Bifunction hemivariational inequalities

In this paper, we introduce and consider a new class of variational inequalities, which is called the bifunction hemivariational inequality. This new class includes several classes of variational inequalities as special cases. A number of iterative methods for solving bifunction hemivariational inequalities are suggested and analyzed by using the auxiliary principle technique. We also study the convergence analysis of these iterative methods under some mild conditions. The results obtained in this paper can be considered as a novel application of the auxiliary principle technique.     

Convergence of multisplitting method for a symmetric positive definite matrix

The auxiliary principle is used to suggest and analyze some iterative methods for solving solving hemivariational inequalities under mild conditions. The results obtained in this paper can be considered as a novel application of the auxiliary principle technique. Since hemivariational inequalities include variational inequalities and nonlinear optimization problems as special cases, our results continue to hold-for these problems.     

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.     

Predictor–Corrector Methods for General Regularized Nonconvex Variational Inequalities

This paper is devoted to the study of a new class of nonconvex variational inequalities, named general regularized nonconvex variational inequalities. By using the auxiliary principle technique, a new modified predictor–corrector iterative algorithm for solving general regularized nonconvex variational inequalities is suggested and analyzed. The convergence of the iterative algorithm is established under the partially relaxed monotonicity assumption. As a consequence, the algorithm and results presented in the paper overcome incorrect algorithms and results existing in the literature.     

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.     

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.     

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.     

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.     

广义混合隐拟h变分不等式的解的存在性与算法

本文在Hilbert空间中,引入了一类广义混合隐拟h变分不等式.运用变分原理,给出了广义混合隐拟h变分不等式逼近解的迭代算法,证明了这类变分不等式解的存在性定理,同时,得到迭代序列的收敛性.并改进和推广了[6~8]一些已知结果.     

一般多值混合隐拟变分不等式的解的存在性与算法

引入了实Hilbert空间中一类新的一般多值混合隐拟变分不等式.它概括了丁协平教授引入与研究过的熟知的广义混合隐拟变分不等式类成特例.运用辅助变分原理技巧来解这类一般多值混合隐拟变分不等式.首先,定义了具真凸下半连续的二元泛函的新的辅助变分不等式,并选取了一适当的泛函,使得其唯一的最小值点等价于此辅助变分不等式的解.其次,利用此辅助变分不等式,构造了用于计算一般多值混合隐拟变分不等式逼近解的新的迭代算法.在此,等价性保证了算法能够生成一列逼近解.最后,证明了一般多值混合隐拟变分不等式解的存在性与逼近解的收敛性.而且,给算法提供了新的收敛判据.因此,结果对M.A.Noor提出的公开问题给出了一个肯定答案,并推广和改进了关于各种变分不等式与补问题的早期与最近的结果,包括最近文献中涉及单值与集值映象的有关混合变分不等式、混合拟变不等式与拟补问题的相应结果.     

Regularized Nonconvex Mixed Variational Inequalities: Auxiliary Principle Technique and Iterative Methods

In this paper, we turn our attention to formulating and studying a new class of variational inequalities in a nonconvex setting, called regularized nonconvex mixed variational inequalities. By using the auxiliary principle technique, some new predictor corrector methods for solving such class of regularized nonconvex mixed variational inequalities are suggested and analyzed. The study of convergence analysis of the proposed iterative algorithms requires either pseudomonotonicity or partially mixed relaxed and strong monotonicity of the operator involved in regularized nonconvex mixed variational inequalities. As a consequence of our main results, we provide the correct versions of the algorithms and results presented in the literature.     

Hemiequilibrium-like problems

In this paper, we introduce and consider a new class of equilibrium problems, known as hemiequilibrium-like problems. This new class includes hemiequilibrium, equilibrium-like problems and several classes of variational inequalities as special cases. A number of iterative methods for solving hemiequilibrium-like problems are suggested and analyzed by using the auxiliary principle technique. We also study the convergence analysis of these iterative methods under some mild conditions. The results obtained in this paper can be considered as a novel application of the auxiliary principle technique.     

Generalized nonlinear variational‐like inequalities in reflexive Banach spaces

In this paper, we introduce and study a new class of generalized nonlinear mixed variational‐like inequalities in reflexive Banach spaces. By applying the auxiliary principle technique and the minimax inequality, we establish the existence and uniqueness theorems for solutions of generalized nonlinear mixed variational‐like inequalities. Copyright © 2010 John Wiley & Sons, Ltd.     

Proximal Methods for Mixed Quasivariational Inequalities

A proximal method for solving mixed quasivariational inequalities is suggested and analyzed by using the auxiliary principle technique. We show that the convergence of the proposed method requires only the pseudomonotonicity, which is a weaker condition than monotonicity. Since mixed quasivariational inequalities include variational and complementarity problems as special cases, the result proved in this paper continues to hold for these problems.     

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.     

Iterative Schemes for Nonconvex Variational Inequalities

In this paper, we suggest and analyze some iterative methods for solving nonconvex variational inequalities using the auxiliary principle technique, the convergence of which requires either only pseudomonotonicity or partially relaxed strong monotonicity. Our proofs of convergence are very simple. As special cases, we obtain earlier results for solving general variational inequalities involving convex sets.     

General algorithm for variational inequalities

In this paper, we consider a general auxiliary principle technique to suggest and analyze a novel and innovative iterative algorithm for solving variational inequalities and optimization problems. We also discuss the convergence criteria.     

Auxiliary Principle Technique for Solving Bifunction Variational Inequalities

In this paper, we use the auxiliary principle technique to suggest and analyze an implicit iterative method for solving bifunction variational inequalities. We also study the convergence criteria of this new method under pseudomonotonicity condition.     

On general hemiequilibrium problems

In this paper, we introduce and analyze a new class of equilibrium problems known as general hemiequilibrium problems. It is shown that this class includes hemiequilibrium problems, hemivariational inequalities and complementarity problems as special cases. We use the auxiliary principle techniques to suggest some iterative-type methods for solving multivalued hemiequilibrium problems. We also analyze the convergence analysis of these new iterative methods under some mild conditions. As special cases, we obtain several new and known methods for solving variational inequalities and equilibrium problems.     

On interative 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.