Mathematical Programming

, Volume 74, Issue 3, pp 333–350

Implicit multifunction theorems for the sensitivity analysis of variational conditions

  • A. B. Levy

DOI: 10.1007/BF02592203

Cite this article as:
Levy, A.B. Mathematical Programming (1996) 74: 333. doi:10.1007/BF02592203


We study implicit multifunctions (set-valued mappings) obtained from inclusions of the form 0∈M(p,x), whereM 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 multifunctionM. 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.


Implicit mapping Sensitivity analysis Variational condition Upper Lipschitz continuity 

Copyright information

© The Mathematical Programming Society, Inc. 1996

Authors and Affiliations

  • A. B. Levy
    • 1
  1. 1.Department of MathematicsBowdoin CollegeBrunswickUSA