Interior Proximal Method for Variational Inequalities: Case of Nonparamonotone Operators

For variational inequalities characterizing saddle points of Lagrangians associated with convex programming problems in Hilbert spaces, the convergence of an interior proximal method based on Bregman distance functionals is studied. The convergence results admit a successive approximation of the variational inequality and an inexact treatment of the proximal iterations.An analogous analysis is performed for finite-dimensional complementarity problems with multi-valued monotone operators.     

A New Inexactness Criterion for Approximate Logarithmic-Quadratic Proximal Methods

Recently,a class of logarithmic-quadratic proximal(LQP)methods was intro- duced by Auslender,Teboulle and Ben-Tiba.The inexact versions of these methods solve the sub-problems in each iteration approximately.In this paper,we present a practical inexactness criterion for the inexact version of these methods.     

On Some Properties of Generalized Proximal Point Methods for Variational Inequalities

We discuss here generalized proximal point methods applied to variational inequality problems. These methods differ from the classical point method in that a so-called Bregman distance substitutes for the Euclidean distance and forces the sequence generated by the algorithm to remain in the interior of the feasible region, assumed to be nonempty. We consider here the case in which this region is a polyhedron (which includes linear and nonlinear programming, monotone linear complementarity problems, and also certain nonlinear complementarity problems), and present two alternatives to deal with linear equality constraints. We prove that the sequences generated by any of these alternatives, which in general are different, converge to the same point, namely the solution of the problem which is closest, in the sense of the Bregman distance, to the initial iterate, for a certain class of operators. This class consists essentially of point-to-point and differentiable operators such that their Jacobian matrices are positive semidefinite (not necessarily symmetric) and their kernels are constant in the feasible region and invariant through symmetrization. For these operators, the solution set of the problem is also a polyhedron. Thus, we extend a previous similar result which covered only linear operators with symmetric and positive-semidefinite matrices.     

Enlargement of Monotone Operators with Applications to Variational Inequalities

Given a point-to-set operator T, we introduce the operator T defined as T(x)= {u: u – v, x – y – for all y Rn, v T(y)}. When T is maximal monotone T inherits most properties of the -subdifferential, e.g. it is bounded on bounded sets, T(x) contains the image through T of a sufficiently small ball around x, etc. We prove these and other relevant properties of T, and apply it to generate an inexact proximal point method with generalized distances for variational inequalities, whose subproblems consist of solving problems of the form 0 H(x), while the subproblems of the exact method are of the form 0 H(x). If k is the coefficient used in the kth iteration and the k's are summable, then the sequence generated by the inexact algorithm is still convergent to a solution of the original problem. If the original operator is well behaved enough, then the solution set of each subproblem contains a ball around the exact solution, and so each subproblem can be finitely solved.     

Using the Banach Contraction Principle to Implement the Proximal Point Method for Multivalued Monotone Variational Inequalities

We apply the Banach contraction-mapping fixed-point principle for solving multivalued strongly monotone variational inequalities. Then, we couple this algorithm with the proximal-point method for solving monotone multivalued variational inequalities. We prove the convergence rate of this algorithm and report some computational results.This work was completed during the stay of the second author at the Department of Mathematics, University of Namur, Namur, Belgium, 2003.     

A Logarithmic-Quadratic Proximal Prediction-Correction Method for Structured Monotone Variational Inequalities

Inspired by the Logarithmic-Quadratic Proximal (LQP) method for variational inequalities, we present a prediction-correction method for structured monotone variational inequalities. Each iteration of the new method consists of a prediction and a correction. Both the predictor and the corrector are obtained easily with tiny computational load. In particular, the LQP system that appears in the prediction is approximately solved under significantly relaxed inexactness restriction. Global convergence of the new method is proved under mild assumptions. In addition, we present a self-adaptive version of the new method that leads to easier implementations. Preliminary numerical experiments for traffic equilibrium problems indicate that the new method is effectively applicable in practice. Presented at the 6th International conference on Optimization: Techniques and Applications, Ballarat Australia, December 9–11, 2004. This author was supported by NSFC Grant 10571083, the MOEC grant 20020284027 and Jiangsu NSF grant BK2002075     

Proximal Alternating Directions Method for Structured Variational Inequalities

In the alternating directions method, the relaxation factor by Glowinski is useful in practical computations for structured variational inequalities. This paper points out that the same restriction region of the relaxation factor is also valid in the proximal alternating directions method. The research was supported by the NSFC of China Grant 10571083 and MOEC Grant 20060284001. The author thanks the anonymous referees for valuable suggestions.     

The Proximal Point Method for Nonmonotone Variational Inequalities

We consider an application of the proximal point method to variational inequality problems subject to box constraints, whose cost mappings possess order monotonicity properties instead of the usual monotonicity ones. Usually, convergence results of such methods require the additional boundedness assumption of the solutions set. We suggest another approach to obtaining convergence results for proximal point methods which is based on the assumption that the dual variational inequality is solvable. Then the solutions set may be unbounded. We present classes of economic equilibrium problems which satisfy such assumptions.     

Extended Projection Methods for Monotone Variational Inequalities

In this paper, we prove that each monotone variational inequality is equivalent to a two-mapping variational inequality problem. On the basis of this fact, a new class of iterative methods for the solution of nonlinear monotone variational inequality problems is presented. The global convergence of the proposed methods is established under the monotonicity assumption. The conditions concerning the implementability of the algorithms are also discussed. The proposed methods have a close relationship to the Douglas–Rachford operator splitting method for monotone variational inequalities.     

Proximal Methods for Mixed Variational Inequalities

A proximal point method for solving mixed variational inequalities is suggested and analyzed by using the auxiliary principle technique. It is shown that the convergence of the proposed method requires only the pseudomonotonicity of the operator, which is a weaker condition than monotonicity. As special cases, we obtain various known and new results for solving variational inequalities and related problems. Our proof of convergence is very simple as compared with other methods.