STRONGLY CONVERGENT INERTIAL FORWARD-BACKWARD-FORWARD ALGORITHM WITHOUT ON-LINE RULE FOR VARIATIONAL INEQUALITIES

This paper studies a strongly convergent inertial forward-backward-forward algorithm for the variational inequality problem in Hilbert spaces.In our convergence analysis,we do not assume the on-line rule of the inertial parameters and the iterates,which have been assumed by several authors whenever a strongly convergent algorithm with an inertial extrapolation step is proposed for a variational inequality problem.Consequently,our proof arguments are different from what is obtainable in the relevant literature.Finally,we give numerical tests to confirm the theoretical analysis and show that our proposed algorithm is superior to related ones in the literature.

A RELAXED INERTIAL FACTOR OF THE MODIFIED SUBGRADIENT EXTRAGRADIENT METHOD FOR SOLVING PSEUDO MONOTONE VARIATIONAL INEQUALITIES IN HILBERT SPACES

In this paper,we investigate pseudomonotone and Lipschitz continuous variational inequalities in real Hilbert spaces.For solving this problem,we propose a new method that combines the advantages of the subgradient extragradient method and the projection contraction method.Some very recent papers have considered different inertial algorithms which allowed the inertial factor is chosen in[0;1].The purpose of this work is to continue working in this direction,we propose another inertial subgradient extragradient method that the inertial factor can be chosen in a special case to be 1.Under suitable mild conditions,we establish the weak convergence of the proposed algorithm.Moreover,linear convergence is obtained under strong pseudomonotonicity and Lipschitz continuity assumptions.Finally,some numerical illustrations are given to confirm the theoretical analysis.

Regularization Methods to Approximate Solutions of Variational Inequalities

In this paper, we study the regularization methods to approximate the solutions of the variational inequalities with monotone hemi-continuous operator having perturbed operators arbitrary. Detail, we shall study regularization methods to approximate solutions of following variational inequalities: and with operator A being monotone hemi-continuous form real Banach reflexive X into its dual space X*, but instead of knowing the exact data (y0, A), we only know its approximate data satisfying certain specified conditions and D is a nonempty convex closed subset of X;the real function f defined on X is assumed to be lower semi-continuous, convex and is not identical to infinity. At the same time, we will evaluate the convergence rate of the approximate solution. The regularization methods here are different from the previous ones.

An improved proximal-based decomposition method for structured monotone variational inequalities

The proximal-based decomposition method was originally proposed by Chen and Teboulle （Math. Programming, 1994, 64：81-101 for solving corrvex minimization problems. This paper extends it to solving monotone variational inequalities associated with separable structures with the improvements that the restrictive assumptions on the involved parameters are much relaxed, and thus makes it practical to solve the subproblems easily. Without additional assumptions, global convergence of the new method is proved under the same mild assumptions on the problem＇s data as the original method.

SYSTEM OF GENERALIZED VECTOR VARIATIONAL INEQUALITIES

In this paper, the authors introduce and study system of generalized vector variational inequalities. Under suitable conditions, the existence of solutions for system of generalized vector variational inequalities is presented by Kakutani-Fan-Glicksberg fixed point theorem.

NEW SIMPLE SMOOTH MERIT FUNCTION FOR BOX CONSTRAINED VARIATIONAL INEQUALITIES AND DAMPED NEWTON TYPE METHOD

By introducing a smooth merit function for the median function, a new smooth merit function for box constrained variational inequalities （BVIs） was constructed. The function is simple and has some good differential properties. A damped Newton type method was presented based on it. Global and local superlinear/ quadratic convergence results were obtained under mild conditions, and the finite termination property was also shown for the linear BVIs. Numerical results suggest that the method is efficient and promising.

AN ITERATIVE METHOD FOR THE DISCRETE PROBLEMS OF A CLASS OF ELLIPTICAL VARIATIONAL INEQUALITIES

Based on the nonlinear characiers of the discrete problems of some ellipticalvariational inequalities, this paper presents a numerical iterative method, the schemesof which are pithy and converge rapidly The new method possesses a high efficiency. insolving such applied engineering problems as obstacle problems and .free boundary.problems arising in fluid lubrications.

Three-step relaxed hybrid steepest-descent methods for variational inequalities

The classical variational inequality problem with a Lipschitzian and strongly monotone operator on a nonempty closed convex subset in a real Hilbert space is studied. A new three-step relaxed hybrid steepest-descent method for this class of variational inequalities is introduced. Strong convergence of this method is established under suitable assumptions imposed on the algorithm parameters.

THE EXISTENCE AND UNIQUENESS OF SOLUTIONS OF GENERALIZED VARIATIONAL INEQUALITIES ARISING FROM ELASTICITY WITH FRICTION

The existence and uniqueness of solutions of generalized variational inequalities arising from elasticity with friction, which is equivalent to corresponding elemental problems, is elucidated in detail, and then FEM approximation and discrete methods are proposed.

