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.     

On General Mixed Quasivariational Inequalities

In this paper, we suggest and analyze several iterative methods for solving general mixed quasivariational inequalities by using the technique of updating the solution and the auxiliary principle. It is shown that the convergence of these methods requires either the pseudomonotonicity or the partially relaxed strong monotonicity of the operator. Proofs of convergence is very simple. Our new methods differ from the existing methods for solving various classes of variational inequalities and related optimization problems. Various special cases are also discussed.     

Iterative Methods for General Mixed Quasivariational Inequalities

It is well known that mixed quasivariational inequalities are equivalent to implicit fixed-point problems. We use this alternative equivalent formulation to suggest and analyze a new self-adaptive resolvent method for solving mixed quasivariational inequalities in conjunction with a technique updating the solution. We show that the convergence of this method requires pseudomonotonicity, which is a weaker condition than monotonicity. Since mixed quasivariational inequalities include various classes of variational inequalities as special cases, our results continue to hold for these problems.     

Generalized Mixed Variational Inequalities and Resolvent Equations

In this paper, we introduce and study a new class of variational inequalities, which is called the generalized mixed variational inequality. Using essentially the resolvent operator concept, we establish the equivalence between the generalized mixed variational inequalities and the system of resolvent equations. This equivalence is used to suggest a number of new iterative algorithms for solving the variational inequalities. Several special cases are discussed which can be obtained from the main results of this paper.     

Set-Valued Resolvent Equations and Mixed Variational Inequalities

In this paper, we introduce and study a new class of variational inequalities, which is called the generalized set-valued mixed variational inequality. The resolvent operator technique is used to establish the equivalence among generalized set-valued variational inequalities, fixed point problems, and the generalized set-valued resolvent equations. This equivalence is used to study the existence of a solution of set-valued variational inequalities and to suggest a number of iterative algorithms for solving variational inequalities and related optimization problems. The results proved in this paper represent a significant refinement and improvement of the previously known results in this area.     

伪单调广义混合变分不等式的改进k步预解算法

在算子的分裂技巧基础上介绍了求解伪单调广义混合变分不等式的改进五步预解算法，算法的收敛性只要求算子的g-伪单调和g—Lipschitz连续性，算子的伪单调比单调更弱．本文的新算法推广了文献中某些已有的结果．     

关于求完全广义强的非线性拟变分不等式的逼近解的迭代算法

本文研究求完全广义强的非线性拟变分不等式的逼近解的迭代算法。概括了该须域中作为特例的若干已知结果。我们的结果是Siddiqi与Ansari.Ding及Zeng的结果的推广和改进。     

Well-Posedness and L-Well-Posedness for Quasivariational Inequalities

In this paper, two concepts of well-posedness for quasivariational inequalities having a unique solution are introduced. Some equivalent characterizations of these concepts and classes of well-posed quasivariational inequalities are presented. The corresponding concepts of well-posedness in the generalized sense are also investigated for quasivariational inequalities having more than one solution The author is grateful to an anonymous referee for valuable comments.     

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.     

Variance-Based Single-Call Proximal Extragradient Algorithms for Stochastic Mixed Variational Inequalities

Journal of Optimization Theory and Applications - In the study of stochastic variational inequalities, the extragradient algorithms attract much attention. However, such schemes require two...     

广义混合拟变分不等式解的存在性及算法

应用辅助变分不等式技巧研究一类广义混合拟变分不等式解的存在性和迭代算法.所得到的结果回答了Noor提出的公开问题,改进和推广了一些较近的已知结果.     

Scalarization and Kuhn-Tucker-Like Conditions for Weak Vector Generalized Quasivariational Inequalities

We consider a weak vector generalized quasivariational inequality. By introducing a method of scalarization which does not require any assumption on the data and by using previous results of the authors concerning scalar generalized quasivariational inequalities, we present Kuhn-Tucker-like conditions for this problem in the case in which the set-valued operator of the constraints is defined by a finite number of inequalities     

Gap Functions for Quasivariational Inequalities and Generalized Nash Equilibrium Problems

The gap function (or merit function) is a classic tool for reformulating a Stampacchia variational inequality as an optimization problem. In this paper, we adapt this technique for quasivariational inequalities, that is, variational inequalities in which the constraint set depends on the current point. Following Fukushima (J. Ind. Manag. Optim. 3:165–171, 2007), an axiomatic approach is proposed. Error bounds for quasivariational inequalities are provided and an application to generalized Nash equilibrium problems is also considered.     

Quasimonotone Quasivariational Inequalities: Existence Results and Applications

A quasivariational inequality is a variational inequality in which the constraint set depends on the variable. Based on fixed point techniques, we prove various existence results under weak assumptions on the set-valued operator defining the quasivariational inequality, namely quasimonotonicity and lower or upper sign-continuity. Applications to quasi-optimization and traffic network are also considered.     

集值混合变分不等式的一类自适应惯性投影次梯度算法

本文在实Hilbert空间上引入了一类求解集值混合变分不等式新的自适应惯性投影次梯度算法.在集值映射T为f-强伪单调或单调的条件下,我们证明了由该自适应惯性投影次梯度算法所产生的序列强收敛于集值混合变分不等式问题的的唯一解.     

Technical Note Discontinuous Implicit Quasivariational Inequalities in Normed Spaces

In this paper, we consider an implicit quasivariational inequality without continuity assumptions in normed spaces. The main result (Theorem 2.1) provides an infinite-dimensional version of Theorem 3.2 in Ref. 1. To achieve such a goal, we employ Theorem 3.2 in Ref. 1 and the technique of Cubiotti in Ref. 2. In particular, Theorem 3.1 covers a recent result of Cubiotti (Theorem 3.1 of Ref. 2) as a special case. Communicated by F. Giannessi This research was partially supported by the National Science Council of Taiwan, ROC.     

广义集值强非线性混合似变分不等式的辅助原理和三步迭代算法

使用辅助原理技巧研究了一类广义集值强非线性混合变分不等式．证明了此类集值强非线性混合变分不等式辅助问题解的存在性和唯一性;构建了一个新的三步迭代算法,通过辅助原理技巧,构建并计算此类非线性混合变分不等式的近似解,进一步证明非线性混合变分不等式解的存在性以及由算法产生的三个序列的收敛性．所得结论推广了近年来许多混合变分不等式和准变分不等式以及他们的有关结果．     

An Extension of a Class of Iterative Procedures for Nonlinear Quasivariational Inequalities

Consider the convergence of the projection methods based on an extension of a special class of algorithms for the approximation--solvability of the following class of nonlinear quasivariational inequality (NQVI) problems: find an element such that and