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Implicit multifunction theorems for the sensitivity analysis of variational conditions
 A. B. Levy
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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.
References
[1]
J.P. Aubin and H. Frankowska,Setvalued Analysis (Birkhäuser, Basel, 1990).MATH
[2]
J.F. Bonnans, Local analysis of Newtontype methods for variational inequalities and nonlinear programming,Applied Mathematics and Optimization 29 (1994) 161–186.MATHCrossRefMathSciNet
[3]
A.L. Dontchev, Implicit function theorems for generalized equations,Mathematical Programming 70 (1995) 91–106.MathSciNet
[4]
A.L. Dontchev and W.W. Hager, On Robinson's implicit function theorem, in:SetValued Analysis and Differential Inclusions (Birkhäuser, Basel, 1991).
[5]
A.L. Dontchev and W.W. Hager, Lipschitzian stability in nonlinear control and optimization,SIAM Journal on Control and Optimization 31 (1993) 569–603.MATHCrossRefMathSciNet
[6]
A.L. Dontchev and W.W. Hager, Implicit functions. Lipschitz maps, and stability in optimization,Mathematics of Operations Research 19 (1994) 753–768.MATHMathSciNet
[7]
A.L. Dontchev, Characterizations of Lipschitz stability in optimization, in: R. Lucchetti and J. Revaslki, eds.,Recent Developments in Wellposed Problems (Kluwer Academic Publishers, Dordrecht, 1995). pp. 95–115.
[8]
A.V. Fiacco and J. Kyparisis, Sensitivity analysis in nonlinear programming under second order assumptions, in: A.V. Balakrishnan and E.M. Thoma, eds.,Lecture Notes in Control and Information Sciences, Vol. 66 (Springer, Berlin, 1985) pp. 74–97.
[9]
A.J. King and R.T. Rockafellar, Sensitivity analysis for nonsmooth generalized equations,Mathematical Programming 55 (1992) 193–212.CrossRefMathSciNet
[10]
D. Klatte, Nonlinear optimization under data perturbations, in: W. Krabs and J. Zowe, eds.,Modern Methods of Optimization (Springer, Berlin, 1992) pp. 204–235.
[11]
J. Kyparisis, Parametric variational inequalities with multivalued solution sets,Mathematics of Operations Research 17 (1992) 341–364.MATHMathSciNet
[12]
A.B. Levy and R.T. Rockafellar, Sensitivity analysis of solutions to generalized equations,Transactions of the American Mathematical Society 345 (1994) 661–671.MATHCrossRefMathSciNet
[13]
A.B. Levy and R.T. Rockafellar, Sensitivity of solutions in nonlinear programming problems with nonunique multipliers, in: D.Z. Du, L. Qi and R.S. Womersley, eds.,Recent Advances in Nonsmooth Optimization (World Scientific, Singapore, 1995) pp. 215–223.
[14]
A.B. Levy and R.T. Rockafellar, Variational conditions and the protodifferentiation of partial subgradient mappings, to appear inNonlinear Analysis, Theory, Methods, and Applications 26 (1996) 1951–1964.
[15]
A.B. Levy and R.T. Rockafellar, Protoderivatives and the geometry of solution mappings in nonlinear programming, to appear inProceedings of EriceConference (1995).
[16]
J.S. Pang, A degreetheoretic approach to parametric nonsmooth equations with multivalued perturbed solution sets,Mathematical Programming 62 (1993) 359–383.CrossRefMathSciNet
[17]
J.S. Pang, Necessary and sufficient conditions for solution stability of parametric nonsmooth equations, in: D.Z. Du, L. Qi and R.S. Womersley, eds.,Recent Advances in Nonsmooth Optimization (World Scientific, Singapore, 1995) pp. 261–288.
[18]
M.S. Gowda and J.S. Pang, Stability analysis of variational inequalities and nonlinear complementarity problems, via the mixed linear complementarity problem and degree theory,Mathematics of Operations Research 19 (1994) 831–879.MATHMathSciNet
[19]
Y. Qiu and T.L. Magnanti, Sensitivity analysis for variational inequalities,Mathematics of Operations Research 17 (1992) 61–76.MATHMathSciNetCrossRef
[20]
S.M. Robinson, Generalized equations and their solutions, Part I: Basic theory,Mathematical Programming Study 10 (1979) 128–141.MATH
[21]
S.M. Robinson, Strongly regular generalized equations,Mathematics of Operations Research 5 (1980) 43–62.MATHMathSciNet
[22]
S.M. Robinson, Generalized equations and their solutions. Part II: Applications to nonlinear programming,Mathematical Programming Study 19 (1982) 200–221.MATH
[23]
S.M. Robinson, Local structure of feasible sets in nonlinear programming, Part III: Stability and sensitivity,Mathematical Programming Study 30 (1987) 45–66.MATH
[24]
S.M. Robinson, An implicitfunction theorem for a class of nonsmooth functions,Mathematics of Operations Research 16 (1991) 292–309.MATHMathSciNet
[25]
R.T. Rockafellar, Protodifferentiability of setvalued mappings and its applications in optimization, in: H. Attouch, J.P. Aubin, F.H. Clarke and I. Ekeland, eds.,Analyse Non Linéaire (GauthierVillars, Paris, 1989) pp. 449–482.
[26]
R.T. Rockafellar, Nonsmooth analysis and parametric optimization, in: A. Cellina, ed.,Methods of Nonconvex Analysis, Lecture Notes in Mathematics, Vol. 1446 (Springer, Berlin, 1990) pp. 137–151.CrossRef
[27]
A. Shapiro, Sensitivity analysis of nonlinear programs and differentiability properties of metric projections,SIAM Journal on Control and Optimization 26 (1988) 628–645.MATHCrossRefMathSciNet
 Title
 Implicit multifunction theorems for the sensitivity analysis of variational conditions
 Journal

Mathematical Programming
Volume 74, Issue 3 , pp 333350
 Cover Date
 19960901
 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