Quasimonotone variational inequalities in Banach spaces

Various existence results for variational inequalities in Banach spaces are derived, extending some recent results by Cottle and Yao. Generalized monotonicity as well as continuity assumptions on the operatorf are weakened and, in some results, the regularity assumptions on the domain off are relaxed significantly. The concept of inner point for subsets of Banach spaces proves to be useful.This work was completed while the first author was visiting the Graduate School of Management of the University of California, Riverside. The author wishes to thank the School for its hospitality.     

On generalized variational inequalities

We obtain a new version of the minimax inequality of Ky Fan. As an application, an existence result for the generalized variational inequality problem with set-valued mappings defined on noncompact sets in Hausdorff topological vector spaces is given. Also, some existence results for the generalized variational inequality problem for quasimonotone and pseudomonotone mappings are obtained. Dedicated to the memory of T. Rapcsák.     

Further applications of a splitting algorithm to decomposition in variational inequalities and convex programming

A classical method for solving the variational inequality problem is the projection algorithm. We show that existing convergence results for this algorithm follow from one given by Gabay for a splitting algorithm for finding a zero of the sum of two maximal monotone operators. Moreover, we extend the projection algorithm to solveany monotone affine variational inequality problem. When applied to linear complementarity problems, we obtain a matrix splitting algorithm that is simple and, for linear/quadratic programs, massively parallelizable. Unlike existing matrix splitting algorithms, this algorithm converges under no additional assumption on the problem. When applied to generalized linear/quadratic programs, we obtain a decomposition method that, unlike existing decomposition methods, can simultaneously dualize the linear constraints and diagonalize the cost function. This method gives rise to highly parallelizable algorithms for solving a problem of deterministic control in discrete time and for computing the orthogonal projection onto the intersection of convex sets.This research is partially supported by the U.S. Army Research Office, contract DAAL03-86-K-0171 (Center for Intelligent Control Systems), and by the National Science Foundation under grant NSF-ECS-8519058.Thanks are due to Professor J.-S. Pang for his helpful comments.     

Stochastic variational inequalities of parabolic type

Existence and uniqueness of strong solutions of stochastic partial differential equations of parabolic type with reflection (e.g., the solutions are never allowed to be negative) is proved. The problem is formulated as a stochastic variational inequality and then compactness is used to derive the result, but the method requires the space dimension to be one.This research was supported by NSERC under Grant No. 8051.     

Projection-proximal methods for general variational inequalities

In this paper, we consider and analyze some new projection-proximal methods for solving general variational inequalities. The modified methods converge for pseudomonotone operators which is a weaker condition than monotonicity. The proposed methods include several new and known methods as special cases. Our results can be considered as a novel and important extension of the previously known results. Since the general variational inequalities include the quasi-variational inequalities and implicit complementarity problems as special cases, results proved in this paper continue to hold for these problems.     

On generalized Nash games and variational inequalities

We show that for a large class of problems a generalized Nash equilibrium can be calculated by solving a variational inequality. We analyze what solutions are found by this reduction procedure and hint at possible applications.     

抛物型变分不等式解的支集的瞬间收缩性

研究了下面的抛物型变分不等式v≥0,(ut-Δu+b(x,t)up)(v-u)≥f(v-u)a.e.,(x,t)∈RN×(0,T],u≥0,(x,t)∈RN×(0,T],u(x,0)=u0(x),x∈RN的解的存在惟一性,以及解的支集的瞬间收缩性.     

Benders decomposition for a class of variational inequalities

This paper examines Benders decomposition for a useful class of variational inequality (VI) problems that can model, e.g., economic equilibrium, games or traffic equilibrium. The dual of the given VI is defined. Benders decomposition of the original VI is derived by applying a Dantzig–Wolfe decomposition procedure to the dual of the given VI, and converting the dual forms of the Dantzig–Wolfe master and subproblems to their primal forms. The master problem VI includes a new cut at each iteration, with information from the latest subproblem VI, which is solved by fixing the “difficult” variables at values determined by the previous master problem. A scalar parameter called the convergence gap is calculated at each iteration; a negative value is equivalent to the algorithm making progress in that the last master problem solution is made infeasible by the new cut. Under mild conditions, the convergence gap approaches zero in the limit of many iterations. With a more restrictive condition that still admits many useful models, a zero value of the convergence gap implies that the master problem has found a solution of the VI. A small model of competitive equilibrium of three commodities in two regions serves as an illustration.     

