Lipschitzian stability of parametric variational inequalities over perturbed polyhedral convex sets

Abstract

In this paper we investigate the Lipschitz-like property of the solution mapping of parametric variational inequalities over perturbed polyhedral convex sets. By establishing some lower and upper estimates for the coderivatives of the solution mapping, among other things, we prove that the solution mapping could not be Lipschitz-like around points where the positive linear independence condition is invalid. Our analysis is based heavily on the Mordukhovich criterion (Mordukhovich in Variational Analysis and Generalized Differentiation. vol. I: Basic Theory, vol. II: Applications. Springer, Berlin, 2006) of the Lipschitz-like property for set-valued mappings between Banach spaces and recent advances in variational analysis. The obtained result complements the corresponding ones of Nam (Nonlinear Anal 73:2271–2282, 2010) and Qui (Nonlinear Anal 74:1674–1689, 2011).