Weak convergence theorem for zero points of inverse strongly monotone mapping and fixed points of nonexpansive mapping in Hilbert space

We know that variational inequality problem is very important in the nonlinear analysis. For a variational inequality problem defined over a nonempty fixed point set of a nonexpansive mapping in Hilbert space, the strong convergence theorem has been proposed by I. Yamada. The algorithm in this theorem is named the hybrid steepest descent method. Based on this method, we propose a new weak convergence theorem for zero points of inverse strongly monotone mapping and fixed points of nonexpansive mapping in Hilbert space. Using this result, we obtain some new weak convergence theorems which are useful in nonlinear analysis and optimization problem.     

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.     

On extragradient-viscosity methods for solving equilibrium and fixed point problems in a Hilbert space

Abstract

In this paper, new numerical algorithms are introduced for finding the solution of a variational inequality problem whose constraint set is the common elements of the set of fixed points of a demicontractive mapping and the set of solutions of an equilibrium problem for a monotone mapping in a real Hilbert space. The strong convergence of the iterates generated by these algorithms is obtained by combining a viscosity approximation method with an extragradient method. First, this is done when the basic iteration comes directly from the extragradient method, under a Lipschitz-type condition on the equilibrium function. Then, it is shown that this rather strong condition can be omitted when an Armijo-backtracking linesearch is incorporated into the extragradient iteration. The particular case of variational inequality problems is also examined.     

A globally convergent Newton method for solving strongly monotone variational inequalities

Variational inequality problems have been used to formulate and study equilibrium problems, which arise in many fields including economics, operations research and regional sciences. For solving variational inequality problems, various iterative methods such as projection methods and the nonlinear Jacobi method have been developed. These methods are convergent to a solution under certain conditions, but their rates of convergence are typically linear. In this paper we propose to modify the Newton method for variational inequality problems by using a certain differentiable merit function to determine a suitable step length. The purpose of introducing this merit function is to provide some measure of the discrepancy between the solution and the current iterate. It is then shown that, under the strong monotonicity assumption, the method is globally convergent and, under some additional assumptions, the rate of convergence is quadratic. Limited computational experience indicates the high efficiency of the proposed method.     

A unifying geometric solution framework and complexity analysis for variational inequalities

In this paper, we propose a concept of polynomiality for variational inequality problems and show how to find a near optimal solution of variational inequality problems in a polynomial number of iterations. To establish this result, we build upon insights from several algorithms for linear and nonlinear programs (the ellipsoid algorithm, the method of centers of gravity, the method of inscribed ellipsoids, and Vaidya's algorithm) to develop a unifying geometric framework for solving variational inequality problems. The analysis rests upon the assumption of strong-f-monotonicity, which is weaker than strict and strong monotonicity. Since linear programs satisfy this assumption, the general framework applies to linear programs.Preparation of this paper was supported, in part, by NSF Grant 9312971-DDM from the National Science Foundation.     

Forcing strong convergence of Korpelevich’s method in Banach spaces with its applications in game theory

We develop a variant of Korpelevich’s method for solving variational inequality problems with pseudomonotone operators in Banach spaces. We establish the strong convergence of the sequence generated by the method under reasonable assumptions on the problem data. Finally, we justify the motivation of our theory by including compelling examples of infinite dimensional variational inequality problems for which our method is applicable.     

Strong convergence theorem by a new hybrid method for equilibrium problems and variational inequality problems

In this paper, we prove a strong convergence theorem for finding a common element of the set of solutions of an equilibrium problem and the set of the solutions of the variational inequality problem by using a new hybrid method. We obtain a new result for finding a solution of an equilibrium problem and the solutions of the variational inequality problem.     

Descent method for nonsmooth variational inequalities

A descent method with a gap function is proposed for a finite-dimensional variational inequality with nonintegrable and nonsmooth mapping. The convergence of the method with line search is established under strong monotonicity conditions on the underlying mapping. Published in Russian in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2006, Vol. 46, No. 7, pp. 1251–1257. This article was translated by the author.     

Numerical solution of the zero-head seepage problem by discretization of the variational inequality

The problem of zero-head seepage through a cutoff is reduced to solving a variational inequality which is discretized by the finite element method. The discrete variational inequality is solved by a two-layer iterative process. A rate of convergence bound is obtained for the approximate solution and the optimal parameters of the two-layer iterative process are determined. A numerical experiment supports the theoretical results.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 58, pp. 45–56, 1986.     

BOUNDARY ELEMENT APPROXIMATION OF THE SEMI-DISCRETE PARABOLIC VARIATIONAL INEQUALITIES OF THE SECOND KIND

The boundary element approximation of the parabolic variational inequalities of the second kind is discussed. First, the parabolic variational inequalities of the second kind can be reduced to an elliptic variational inequality by using semidiscretization and implicit method in time; then the existence and uniqueness for the solution of nonlinear non-differentiable mixed variational inequality is discussed. Its corresponding mixed boundary variational inequality and the existence and uniqueness of its solution are yielded. This provides the theoretical basis for using boundary element method to solve the mixed vuriational inequality.     

Convergence of an extragradient-type method for variational inequality with applications to optimal control problems

Numerical Algorithms - Our aim in this paper is to introduce an extragradient-type method for solving variational inequality with uniformly continuous pseudomonotone operator. The strong...     