Modified descent-projection method for solving variational inequalities

In this paper, we propose a modified descent-projection method for solving variational inequalities. The method makes use of a descent direction to produce the new iterate and can be viewed as an improvement of the descent-projection method by using a new step size. Under certain conditions, the global convergence of the proposed method is proved. In order to demonstrate the efficiency of the proposed method, we provide numerical results for a traffic equilibrium problems.     

Merit functions and error bounds for generalized variational inequalities

We consider the generalized variational inequality and construct certain merit functions associated with this problem. In particular, those merit functions are everywhere nonnegative and their zero-sets are precisely solutions of the variational inequality. We further use those functions to obtain error bounds, i.e., upper estimates for the distance to solutions of the problem.     

On the solution existence of pseudomonotone variational inequalities

As shown by Thanh Hao [Acta Math. Vietnam 31, 283–289, 2006], the solution existence results established by Facchinei and Pang [Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. I (Springer, Berlin, 2003) Prop. 2.2.3 and Theorem 2.3.4] for variational inequalities (VIs) in general and for pseudomonotone VIs in particular, are very useful for studying the range of applicability of the Tikhonov regularization method. This paper proposes some extensions of these results of Facchinei and Pang to the case of generalized variational inequalities (GVI) and of variational inequalities in infinite-dimensional reflexive Banach spaces. Various examples are given to analyze in detail the obtained results. B. T. Kien: On leave from Hanoi University of Civil Engineering. The online version of the original article can be found at .     

Estimates of approximate solutions and well-posedness for variational inequalities

The purpose of this paper is to estimate the approximate solutions for variational inequalities. In terms of estimate functions, we establish some estimates of the sizes of the approximate solutions from outside and inside respectively. By considering the behaviors of estimate functions, we give some characterizations of the well-posedness for variational inequalities. This work was partially supported by the Basic and Applied Research Projection of Sichuan Province (05JY029-009-1) and the National Natural Science Foundation of China (10671135).     

A continuation method for monotone variational inequalities

This paper presents a continuation method for monotone variational inequality problems based on a new smooth equation formulation. The existence, uniqueness and limiting behavior of the path generated by the method are analyzed.This work was supported by the National Science Foundation Presidential Young Investigator Award ECE-8552773 and by a grant from the Burlington Northern Railroad.     

Convergence of stationary sequences for variational inequalities with maximal monotone operators

LetT be a maximal monotone operator defined on ^N . In this paper we consider the associated variational inequality 0 T(x ^*) and stationary sequences {x _k ^* for this operator, i.e., satisfyingT(x _k ^* 0. The aim of this paper is to give sufficient conditions ensuring that these sequences converge to the solution setT ^–1(0) especially when they are unbounded. For this we generalize and improve the directionally local boundedness theorem of Rockafellar to maximal monotone operatorsT defined on ^N .     

Degree theory for generalized variational inequalities and applications

In this paper, a degree theory for finite dimensional generalized variational inequalities is built and employed to prove some results on solution existence and solution stability.     

Mixed projection methods for systems of variational inequalities

Let H be a real Hilbert space. Let be bounded and continuous mappings where D(F) and D(K) are closed convex subsets of H. We introduce and consider the following system of variational inequalities: find [u ^*,v ^*]∈D(F) × D(K) such that This system of variational inequalities is closely related to a pseudomonotone variational inequality. The well-known projection method is extended to develop a mixed projection method for solving this system of variational inequalities. No invertibility assumption is imposed on F and K. The operators K and F also need not be defined on compact subsets of H.