A self-adaptive method for solving general mixed variational inequalities

The general mixed variational inequality containing a nonlinear term φ is a useful and an important generalization of variational inequalities. The projection method cannot be applied to solve this problem due to the presence of the nonlinear term. To overcome this disadvantage, Noor [M.A. Noor, Pseudomonotone general mixed variational inequalities, Appl. Math. Comput. 141 (2003) 529-540] used the resolvent equations technique to suggest and analyze an iterative method for solving general mixed variational inequalities. In this paper, we present a new self-adaptive iterative method which can be viewed as a refinement and improvement of the method of Noor. Global convergence of the new method is proved under the same assumptions as Noor's method. Some preliminary computational results are given.     

Algorithms for solutions of extended general mixed variational inequalities and fixed points

In this paper, we introduce and study a new class of extended general nonlinear mixed variational inequalities and a new class of extended general resolvent equations and establish the equivalence between the extended general nonlinear mixed variational inequalities and implicit fixed point problems as well as the extended general resolvent equations. Then by using this equivalent formulation, we discuss the existence and uniqueness of solution of the problem of extended general nonlinear mixed variational inequalities. Applying the aforesaid equivalent alternative formulation and a nearly uniformly Lipschitzian mapping S, we construct some new resolvent iterative algorithms for finding an element of set of the fixed points of nearly uniformly Lipschitzian mapping S which is the unique solution of the problem of extended general nonlinear mixed variational inequalities. We study convergence analysis of the suggested iterative schemes under some suitable conditions. We also suggest and analyze a class of extended general resolvent dynamical systems associated with the extended general nonlinear mixed variational inequalities and show that the trajectory of the solution of the extended general resolvent dynamical system converges globally exponentially to the unique solution of the extended general nonlinear mixed variational inequalities. The results presented in this paper extend and improve some known results in the literature.     

Mixed quasi complementarity problems in topological vector spaces

In this paper, we introduce and consider a new class of complementarity problems, which is called the mixed quasi complementarity problems in a topological vector space. We show that the mixed quasi complementarity problems are equivalent to the mixed quasi variational inequalities. Using the KKM mapping theorem, we study the existence of a solution of the mixed quasi variational inequalities and mixed quasi complementarity problems. Several special cases are also discussed. Results obtained in this paper can be viewed as extension and generalization of the previously known results.     

Self-adaptive methods for mixed quasi-variational inequalities

It is well known that the variational inequalities involving the nonlinear term φ are equivalent to the fixed-point problems and the resolvent equations. In this paper, we use these alternative equivalent formulations to suggest and analyze some new self-adaptive iterative methods for solving mixed quasi-variational inequalities. Our results can be viewed as significant extensions of the previously known results for mixed quasi-variational inequalities. An example is given to illustrate the efficiency of the proposed method.     

On a system of general mixed variational inequalities

In this paper, we introduce and consider a new system of general mixed variational inequalities involving three different operators. Using the resolvent operator technique, we establish the equivalence between the general mixed variational inequalities and the fixed point problems. We use this equivalent formulation to suggest and analyze some new explicit iterative methods for this system of general mixed variational inequalities. We also study the convergence analysis of the new iterative method under certain mild conditions. Since this new system includes the system of mixed variational inequalities involving two operators, variational inequalities and related optimization problems as special cases, results obtained in this paper continue to hold for these problems. Our results can be viewed as a refinement and improvement of the previously known results for variational inequalities.     

Generalized set-valued mixed nonlinear quasi variational inequalities

In this paper we introduce and study a number of new classes of quasi variational inequalities. Using essentially the projection technique and its variant forms we prove that the generalized set-valued mixed quasivariational inequalities are equivalent to the fixed point problem and the Wiener-Hopf equations (normal maps). This equivalence enables us to suggest a number of iterative algorithms for solving the generalized variational inequalities. As a special case of the generalized set-valued mixed quasi variational inequalities, we obtain a class of quasi variational inequalities studied by Siddiqi, Husain and Kazmi [35], but there are several inaccuracies in their formulation of the problem, the statement and the proofs of their results. We have removed these inaccuracies. The correct formulation of their results can be obtained as special cases from our main results.     

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.     

Merit functions for general mixed quasi-variational inequalities

In this paper, we present some merit functions for general mixed quasi-variational inequalities, and we obtain the equivalent optimization problems to general mixed quasi-variational inequalities. Since the general mixed quasi-variational inequalities include general variational inequalities, quasi-variational inequalities and nonlinear (implicit) complementarity problems as special cases, our results continue to hold for these problems. In this respect, results obtained in this paper represent an extension of previously known results.     

Projection methods for a generalized system of nonconvex variational inequalities with different nonlinear operators

In this paper, we introduce and consider a new generalized system of nonconvex variational inequalities with different nonlinear operators. We establish the equivalence between the generalized system of nonconvex variational inequalities and the fixed point problems using the projection technique. This equivalent alternative formulation is used to suggest and analyze a general explicit projection method for solving the generalized system of nonconvex variational inequalities. Our results can be viewed as a refinement and improvement of the previously known results for variational inequalities.     