SINGLE PROJECTION ALGORITHM FOR VARIATIONAL INEQUALITIES IN BANACH SPACES WITH APPLICATION TO CONTACT PROBLEM

We study the single projection algorithm of Tseng for solving a variational inequality problem in a 2-uniformly convex Banach space. The underline cost function of the variational inequality is assumed to be monotone and Lipschitz continuous. A weak convergence result is obtained under reasonable assumptions on the variable step-sizes. We also give the strong convergence result for when the underline cost function is strongly monotone and Lipchitz continuous. For this strong convergence case, the proposed method does not require prior knowledge of the modulus of strong monotonicity and the Lipschitz constant of the cost function as input parameters, rather, the variable step-sizes are diminishing and non-summable. The asymptotic estimate of the convergence rate for the strong convergence case is also given. For completeness, we give another strong convergence result using the idea of Halpern's iteration when the cost function is monotone and Lipschitz continuous and the variable step-sizes are bounded by the inverse of the Lipschitz constant of the cost function.Finally, we give an example of a contact problem where our proposed method can be applied.     

Projected subgradient techniques and viscosity methods for optimization with variational inequality constraints

In this paper, we propose an easily implementable algorithm in Hilbert spaces for solving some classical monotone variational inequality problem over the set of solutions of mixed variational inequalities. The proposed method combines two strategies: projected subgradient techniques and viscosity-type approximations. The involved stepsizes are controlled and a strong convergence theorem is established under very classical assumptions. Our algorithm can be applied for instance to some mathematical programs with complementarity constraints.     

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.     

Nonsmooth Multiobjective Problems and Generalized Vector Variational Inequalities Using Quasi-Efficiency

In this paper, a multiobjective problem with a feasible set defined by inequality, equality and set constraints is considered, where the objective and constraint functions are locally Lipschitz. Here, a generalized Stampacchia vector variational inequality is formulated as a tool to characterize quasi- or weak quasi-efficient points. By using two new classes of generalized convexity functions, under suitable constraint qualifications, the equivalence between Kuhn–Tucker vector critical points, solutions to the multiobjective problem and solutions to the generalized Stampacchia vector variational inequality in both weak and strong forms will be proved.     

A hybrid extragradient method for approximating the common solutions of a variational inequality, a system of variational inequalities, a mixed equilibrium problem and a fixed point problem

In this paper, we give a hybrid extragradient iterative method for finding the approximate element of the common set of solutions of a generalized equilibrium problem, a system of variational inequality problems, a variational inequality problem and a fixed point problem for a strictly pseudocontractive mapping in a real Hilbert space. Further we establish a strong convergence theorem based on this method. The results presented in this paper improves and generalizes the results given in Yao et al. [36] and Ceng et al. [7], and some known corresponding results in the literature.     

Decomposition algorithm for convex differentiable minimization

In this paper, we propose a decomposition algorithm for convex differentiable minimization. This algorithm at each iteration solves a variational inequality problem obtained by adding to the gradient of the cost function a strongly proximal related function. A line search is then performed in the direction of the solution to this variational inequality (with respect to the original cost). If the constraint set is a Cartesian product ofm sets, the variational inequality decomposes intom coupled variational inequalities, which can be solved in either a Jacobi manner or a Gauss-Seidel manner. This algorithm also applies to the minimization of a strongly convex (possibly nondifferentiable) cost subject to linear constraints. As special cases, we obtain the GP-SOR algorithm of Mangasarian and De Leone, a diagonalization algorithm of Feijoo and Meyer, the coordinate descent method, and the dual gradient method. This algorithm is also closely related to a splitting algorithm of Gabay and a gradient projection algorithm of Goldstein and of Levitin-Poljak, and has interesting applications to separable convex programming and to solving traffic assignment problems.This work was partially supported by the US Army Research Office Contract No. DAAL03-86-K-0171 and by the National Science Foundation Grant No. ECS-85-19058. The author thanks the referees for their many helpful comments, particularly for suggesting the use of a general functionH instead of that given by (4).     

On solvability of variational inequalities with discontinuous semimonotone operators

By using the method of monotone operators, a theorem on the existence of the solution with a special property is obtained for an elliptic variational inequality with discontinuous semimonotone operator; this theorem is then used to prove the existence of a semicorrect solution of a variational inequality with a differential semilinear high-order operator of elliptic type with a nonsymmetric linear part and discontinuous nonlinearity.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 3, pp. 443–447, March, 1993.     

Iterative methods for solving equilibrium problems, variational inequalities and fixed points of nonexpansive semigroups

In this work, strong convergence theorems by the viscosity approximation method associated with Meir–Keeler contractions are established for solving fixed point problems of a nonexpansive semigroup, a system of equilibrium problems and variational inequality problems in a real Hilbert space. Further, applications related to commutative semigroup are obtained.     

A Modified Alternating Direction Method for Variational Inequality Problems

The alternating direction method is an attractive method for solving large-scale variational inequality problems whenever the subproblems can be solved efficiently. However, the subproblems are still variational inequality problems, which are as structurally difficult to solve as the original one. To overcome this disadvantage, in this paper we propose a new alternating direction method for solving a class of nonlinear monotone variational inequality problems. In each iteration the method just makes an orthogonal projection to a simple set and some function evaluations. We report some preliminary computational results to illustrate the efficiency of the method. Accepted 4 May 2001. Online publication 19 October, 2001.