A Logarithmic-Quadratic Proximal Method for Variational Inequalities

We present a new method for solving variational inequalities on polyhedra. The method is proximal based, but uses a very special logarithmic-quadratic proximal term which replaces the usual quadratic, and leads to an interior proximal type algorithm. We allow for computing the iterates approximately and prove that the resulting method is globally convergent under the sole assumption that the optimal set of the variational inequality is nonempty.     

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.     

Hilbert空间中的广义非线性变分不等式组的投影算法

利用投影技术讨论了Hilbert空间中一类含松弛伪上强制映射的广义非线性变分不等式组的逼近解及其收敛性,所得到结果推广和统一了系列最新结果.     

Proximal Point Algorithms for General Variational Inequalities

This paper presents a unified framework of proximal point algorithms (PPAs) for solving general variational inequalities (GVIs). Some existing PPAs for classical variational inequalities, including both the exact and inexact versions, are extended to solving GVIs. Consequently, several new PPA-based algorithms are proposed. M. Li was supported by NSFC Grant 10571083 and SRFDP Grant 200802861031. L.Z. Liao was supported in part by grants from Hong Kong Baptist University and the Research Grant Council of Hong Kong. X.M. Yuan was supported in part by FRG/08-09/II-40 from Hong Kong Baptist University and NSFC Grant 10701055.     

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.     

On a Generalized System for Relaxed Cocoercive Variational Inequalities and Projection Methods

Verma introduced a system of nonlinear variational inequalities and proposed projection methods to solve it. This system reduces to a variational inequality problem under certain conditions. So, at least in form, it can be regarded as a extension of a variational inequality problem. In this note, we show that solving this system coincides exactly with solving a variational inequality problem. Therefore, we conclude that it suffices to study the corresponding variational inequalities.This work was supported by the National Natural Science Foundation of China, Grant 10571134.Communicated by M. J. Balas     

Pseudomonotone Variational Inequalities: Convergence of Proximal Methods

In this paper, we study the convergence of proximal methods for solving pseudomonotone (in the sense of Karamardian) variational inequalities. The main result is given in the finite-dimensional case, but we show that we still obtain convergence in an infinite-dimensional Hilbert space under a strong pseudomonotonicity or a pseudo-Dunn assumption on the operator involved in the variational inequality problem.     

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.     

Weighted Variational Inequalities

In this paper, we introduce weighted variational inequalities over product of sets and system of weighted variational inequalities. It is noted that the weighted variational inequality problem over product of sets and the problem of system of weighted variational inequalities are equivalent. We give a relationship between system of weighted variational inequalities and systems of vector variational inequalities. We define several kinds of weighted monotonicities and establish several existence results for the solution of the above-mentioned problems under these weighted monotonicities. We introduce also the weighted generalized variational inequalities over product of sets, that is, weighted variational inequalities for multivalued maps and systems of weighted generalized variational inequalities. Extensions of weighted monotonicities for multivalued maps are also considered. The existence of a solution of weighted generalized variational inequalities over product of sets is also studied. The existence results for a solution of weighted generalized variational inequality problem give also the existence of solutions of systems of generalized vector variational inequalities. The first and third author express their thanks to the Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia for providing excellent research facilities. The authors are also grateful to the referees for comments and suggestions improving the final draft of this paper.     

Generalized Vector Variational Inequalities

In this paper, we introduce a generalized vector variational inequality problem (GVVIP) which extends and unifies vector variational inequalities as well as classical variational inequalities in the literature. The concepts of generalized C-pseudomonotone and generalized hemicontinuous operators are introduced. Some existence results for GVVIP are obtained with the assumptions of generalized C-pseudomonotonicity and generalized hemicontinuity. These results appear to be new and interesting. New existence results of the classical variational inequality are also obtained.     

Generalized System for Relaxed Cocoercive Variational Inequalities and Projection Methods

Let K be a nonempty closed convex subset of a real Hilbert space H. The approximate solvability of a system of nonlinear variational inequality problems, based on the convergence of projection methods, is discussed as follows: find an element (x*, y*)K×K such that

Improvements of Some Projection Methods for Monotone Nonlinear Variational Inequalities

In this paper, we study the relationship of some projection-type methods for monotone nonlinear variational inequalities and investigate some improvements. If we refer to the Goldstein–Levitin–Polyak projection method as the explicit method, then the proximal point method is the corresponding implicit method. Consequently, the Korpelevich extragradient method can be viewed as a prediction-correction method, which uses the explicit method in the prediction step and the implicit method in the correction step. Based on the analysis in this paper, we propose a modified prediction-correction method by using better prediction and correction stepsizes. Preliminary numerical experiments indicate that the improvements are significant.     

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.     

Local Convergence Behavior of Some Projection-Type Methods for Affine Variational Inequalities

In this paper, we study the local convergence behavior of four projection-type methods for the solution of the affine variational inequality (AVI) problem. It is shown that, if the sequence generated by one of the methods converges to a nondegenerate KKT point of the AVI problem, then after a finite number of iterations, some index sets in the dual variables at each iterative point coincide with the index set of the active constraints in the primal variables at the KKT point. As a consequence, we find that, after finitely many iterations, the four methods need not compute projections and their iterative equations are of reduced dimension.     

Some Methods Based on the D-Gap Function for Solving Monotone Variational Inequalities

The D-gap function has been useful in developing unconstrained descent methods for solving strongly monotone variational inequality problems. We show that the D-gap function has certain properties that are useful also for monotone variational inequality problems with bounded feasible set. Accordingly, we develop two unconstrained methods based on them that are similar in spirit to a feasible method of Zhu and Marcotte based on the regularized-gap function. We further discuss a third method based on applying the D-gap function to a regularized problem. Preliminary numerical experience is also reported.     

Equilibrium Problems with Generalized Monotone Bifunctions and Applications to Variational Inequalities

This paper attempts to generalize and unify several new results that have been obtained in the ongoing research area of existence of solutions for equilibrium problems. First, we propose sufficient conditions, which include generalized monotonicity and weak coercivity conditions, for existence of equilibrium points. As consequences, we generalize various recent theorems on the existence of such solutions. For applications, we treat some generalized variational inequalities and complementarity problems. In addition, considering penalty functions, we study the position of a selected solution by relying on the viscosity principle.     

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.     

一类积集上的广义向量变分不等式

研究一类积集上具某种权向量的广义向量变分不等式组及其广义向量变分不等式的有关问题,刻画它们之间解的相互关系.在映射的次连续性和关于某向量广义单调性的条件下,利用集值映射的不动点定理,对所讨论的几种类型的广义向量变分不等式给出解的存在性.     

Variational inequalities with nonmonotone operators

In this paper, existence results on variational inequalities and generalized variational inequalities for some nonmonotone operators over closed convex subsets of a real reflexive Banach space are proved. In particular, some surjectivity results and applications to complementarity and generalized complementarity problems are given.This work was partially supported by the National Science Council of the Republic of China under Contracts NSC 81-0208-M-007-34 and NSC 82-0208-M-110-023.     

Proximal Point Methods and Nonconvex Optimization

The goal of this paper is to discover some possibilities for applying the proximal point method to nonconvex problems. It can be proved that – for a wide class of problems – proximal regularization performed with appropriate regularization parameters ensures convexity of the auxiliary problems and each accumulation point of the method satisfies the necessary optimality conditions.