New inexact implicit method for general mixed quasi variational inequalities

In this paper, we suggest and analyze a new self-adaptive inexact implicit method with a variable parameter for general mixed quasi variational inequalities, where the skew-symmetry of the nonlinear bifunction plays a crucial part in the convergence analysis of this method. We use a self-adaptive technique to adjust parameter ρ at each iteration. The global convergence of the proposed method is proved under some mild conditions. Preliminary numerical results indicate that the self-adaptive adjustment rule is necessary in practice. Muhammad Aslam Noor is supported by the Higher Education Commission, Pakistan, through research grant No: 1-28/HEC/HRD/2005/90.     

On General Mixed Variational Inequalities

In this paper, we introduce and consider a new class of mixed variational inequalities, which is called the general mixed variational inequality. Using the resolvent operator technique, we establish the equivalence between the general mixed variational inequalities and the fixed-point problems as well as resolvent equations. We use this alternative equivalent formulation to suggest and analyze some iterative methods for solving the general mixed variational inequalities. We study the convergence criteria of the suggested iterative methods under suitable conditions. Using the resolvent operator technique, we also consider the resolvent dynamical systems associated with the general mixed variational inequalities. We show that the trajectory of the dynamical system converges globally exponentially to the unique solution of the general mixed variational inequalities. Our methods of proofs are very simple as compared with others’ techniques. Results proved in this paper may be viewed as a refinement and important generalizations of the previous known results.     

Extended general variational inequalities

In this work, we introduce and consider a new class of general variational inequalities involving three nonlinear operators, which is called the extended general variational inequalities. Noor [M. Aslam Noor, Projection iterative methods for extended general variational inequalities, J. Appl. Math. Comput. (2008) (in press)] has shown that the minimum of nonconvex functions can be characterized via these variational inequalities. Using a projection technique, we establish the equivalence between the extended general variational inequalities and the general nonlinear projection equation. This equivalent formulation is used to discuss the existence of a solution of the extended general variational inequalities. Several special cases are also discussed.     

Some iterative schemes for general mixed variational inequalities

In this paper, we consider a class of variational inequalities which is called the general mixed variational inequality. It is known that the general mixed variational inequalities are equivalent to the fixed point problems. This equivalent formulation is used to suggest and analyze some three-step iterative schemes for finding the common element of the set of fixed points of a nonexpansive mappings and the set of solutions of the mixed variational inequalities. We also study the convergence criteria of three-step iterative method under some mild conditions. Our results include the previous results as special cases and may be considered as an improvement and refinement of the previously known results.     

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.     

Iterative methods for solving general quasi-variational inequalities

In this paper, we introduce a new class of variational inequalities, which is called the general quasi-variational inequality. We establish the equivalence among the general quasi variational inequality and implicit fixed point problems and the Wiener–Hopf equations. We use this equivalent formulation to discuss the existence of a solution of the general quasi-variational inequality. This equivalent formulation is used to suggest and analyze some iterative algorithms for solving the general quasi-variational inequality. We also discuss the convergence analysis of these iterative methods. Several special cases are also discussed.     

Optimal control of multivalued quasi variational inequalities

Optimal control of various variational problems has been an area of active research. On the other hand, in recent years many important models in mechanics and economics have been formulated as multi-valued quasi variational inequalities. The primary objective of this work is to study optimal control of the general nonlinear problems of this type. Under suitable conditions, we ensure the existence of an optimal control for a quasi variational inequality with multivalued pseudo-monotone maps. Convergence behavior of the control is studied when the data for the state quasi variational inequality is contaminated by some noise. Some possible applications are discussed.     

Iterative algorithms for a new class of extended general nonconvex set-valued variational inequalities

Some new classes of extended general nonconvex set-valued variational inequalities and the extended general Wiener-Hopf inclusions are introduced. By the projection technique, equivalence between the extended general nonconvex set-valued variational inequalities and the fixed point problems as well as the extended general nonconvex Wiener-Hopf inclusions is proved. Then by using this equivalent formulation, we discuss the existence of solutions of the extended general nonconvex set-valued variational inequalities and construct some new perturbed finite step projection iterative algorithms with mixed errors for approximating the solutions of the extended general nonconvex set-valued variational inequalities. We also verify that the approximate solutions obtained by our algorithms converge to the solutions of the extended general nonconvex set-valued variational inequalities. The results presented in this paper extend and improve some known results from the literature.     

Self-adaptive methods for general variational inequalities

It is well known that the general variational inequalities are equivalent to the fixed point problems and the Wiener-Hopf equations. In this paper, we use these alternative equivalent formulations to suggest and analyze some new self-adaptive iterative methods for solving the general variational inequalities. Our results can be viewed as a significant extension of the previously known results for variational inequalities. An example is given to illustrate the efficiency of the proposed method.     

Resolvent Algorithms for Mixed Quasivariational Inequalities

It is well known that mixed quasivariational inequalities are equivalent to implicit resolvent equations. We use this alternative equivalent formulation to suggest and analyze a new modified resolvent method for solving mixed quasivariational inequalities and related problems. We show that the convergence of this modified method requires only pseudomonotonicity, which is a weaker condition than monotonicity. Since mixed quasivariational inequalities include various classes of variational inequalities as special cases, our result continues to hold for these problems.     

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.