Strong Lipschitz Stability of Stationary Solutions for Nonlinear Programs and Variational Inequalities

Abstract

The stationary solution map X of a canonically perturbed nonlinear program or variational condition is studied. We primarily aim at characterizing X to be locally single-valued and Lipschitz near some stationary point x0 of an initial problem, where our focus is on characterizations which are explicitly given in terms of the original functions and assigned quadratic problems. Since such criteria are closely related to a nonsingularity property of the strict graphical derivative of X, explicit formulas for this derivative are presented, too. It turns out that even for convex polynomial problems our stability (and the Aubin property) does not depend only on the derivatives, up to some fixed order, of the problem functions at x0 . This is in contrast to various other stability concepts. Further, we clarify completely the relations to Kojima's strong stability and present simplifications for linearly and certain linearly quadratically constrained problems, convex programs, and for the map of global minimizers as well.