IMPLICIT MULTIFUNCTION THEOREMS FOR THE SENSITIVITY ANALYSIS OF VARIATIONAL CONDITIONS

We study implicit multifunctions (set-valued mappings) obtained from i nclusions of the form 0 is an element of M(p, x), where M is a multifu nction. Our basic implicit multifunction theorem provides an approxima tion for a generalized derivative of the implicit multifunction in ter ms of the derivative of the multifunction M. Our primary focus is on t hree special cases of inclusions 0 is an element of M(p, x) which repr esent different kinds of generalized variational inequalities, called ''variational conditions''. Appropriate versions of our basic implicit multifunction theorem yield approximations for generalized derivative s of the solutions to each kind of variational condition. We character ize a well-known generalized Lipschitz property in terms of generalize d derivatives, and use our implicit multifunction theorems to state su fficient conditions (and necessary in one case) for solutions of varia tional conditions to possess this Lipschitz property. We apply our res ults to a general parameterized nonlinear programming problem, and der ive a new second-order condition which guarantees that the stationary points associated with the Karush-Kuhn-Tucker conditions exhibit gener alized Lipschitz continuity with respect to the parameter.