Journal ArticleDOI
A Generalized Proximal Point Algorithm for the Variational Inequality Problem in a Hilbert Space
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It is proved that the sequence converges (weakly) if and only if the problem has solutions, in which case the weak limit is a solution, and if the solution does not have solutions, then the sequence is unbounded.Abstract:
We consider a generalized proximal point method for solving variational inequality problems with monotone operators in a Hilbert space. It differs from the classical proximal point method (as discussed by Rockafellar for the problem of finding zeroes of monotone operators) in the use of generalized distances, called Bregman distances, instead of the Euclidean one. These distances play not only a regularization role but also a penalization one, forcing the sequence generated by the method to remain in the interior of the feasible set so that the method becomes an interior point one. Under appropriate assumptions on the Bregman distance and the monotone operator we prove that the sequence converges (weakly) if and only if the problem has solutions, in which case the weak limit is a solution. If the problem does not have solutions, then the sequence is unbounded. We extend similar previous results for the proximal point method with Bregman distances which dealt only with the finite dimensional case and which applied only to convex optimization problems or to finding zeroes of monotone operators, which are particular cases of variational inequality problems.read more
Citations
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Journal ArticleDOI
Bregman Monotone Optimization Algorithms
TL;DR: A systematic investigation of the notion of Bregman monotonicity leads to a simplified analysis of numerous algorithms and to the development of a new class of parallel block-iterative surrogate BRegman projection schemes.
Journal ArticleDOI
Enlargement of Monotone Operators with Applications to Variational Inequalities
TL;DR: In this paper, a point-to-set operator Te defined as Te(x) is introduced, which inherits most properties of the e-subdifferential, e.g., it is bounded on bounded sets, it contains the image through T of a sufficiently small ball around x, etc., and apply it to generate an inexact proximal point method with generalized distances for variational inequalities.
Journal ArticleDOI
An Inexact Hybrid Generalized Proximal Point Algorithm and Some New Results on the Theory of Bregman Functions
TL;DR: A new Bregman-function-based algorithm which is a modification of the generalized proximal point method for solving the variational inequality problem with a maximal monotone operator and eliminates the assumption of pseudomonotonicity, which was standard in proving convergence for paramonotone operators.
Journal ArticleDOI
Approximate iterations in Bregman-function-based proximal algorithms
TL;DR: This paper establishes convergence of generalized Bregman-function-based proximal point algorithms when the iterates are computed only approximately, and the accuracy conditions on the iterate resemble those required for the classical “linear” proximal Point algorithm, but are slightly stronger.
Journal ArticleDOI
A Logarithmic-Quadratic Proximal Method for Variational Inequalities
TL;DR: This work allows for computing the iterates approximately and proves that the resulting method is globally convergent under the sole assumption that the optimal set of the variational inequality is nonempty.
References
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Book
An introduction to variational inequalities and their applications
TL;DR: In this paper, the SIAM edition Preface Glossary of notations Introduction Part I. Variational Inequalities in Rn Part II. Free Boundary Problems Governed by Elliptic Equations and Systems Part VII. A One Phase Stefan Problem Bibliography Index.
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Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert
TL;DR: In this article, Operateurs Maximaux Monotones: Et Semi-Groupes De Contractions Dans Les Espaces De Hllbert are described and discussed. But the focus is not on the performance of the operators.
Journal ArticleDOI
Monotone Operators and the Proximal Point Algorithm
TL;DR: In this paper, the proximal point algorithm in exact form is investigated in a more general form where the requirement for exact minimization at each iteration is weakened, and the subdifferential $\partial f$ is replaced by an arbitrary maximal monotone operator T.
Journal ArticleDOI
The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming
TL;DR: This method can be regarded as a generalization of the methods discussed in [1–4] and applied to the approximate solution of problems in linear and convex programming.
Journal ArticleDOI
Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications
Patrick T. Harker,Jong-Shi Pang +1 more
TL;DR: The field of finite-dimensional variational inequality and complementarity problems has seen a rapid development in its theory of existence, uniqueness and sensitivity of solution(s), in the theory of algorithms, and in the application of these techniques to transportation planning, regional science, socio-economic analysis, energy modeling, and game theory as mentioned in this paper.