A Class of Differential Vector Variational Inequalities in Finite Dimensional Spaces

In this paper, we introduce and study a class of differential vector variational inequalities in finite dimensional Euclidean spaces. We establish a relationship between differential vector variational inequalities and differential scalar variational inequalities. Under various conditions, we obtain the existence and linear growth of solutions to the scalar variational inequalities. In particular we prove existence theorems for Carathéodory weak solutions of the differential vector variational inequalities. Furthermore, we give a convergence result on Euler time-dependent procedure for solving the initial-value differential vector variational inequalities.     

Splitting Algorithms for General Pseudomonotone Mixed Variational Inequalities

In this paper, we suggest and analyze a number of resolvent-splitting algorithms for solving general mixed variational inequalities by using the updating technique of the solution. The convergence of these new methods requires either monotonicity or pseudomonotonicity of the operator. Proof of convergence is very simple. Our new methods differ from the existing splitting methods for solving variational inequalities and complementarity problems. The new results are versatile and are easy to implement.     

Some Predictor-Corrector Algorithms for Multivalued Variational Inequalities

In this paper, we use the auxiliary principle technique to suggest a new class of predictor-corrector algorithms for solving multivalued variational inequalities. The convergence of the proposed methods requires only the partially-relaxed strong monotonicity of the operator, which is weaker than cocoercivity. As special cases, we obtain a number of known and new results for solving various classes of variational inequalities.     

A class of new iterative methods for general mixed variational inequalities

In this paper, we use the auxiliary principle technique to suggest a class of predictor-corrector methods for solving general mixed variational inequalities. The convergence of the proposed methods only requires the partially relaxed strongly monotonicity of the operator, which is weaker than co-coercivity. As special cases, we obtain a number of known and new results for solving various classes of variational inequalities and related problems.     

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.     

A predictor-corrector algorithm for general variational inequalities

In this paper, we suggest a new predictor-corrector algorithm for solving general variational inequalities by using the auxiliary principle technique. The convergence of the proposed method only requires the partially relaxed strong monotonicity of the operator, which is weaker than co-coercivity. As special cases, we obtain a number of known and new results for solving various classes of variational inequalities.     

变分不等式的一类二次投影算法

通过构造的一类严格分离当前点与解集的超平面得到了一类解伪单调变分不等式的修正二次投影算法,该算法对He Yiran的算法进行了修正.从而建立了解伪单调变分不等式二次投影算法的一种框架结构.证明了该算法生成的无穷序列具有的全局收敛性,在具备某种局部误差界和Lipchitz连续条件下给出了收敛率分析.并给出了该算法的数值演算结果.     

Projection algorithms for the system of mixed variational inequalities in Banach spaces

In this paper, the system of mixed variational inequalities is introduced and considered in Banach spaces, which includes some known systems of variational inequalities and the classical variational inequalities as special cases. Using the projection operator technique, we suggest some iterative algorithms for solving the system of mixed variational inequalities and prove the convergence of the proposed iterative methods under suitable conditions. Our theorems generalize some known results shown recently.     

Modified resolvent splitting algorithms for general mixed variational inequalities

In this paper, we suggest and analyze a number of four-step resolvent splitting algorithms for solving general mixed variational inequalities by using the updating technique of the solution. The convergence of these new methods requires either monotonicity or pseudomonotonicity of the operator. Proof of convergence is very simple. Our new methods differ from the existing splitting methods for solving variational inequalities and complementarity problems. The new results are versatile and are easy to implement.     

Modified Extragradient Methods for a System of Variational Inequalities in Banach Spaces

In this paper, we introduce a new system of general variational inequalities in Banach spaces. We establish the equivalence between this system of variational inequalities and fixed point problems involving the nonexpansive mapping. This alternative equivalent formulation is used to suggest and analyze a modified extragradient method for solving the system of general variational inequalities. Using the demi-closedness principle for nonexpansive mappings, we prove the strong convergence of the proposed iterative method under some suitable conditions.     

Wiener-hopf equations and variational inequalities

In this paper, we show that the general variational inequality problem is equivalent to solving the Wiener-Hopf equations. We use this equivalence to suggest and analyze a number of iterative algorithms for solving general variational inequalities. We also discuss the convergence criteria for these algorithms.     

Some New Iterative Methods for General Mixed Variational Inequalities

In this paper, we use the auxiliary principle technique to suggest a class of predictorcorrector methods for solving general mixed variational inequalities. The convergence of the proposed methods only requires the partially relaxed strongly monotonicity of the operator, which is weaker than co-coercivity. From special cases, we obtain various known and new results for solving various classes of variational inequalities and related problems.AMS Subject Classification (1991): 49J40, 90C33.     

Projection methods for nonconvex variational inequalities

In this paper, we introduce and consider a new class of variational inequalities, which is called the nonconvex variational inequalities. We establish the equivalence between the nonconvex variational inequalities and the fixed-point problems using the projection technique. This equivalent formulation is used to discuss the existence of a solution of the nonconvex variational inequalities. We also use this equivalent alternative formulation to suggest and analyze a new iterative method for solving the nonconvex variational inequalities. We also discuss the convergence of the iterative method under suitable conditions. Our method of proof is very simple as compared with other techniques.     

Projection iterative methods for extended general variational inequalities

In this paper, we introduce and study a new class of variational inequalities involving three operators, which is called the extended general variational inequality. Using the projection technique, we show that the extended general variational inequalities are equivalent to the fixed point and the extended general Wiener-Hopf equations. This equivalent formulation is used to suggest and analyze a number of projection iterative methods for solving the extended general variational inequalities. We also consider the convergence of these new methods under some suitable conditions. Since the extended general variational inequalities include general variational inequalities and related optimization problems as special cases, results proved in this paper continue to hold for these problems.     

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.     

On nonconvex bifunction variational inequalities

In this paper, we introduce and consider a new class of variational inequalities, which is called the nonconvex bifunction variational inequality. We suggest and analyze some iterative methods for solving nonconvex bifunction variational inequalities using the auxiliary principle technique. We prove that the convergence of implicit method requires only pseudomonotonicity, which is weaker condition than monotonicity. Our proof of convergence is very simple. Results proved in this paper may stimulate further research in this dynamic field.     

Extragradient methods for solving nonconvex variational inequalities

In this paper, we introduce and consider a new class of variational inequalities, which are called the nonconvex variational inequalities. Using the projection technique, we suggest and analyze an extragradient method for solving the nonconvex variational inequalities. We show that the extragradient method is equivalent to an implicit iterative method, the convergence of which requires only pseudo-monotonicity, a weaker condition than monotonicity. This clearly improves on the previously known result. Our method of proof is very simple as compared with other techniques.     

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.     

A new predictor-corrector method for noncoercive mixed variational inequalities

In this paper, we use the auxiliary principle technique to suggest and analyze a class of predictor-corrector methods for solving noncoercive mixed variational inequalities. The convergence of the proposed method requires only the partially relaxed strongly monotonicity, which is even weaker than the co-coercivity. As special cases, we obtain a number of new and known results for classical variational inequalities.     

New extragradient-type methods for general variational inequalities

In this paper, we consider and analyze a new class of extragradient-type methods for solving general variational inequalities. The modified methods converge for pseudomonotone operators which is weaker condition than monotonicity. Our proof of convergence is very simple as compared with other methods. The proposed methods include several new and known methods as special cases. Our results present a significant improvement of previously known methods for solving variational inequalities and related optimization problems.