A NEW SELF-ADAPTIVE ITERATIVE METHOD FOR GENERAL MIXED QUASI VARIATIONAL INEQUALITIES

The general mixed quasi variational inequality containing a nonlinear term φ is a useful and an important generalization of variational inequalities. The projection method can not be applied to solve this problem due to the presence of nonlinear term. It is well known that the variational inequalities involving the nonlinear term φ are equivalent to the fixed point problems and resolvent equations. In this article, the authors use these alternative equivalent formulations to suggest and analyze a new self-adaptive iterative method for solving general mixed quasi variational inequalities. Global convergence of the new method is proved. An example is given to illustrate the efficiency of the proposed method.

ON A CLASS OF QUASI VARIATIONAL INEQUALITIES

In this paper,we introduce and study a new class of quasi variational inequalities.Using'essentially the projection technique and its variant forms,we establish the equivalence between generalized nonlinear quasi variational inequalities and the fixed point problems.This equivalence is then used to suggest and analyze a number of new iterative algorithms.These new results include the corresponding known results for generalized quasi variational inequalities as special cases.

ON ITERATIVE ALGORITHMS FOR A CLASS OF NONLINEAR VARIATIONAL INEQUALITIES

In this paper we use the auxiliary principle technique to suggest and analyze novel and innovative iterative algorithms for a class of nonlinear variational inequalities. Several special cases, which can be obtained from our main results, are also discussed.

DIFFERENTIAL VARIATIONAL INEQUALITIES IN INFINITE BANACH SPACES

In this paper,we consider a new differential variational inequality（DVI,for short）which is composed of an evolution equation and a variational inequality in infinite Banach spaces.This kind of problems may be regarded as a special feedback control problem.Based on the Browder＇s theorem and the optimal control theory,we show the existence of solutions to the mentioned problem.

APPROXIMATION OF COMMON SOLUTIONS OF VARIATIONAL INEQUALITIES VIA STRICT PSEUDOCONTRACTIONS

In this paper, a convex feasibility problem is considered. We construct an iterative method to approximate a common element of the solution set of classical variational inequalities and of the fixed point set of a strict pseudocontraction. Strong convergence theorems for the common element are established in the framework of Hilbert spaces.

DIFFERENTIAL INVERSE VARIATIONAL INEQUALITIES IN FINITE DIMENSIONAL SPACES

In this article, a new differential inverse variational inequality is introduced and studied in finite dimensional Euclidean spaces. Some results concerned with the linear growth of the solution set for the differential inverse variational inequalities are obtained under different conditions. Some existence theorems of Caratheodory weak solutions for the differential inverse variational inequality are also established under suitable conditions. An application to the time-dependent spatial price equilibrium control problem is also given.

A PROJECTION-TYPE ALGORITHM FOR SOLVING GENERALIZED MIXED VARIATIONAL INEQUALITIES

We propose a projection-type algorithm for generalized mixed variational in- equality problem in Euclidean space Rn. We establish the convergence theorem for the pro- posed algorithm, provided the multi-valued mapping is continuous and f-pseudomonotone with nonempty compact convex values on dom（f）, where f ： Rn --RU{＋∞} is a proper func- tion. The algorithm presented in this paper generalize and improve some known algorithms in literatures. Preliminary computational experience is also reported.

STRONG CONVERGENCE OF AN INERTIAL EXTRAGRADIENT METHOD WITH AN ADAPTIVE NONDECREASING STEP SIZE FOR SOLVING VARIATIONAL INEQUALITIES

In this work,we investigate a classical pseudomonotone and Lipschitz continuous variational inequality in the setting of Hilbert space,and present a projection-type approximation method for solving this problem.Our method requires only to compute one projection onto the feasible set per iteration and without any linesearch procedure or additional projections as well as does not need to the prior knowledge of the Lipschitz constant and the sequentially weakly continuity of the variational inequality mapping.A strong convergence is established for the proposed method to a solution of a variational inequality problem under certain mild assumptions.Finally,we give some numerical experiments illustrating the performance of the proposed method for variational inequality problems.

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 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.

Generalized system for strongly g-r-pseudomonotonic nonlinear variational inequalities in Hilbert spaces

The approximation solvability of a generalized system for strongly g-r- pseudomonotonic nonlinear variational inequalities in Hilbert spaces is studied based on the convergence of the projection method. The results presented in this paper improve, generalize and unify some recent results in the literature.

KKM Theorems on FC-spaces with Their Applications to Variational Inequalities

A new concept of finitely continuous topological spaces （in short, FC-space） without convexity structure and linear structure was introduced. Some KKM type theorems in noncompact FC-spaces were obtained, and from which, some section theorems and wriational inequality theorems were proved under much weak assumptions. Our results improve and generalize the corresponding conclusions in recent literature.