstract. We review some recent extensions of the so-called generalized empirical likelihood method, when the Kullback distance is replaced by some general convex divergence. We propose to use, instead of empirical likelihood, some regularized form or quasi-empirical likelihood method, corresponding to a convex combination of Kullback and χ 2 discrepancies. We show that for some adequate choice of the weight in this combination, the corresponding quasi-empirical likelihood is Bartlett-correctable. We also establish some non-asymptotic exponential bounds for the confidence regions obtained by using this method. These bounds are derived via bounds for self-normalized sums in the multivariate case obtained in a previous work by the authors. We also show that this kind of results may be extended to process valued infinite dimensional parameters. In this case some known results about selfnormalized processes may be used to control the behavior of generalized empirical likelihood.

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Section: Quasi-kullback or Log-proximal Divergencementioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Empirical Phi-discrepancies and quasi-empirical likelihood: exponential bounds

2015

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“…Examples of these regularizing functionals are the Bregman distances (see, e.g. [1,8,13,14,20,25]), ϕ-divergences ( [26,5,15,18,19,27,28]) and log-quadratic regularizations ( [3,4]). Being interior point methods, it is a basic assumption that the topological interior of C is nonempty.…”

Section: Introductionmentioning

confidence: 99%

“…So, if C = ∅, it holds C ⊂ int C k and hence a regularizing functional can be associated with the set C k . Denote by d k the regularization functional proposed in [3,4] (associated with the set C k with non-empty interior) and by ∇ 1 d k the derivative of d k with respect to its first argument. The subproblems in [29] find an approximate solution x k ∈ int C k of the inclusion where λ k > 0, ∂ ε f is the ε-subdifferential of f [6].…”

Section: Introductionmentioning

confidence: 99%

An inexact interior point proximal method for the variational inequality problem

Burachik¹,

Lopes²,

Silva³

2009

Comput.appl.math.

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Abstract.We propose an infeasible interior proximal method for solving variational inequality problems with maximal monotone operators and linear constraints. The interior proximal method proposed by Auslender, Teboulle and Ben-Tiba [3] is a proximal method using a distance-like barrier function and it has a global convergence property under mild assumptions. However, this method is applicable only to problems whose feasible region has nonempty interior. The algorithm we propose is applicable to problems whose feasible region may have empty interior. Moreover, a new kind of inexact scheme is used. We present a full convergence analysis for our algorithm.Mathematical subject classification: 90C51, 65K10, 47J20, 49J40, 49J52, 49J53.

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Accelerated derivative‐free method for nonlinear monotone equations with an application

Ibrahim

Abubakar

Adamu

2021

Numerical Linear Algebra App

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In optimization theory, to speed up the convergence of iterative procedures, many mathematicians often use the inertial extrapolation method. In this article, based on the three‐term derivative‐free method for solving monotone nonlinear equations with convex constraints [Calcolo, 2016;53(2):133‐145], we design an inertial algorithm for finding the solutions of nonlinear equation with monotone and Lipschitz continuous operator. The convergence analysis is established under some mild conditions. Furthermore, numerical experiments are implemented to illustrate the behavior of the new algorithm. The numerical results have shown the effectiveness and fast convergence of the proposed inertial algorithm over the existing algorithm. Moreover, as an application, we extend this method to solve the LASSO problem to decode a sparse signal in compressive sensing. Performance comparisons illustrate the effectiveness and competitiveness of the method.

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A Logarithmic-Quadratic Proximal Method for Variational Inequalities

Cited by 50 publications

References 11 publications

Empirical Phi-discrepancies and quasi-empirical likelihood: exponential bounds

Empirical Phi-discrepancies and quasi-empirical likelihood: exponential bounds

An inexact interior point proximal method for the variational inequality problem

Accelerated derivative‐free method for nonlinear monotone equations with an application

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