A Condition Number Theorem for Variational Inequalities under Operator Perturbations

Nonlinear variational inequalities in Banach spaces are considered with fixed right-hand side and constraint set. Condition number and distance between inequalities are defined taking into account perturbations of the operators defining the inequalities. It is shown that the distance of suitably restricted inequalities to those which are ill-conditioned is bounded below and above by multiples of the reciprocal of the condition number. This extends known results for variational inequalities when the right-hand side is perturbed, and for optimisation problems.