A Lower Bound for a Condition Number Theorem of Variational Inequalities

Nonlinear variational inequalities in Banach spaces are considered. A suitable notion of condition number with respect to the right-hand side is introduced. A distance among variational inequalities is defined. Based on a new criterion about the Lipschitz property of set-valued mappings, it is shown that the distance to suitably restricted ill-conditioned variational inequalities is bounded from below by the reciprocal of the condition number. By using a similar upper bound of the companion paper "An upper bound for a condition number theorem of variational inequalities", we obtain a full condition number theorem for variational inequalities. The particular case of convex minimization problems is considered. Known results dealing with optimization problems are thereby generalized.