We study implicit multifunctions (set-valued mappings) obtained from inclusions of the form 0 ∈ M(p, x), where M is a multifunction. Our basic implicit multifunction theorem provides an approximation for a generalized derivative of the implicit multifunction in terms of the derivative of the multifunction M. Our primary focus is on three special cases of inclusions 0 ∈ M(p, x) which represent different kinds of generalized variational inequalities, called ‘variational conditions’. Appropriate versions of our basic implicit multifunction theorem yield approximations for generalized derivatives of the solutions to each kind of variational condition. We characterize a well-known generalized Lipschitz property in terms of generalized derivatives, and use our implicit multifunction theorems to state sufficient conditions (and necessary in one case) for solutions of variational conditions to possess this Lipschitz property. We apply our results to a general parameterized nonlinear programming problem, and derive a new second-order condition which guarantees that the stationary points associated with the Karush–Kuhn–Tucker conditions exhibit generalized Lipschitz continuity with respect to the parameter.
