, Volume 74, Issue 3, pp 333350
First online:
Implicit multifunction theorems for the sensitivity analysis of variational conditions
 A. B. LevyAffiliated withDepartment of Mathematics, Bowdoin College
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We study implicit multifunctions (setvalued 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 wellknown 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 secondorder condition which guarantees that the stationary points associated with the KarushKuhnTucker conditions exhibit generalized Lipschitz continuity with respect to the parameter.
Keywords
Implicit mapping Sensitivity analysis Variational condition Upper Lipschitz continuity Title
 Implicit multifunction theorems for the sensitivity analysis of variational conditions
 Journal

Mathematical Programming
Volume 74, Issue 3 , pp 333350
 Cover Date
 199609
 DOI
 10.1007/BF02592203
 Print ISSN
 00255610
 Online ISSN
 14364646
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Implicit mapping
 Sensitivity analysis
 Variational condition
 Upper Lipschitz continuity
 Industry Sectors
 Authors

 A. B. Levy ^{(1)}
 Author Affiliations

 1. Department of Mathematics, Bowdoin College, 04011, Brunswick, ME, USA