A solution differentiability result for evolutionary quasi-variational inequalities

We consider a class of evolutionary quasi-variational inequalities arising in the study of some network equilibrium problems. First we prove the existence and uniqueness of solutions and, subsequently, present a differentiability result based on projection arguments.     

The semismooth Newton method for the solution of quasi-variational inequalities

We consider the application of the globalized semismooth Newton method to the solution of (the KKT conditions of) quasi variational inequalities. We show that the method is globally and locally superlinearly convergent for some important classes of quasi variational inequality problems. We report numerical results to illustrate the practical behavior of the method.     

On first-order quasi-variational inequalities with integral terms

This paper deals with first-order quasi-variational inequalities with integral terms associated with impulsive and switching control of piecewise-deterministic processes. Two formulations of quasi-variational inequalities are studied, characteristic and viscosity, and the relations between them are discussed. As a tool we apply convex analysis methods.     

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.     

Numerical solution of quasi-variational inequalities arising in stochastic game theory

We study the finite-difference approximation for the quasi-variational inequalities for a stochastic game involving discrete actions of the players and continuous and discrete payoff. We prove convergence of iterative schemes for the solution of the discretized quasi-variational inequalities, with estimates of the rate of convergence (via contraction mappings) in two particular cases. Further, we prove stability of the finite-difference schemes, and convergence of the solution of the discrete problems to the solution of the continuous problem as the discretization mesh goes to zero. We provide a direct interpretation of the discrete problems in terms of finite-state, continuous-time Markov processes.     

On a class of discontinuous vector-valued functions and the associated quasi-variational inequalities

We study in detail a class of discontinuous vector-valued functions defined on a closed convex subset of Rn, which was introduced by B. Ricceri [7] and which is very useful in the theory of variational inequalities. The results are used to give a new proof for the existence theorem due to P. Cubiotti [3]. The proof allows us to have a better understanding of quasi-variational inequalities associated with the abovementioned class of functions.     

Regularization of quasi-variational inequalities

An ill-posed quasi-variational inequality with contaminated data can be stabilized by employing the elliptic regularization. Under suitable conditions, a sequence of bounded regularized solutions converges strongly to a solution of the original quasi-variational inequality. Moreover, the conditions that ensure the boundedness of regularized solutions, become sufficient solvability conditions. It turns out that the regularization theory is quite strong for quasi-variational inequalities with set-valued monotone maps but restrictive for generalized pseudo-monotone maps. The results are quite general and are applicable to ill-posed variational inequalities, hemi-variational inequalities, inverse problems, and split feasibility problem, among others.     

On an existence theorem for generalized quasi-variational inequalities

We obtain some characteristic properties of a subclass of multifunctions introduced by B. Ricceri and give a new proof for the result of P. Cubiotti on the existence of solutions to generalized quasi-variational inequalities involving multifunctions from the class.     

Internal approximation of quasi-variational inequalities

Summary In this paper the internal approximation of a quasi-variational inequality is considered. An algorithm of Bensoussan-Lions type is proposed for which the convergence is proved. These results are applied to Signorini problem with friction for which two error estimates and numerical examples are also given.     

Non-Zenoness of a class of differential quasi-variational inequalities

The Zeno phenomenon of a switched dynamical system refers to the infinite number of mode switches in finite time. The absence of this phenomenon is crucial to the numerical simulation of such a system by time-stepping methods and to the understanding of the behavior of the system trajectory. Extending a previous result for a strongly regular differential variational inequality, this paper establishes that a certain class of non-strongly regular differential variational inequalities is devoid of the Zeno phenomenon. The proof involves many supplemental results that are of independent interest. Specialized to a frictional contact problem with local compliance and polygonal friction laws, this non-Zenoness result is of fundamental significance and the first of its kind. This work was based on research partially supported by the National Science Foundation under grants DMS-0508986 and IIS-0413227 awarded to Rensselaer Polytechnic Institute, where the original version of the paper was first written. The revision was based on research partially supported by the National Science Foundation under grant DMS awarded to the University of Illinois at Urbana-Champaign.     

Inverse problems for quasi-variational inequalities

In this short note, our aim is to investigate the inverse problem of parameter identification in quasi-variational inequalities. We develop an abstract nonsmooth regularization approach that subsumes the total variation regularization and permits the identification of discontinuous parameters. We study the inverse problem in an optimization setting using the output-least squares formulation. We prove the existence of a global minimizer and give convergence results for the considered optimization problem. We also discretize the identification problem for quasi-variational inequalities and provide the convergence analysis for the discrete problem. We give an application to the gradient obstacle problem.     

Operator inclusions and quasi-variational inequalities

The operator inclusion 0 ∈ A(x) + N(x) is studied. Themain results are concerned with the case where A is a bounded monotone-type operator from a reflexive space to its dual and N is a cone-valued operator. A criterion for this inclusion to have no solutions is obtained. Additive and homotopy-invariant integer characteristics of set-valued maps are introduced. Applications to the theory of quasi-variational inequalities with set-valued operators are given.     

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.This work has been supported in parts by a grant from the German Scientific Foundation and by a grant from the Czech Academy of Sciences.     

Existence theorem for a class of generalized quasi-variational inequalities

In this paper we consider a class of generalized quasi-variational inequalities. The variational problem is studied in the convex set \(X\times Y\) , with \(Y\) bounded and \(X\) unbounded. In the latter settings, we investigate about the solvability of the problem. In particular, by using the perturbation theory, we give an existence result of the solution without requesting any coercivity hypothesis on the operator. Finally, we give an application to the obtained theoretical results in terms of an economic equilibrium problem.     

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.