Set-Valued Analysis

, Volume 7, Issue 1, pp 89–99 | Cite as

Nonsingularity Conditions for Multifunctions

  • A. B. Levy


We discuss three different characterizations of continuity properties for general multifunctions S : Rd ⇒ Rn. Each of these characterizations is given by the same simple nonsingularity condition, but stated in terms of three different generalized derivatives. Two of these characterizations are known, but the third is new to this paper. We discuss how all three have immediate analogues as generalized inverse mapping theorems, and we apply our new characterization to develop a fundamental and very broad sensitivity theorem for solutions to parameterized optimization problems.

coderivative outer graphical derivative strict derivative pseudo-Lipschitz upper Lipschitz inverse mapping theorem optimal solution stationary point sensitivity analysis 


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  1. 1.
    Aubin, J. P.: Lipschitz behavior of solutions to convex minimization problems, Math.Oper.Res. 9 (1984), 87–111.Google Scholar
  2. 2.
    Aubin, J. P. and Frankowska, H.: Set-Valued Analysis, Birkhäuser, Basel, 1990.Google Scholar
  3. 3.
    Bonnans, J. F.: Local analysis of Newton-type methods for variational inequalities and nonlinear programming, Appl.Math.Optim. 29 (1994), 161–186.Google Scholar
  4. 4.
    Dontchev, A. L.: Characterizations of Lipschitz stability in optimization, in: R. Lucchetti and J. Revalski (eds), Recent Developments in Well-Posed Problems, Kluwer Acad. Publ., Dordrecht, 1995, pp. 95–115.Google Scholar
  5. 5.
    Dontchev, A. L. and Hager, W. W.: Implicit functions, Lipschitz maps, and stability in optimization, Math.Oper.Res. 19 (1994), 297–326.Google Scholar
  6. 6.
    Gowda, M. S. and Pang, J.-S.: Stability analysis of variational inequalities and nonlinear complementarity problems, via the mixed linear complementarity problem and degree theory, Math.Oper.Res. 19 (1994), 831–879.Google Scholar
  7. 7.
    King, A. J. and Rockafellar, R. T.: Sensitivity analysis for nonsmooth generalized equations, Math.Programming 55 (1992), 193–212.Google Scholar
  8. 8.
    Kummer, B.: Lipschitzian inverse functions, directional derivatives, and applications in C 1;1 optimization, J.Optim.Theory Appl. 70 (1991).Google Scholar
  9. 9.
    Levy, A. B.: Calm minima in parameterized finite-dimensional optimization, submitted for publication, 1998.Google Scholar
  10. 10.
    Levy, A. B.: Lipschitzian multifunctions and a Lipschitzian inverse mapping theorem, Submitted for publication, 1998.Google Scholar
  11. 11.
    Levy, A. B.: Implicit multifunction theorems for the sensitivity analysis of variational conditions, Math.Programming 74 (1996), 333–350.Google Scholar
  12. 12.
    Levy, A. B., Poliquin, R. A. and Rockafellar, R. T.: Stability of locally optimal solutions, Submitted for publication, 1998.Google Scholar
  13. 13.
    Mordukhovich, B.: Approximation Methods in Problems of Optimization and Control, Nauka, Moscow, 1988.Google Scholar
  14. 14.
    Mordukhovich, B.: Sensitivity analysis in nonsmooth optimization, in: D. A. Field and V. Komkov (eds), Theoretical Aspects of Industrial Design, SIAM J.Appl.Math. 58 (1992), pp. 32–46.Google Scholar
  15. 15.
    Mordukhovich, B.: Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions, Trans.Amer.Math.Soc. 340 (1993), 1–35.Google Scholar
  16. 16.
    Mordukhovich, B.: Generalized differential calculus of nonsmooth and set-valued mappings, J.Math.Anal.Appl. 183 (1994), 250–288.Google Scholar
  17. 17.
    Pang, J.-S.: A degree-theoretic approach to parametric nonsmooth equations with multivalued perturbed solution sets, Math.Programming 62 (1993), 359–383.Google Scholar
  18. 18.
    Pang, J.-S.: 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.Google Scholar
  19. 19.
    Rockafellar, R. T. and Wets, R. J.-B.: Variational Analysis, Springer-Verlag, New York, 1998.Google Scholar

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© Kluwer Academic Publishers 1999

Authors and Affiliations

  • A. B. Levy
    • 1
  1. 1.Department of MathematicsBowdoin CollegeBrunswickU.S.A.

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