Class Of Quasi-variational Inequalities Research Articles

Inertial projection methods for solving general quasi-variational inequalities

In this paper, we consider a new class of quasi-variational inequalities, which is called the general quasi-variational inequality. Using the projection operator technique, we establish the equivalence between the general quasi-variational inequalities and the fixed point problems. We use this alternate formulation to propose some new inertial iterative schemes for solving the general quasi-variational inequalities. The convergence criteria of the new inertial projection methods under some appropriate conditions is investigated. Since the general quasi-variational inequalities include the quasi-variational inequalities, variational inequalities, complementarity problems and the related optimization problems as special cases, our results continue to hold for these problems. It is an interesting problem to compare the efficiency of the proposed methods with other known methods.

Nonsingularity and Stationarity Results for Quasi-Variational Inequalities

The optimality system of a quasi-variational inequality can be reformulated as a non-smooth equation or a constrained equation with a smooth function. Both reformulations can be exploited by algorithms, and their convergence to solutions usually relies on the nonsingularity of the Jacobian, or the fact that the merit function has no nonoptimal stationary points. We prove new sufficient conditions for the absence of nonoptimal constrained or unconstrained stationary points that are weaker than some known ones. All these conditions exploit some properties of a certain matrix, but do not require the nonsingularity of the Jacobian. Further, we present new necessary and sufficient conditions for the nonsingularity of the Jacobian that are based on the signs of certain determinants. Additionally, we consider generalized Nash equilibrium problems that are a special class of quasi-variational inequalities. Exploiting their structure, we also prove some new sufficient conditions for stationarity and nonsingularity results.

Three step iterative algorithms for solving a class of quasi variational inequalities

In this paper, we discuss the existence of a solution of extended general quasi variational inequalities using projection technique. Using the technique of Noor, we suggest and analyze a new class of three-step iterative schemes for solving extended general quasi variational inequalities. Under some certain conditions on operators, we also discuss the convergence of the proposed iterative scheme. Results proved in this paper continue to hold for previously known and new classes of quasi variational inequalities. Ideas and techniques of this paper may stimulate further research in this field. The implementation and comparison of the proposed new algorithms is an interesting problem for further research.

A canonical duality approach for the solution of affine quasi-variational inequalities

We propose a new formulation of the Karush---Kunt---Tucker conditions of a particular class of quasi-variational inequalities. In order to reformulate the problem we use the Fisher---Burmeister complementarity function and canonical duality theory. We establish the conditions for a critical point of the new formulation to be a solution of the original quasi-variational inequality showing the potentiality of such approach in solving this class of problems. We test the obtained theoretical results with a simple heuristic that is demonstrated on several problems coming from the academy and various engineering applications.

On a Smooth Dual Gap Function for a Class of Quasi-Variational Inequalities

A well-known technique for the solution of quasi-variational inequalities (QVIs) consists in the reformulation of this problem as a constrained or unconstrained optimization problem by means of so-called gap functions. In contrast to standard variational inequalities, however, these gap functions turn out to be nonsmooth in general. Here, it is shown that one can obtain an unconstrained optimization reformulation of a class of QVIs with affine operator by using a continuously differentiable dual gap function. This extends an idea from Dietrich (J. Math. Anal. Appl. 235:380---393 [24]). Some numerical results illustrate the practical behavior of this dual gap function approach.

Solving Quasi-Variational Inequalities for Image Restoration with Adaptive Constraint Sets

We consider a class of quasi-variational inequalities (QVIs) for adaptive image restoration, where the adaptivity is described via solution-dependent constraint sets. In previous work we studied both theoretical and numerical issues. While we were able to show the existence of solutions for a relatively broad class of problems, we encountered difficulties concerning uniqueness of the solution as well as convergence of existing algorithms for solving QVIs. In particular, it seemed that with increasing image size the growing condition number of the involved differential operator posed severe problems. In the present paper we prove uniqueness for a larger class of problems, particularly independent of the image size. Moreover, we provide a numerical algorithm with proved convergence. Experimental results support our theoretical findings.

Some Iterative Schemes for Solving Extended General Quasi Variational Inequalities

In this paper, we consider a new class of quasi variational inequalities involving three operators, which is called the extended general quasi variational inequality. It is shown that the extended general quasi variational inequalities are equivalent to the fixed problems. This equivalence is used to suggest and analyze some iterative methods for solving the extended general quasi variational inequalities. Convergence analysis is also considered. We have also shown that the extended general quasi variational inequalities are equivalent to the extended general implicit Wiener-Hopf equations. This alternative formulation is used to suggest and analyze some iterative methods. The convergence analysis of these new methods under some suitable conditions is investigated. Several special cases are discussed. Since the extended general quasi variational inequalities include general variational inequalities, quasi variational inequalities and related optimization problems as special cases, results proved in this paper continue to hold for these problems. Results of this paper may stimulate further research in this fascinating area.

Generalized nonlinear mixed quasi-variational inequalities

In this paper, we introduce and study a new class of quasi-variational inequalities, which is called the generalized nonlinear set-valued mixed quasi-variational inequality. Using the resolvent operator technique for maximal monotone mapping, we construct some new iterative algorithms for solving this class of generalized nonlinear set-valued mixed quasi-variational inequalities. We prove the existence of solution for this kind of generalized nonlinear set-valued mixed quasivariational inequalities without compactness and the convergence of iterative sequences generated by the algorithms. We also discuss the convergence and stability of perturbed iterative algorithm for solving a class of generalized nonlinear mixed quasi-variational inequalities.

Strongly Nonlinear Generalized Equations

We study a class of generalized equations which includes as a special case various classes of quasi variational inequalities introduced by Noor, Isac, Siddiqi, and Ansari. Such problems occur in mathematical programming and in various other branches of pure and applied mathematics. We also propose two iterative schemes for the numerical solution of the problems considered.

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.

A Newton method for a class of quasi-variational inequalities

A variant of the Newton method for nonsmooth equations is applied to solve numerically quasivariational inequalities with monotone operators. For this purpose, we investigate the semismoothness of a certain locally Lipschitz operator coming from the quasi-variational inequality, and analyse the generalized Jacobian of this operator to ensure local convergence of the method. A simplified variant of this approach, applicable to implicit complementarity problems, is also studied. Small test examples have been computed.

Some further generalizations of Ky Fan's minimax inequality and its applications to variational inequalities

The purpose of this paper is to introduce the concept of generalized KKM mapping and to obtain some general version of the famous KKM theorem and Ky Fan's minimax inequality. As applications, we utilize the results presented in this paper to study the saddle point problem and the existence problem of solutions for a class of quasi-variational inequalities. The results obtained in this paper extend and improve some recent results of [1–6].

Quasi variational inequalities

Projection Technique is used to suggest a unified and general iterative algorithm for computing the approximate solution of a new class of quasi variational inequalities. The convergence properties of this algorithm are also considered. Several special cases which can be obtained from the general results are also discussed.

An iterative scheme for a class of quasi variational inequalities

An iterative scheme is given to obtain the approximate solution of a class of quasi variational inequalities. It is shown that the approximate solution obtained by the iterative scheme converges strongly in the Hilbert space to the exact solution. As a special case, we obtain the corresponding iterative scheme for variational inequalities.