Some Novel Aspects of Quasi Variational Inequalities

Abstract

Quasi variational inequalities can be viewed as novel generalizations of the variational inequalities and variational principles, the origin of which can be traced back to Euler, Lagrange, Newton and Bernoulli's brothers. It is well known that quasi-variational inequalities are equivalent to the implicit fixed point problems. We consider this alternative equivalent fixed point formulation to suggest some new iterative methods for solving quasi-variational inequalities and related optimization problems using projection methods, Wiener-Hopf equations, dynamical systems, merit function and nonexpansive mappings. Convergence analysis of these methods is investigated under suitable conditions. Our results present a significant improvement of previously known methods for solving quasi variational inequalities and related optimization problems. Since the quasi variational inequalities include variational inequalities and complementarity problems as special cases. Results obtained in this paper continue to hold for these problems. Some special cases are discussed as applications of the main results. The implementation of these algorithms and comparison with other methods need further efforts.