Mathematical Programming

, Volume 104, Issue 2–3, pp 635–668 | Cite as

Subgradient of distance functions with applications to Lipschitzian stability



The paper is devoted to studying generalized differential properties of distance functions that play a remarkable role in variational analysis, optimization, and their applications. The main object under consideration is the distance function of two variables in Banach spaces that signifies the distance from a point to a moving set. We derive various relationships between Fréchet-type subgradients and limiting (basic and singular) subgradients of this distance function and corresponding generalized normals to sets and coderivatives of set-valued mappings. These relationships are essentially different depending on whether or not the reference point belongs to the graph of the involved set-valued mapping. Our major results are new even for subdifferentiation of the standard distance function signifying the distance between a point and a fixed set in finite-dimensional spaces. The subdifferential results obtained are applied to deriving efficient dual-space conditions for the local Lipschitz continuity of distance functions generated by set-valued mappings, in particular, by those arising in parametric constrained optimization.


Variational analysis and optimization Distance functions Generalized differentiation Lipschitzian stability 

Mathematics Subject Classification (1991)

20E28 20G40 20C20 


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  1. 1.
    Aubin, J.-P.: Lipschitz behavior of solutions to convex minimization problems. Math. Oper. Res. 9, 87–111 (1984)Google Scholar
  2. 2.
    Borwein, J.M., Giles, J.R.: The proximal normal formula in Banach space. Trans. Am. Math. Soc. 302, 371–381 (1987)Google Scholar
  3. 3.
    Borwein, J.M., Zhu, Q.J.: Techniques of Variational Analysis. An Introduction. Springer, New York (to appear)Google Scholar
  4. 4.
    Bounkhel, M.: Scalarization of the normal Fréchet regularity of set-valued mappings. New Zealand J. Math. 33, 1–18 (2004)Google Scholar
  5. 5.
    Bounkhel, M., Thibault, L.: On various notions of regularity of sets in nonsmooth analysis. Nonlinear Anal. 48, 223–246 (2002)CrossRefGoogle Scholar
  6. 6.
    Burke, J., Ferris, M.C., Quian, M.: On the Clarke subdifferential of the distance function of a closed set. J. Math. Anal. Appl. 166, 199–213 (1992)CrossRefGoogle Scholar
  7. 7.
    Clarke, F.H., Ledyaev, Y.S., Stern, R.J., Wolenski, P.R.: Nonsmooth Analysis and Control Theory. Springer, New York, 1998Google Scholar
  8. 8.
    Clarke, F.H., Stern, R.J., Wolenski, P.R.: Proximal smoothness and lower-C2 property. J. Convex Anal. 2, 117–144 (1995)Google Scholar
  9. 9.
    Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)CrossRefGoogle Scholar
  10. 10.
    Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, Volume I and II. Springer, New York, 2003Google Scholar
  11. 11.
    Ioffe, A.D.: Proximal analysis and approximate subdifferentials. J. London Math. Soc. 2, 175–192 (1990)Google Scholar
  12. 12.
    Jourani, A., Thibault, L.: Metric regularity and subdifferential calculus in Banach space. Set-Valued Anal. 3, 87–100 (1995)CrossRefGoogle Scholar
  13. 13.
    Kruger, A.Y.: Epsilon-semidifferentials and epsilon-normal elements. Depon. VINITI 1331, Moscow (1981)Google Scholar
  14. 14.
    Kruger, A.Y., Mordukhovich, B.S.: Extremal points and the Euler equation in nonsmooth optimization. Dokl. Akad. Nauk BSSR 24, 684–687 (1980)Google Scholar
  15. 15.
    Mordukhovich, B.S.: Maximum principle in problems of time optimal control with nonsmooth constraints. J. Appl. Math. Mech. 40, 960–969 (1976)CrossRefGoogle Scholar
  16. 16.
    Mordukhovich, B.S.: Approximation Methods in Problems of Optimization and Control. Nauka, Moscow, 1988Google Scholar
  17. 17.
    Mordukhovich, B.S.: Complete characterizations of openness, metric regularity, and Lipschitzian properties of multifunctions. Trans. Am. Math. Soc. 340, 1–36 (1993)Google Scholar
  18. 18.
    Mordukhovich, B.S.: Coderivative analysis of variational systems. J. Global Optim. 28, 347–362 (2004)CrossRefGoogle Scholar
  19. 19.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, Vol. I: Basic Theory, Vol. II: Applications. Springer, Berlin (to appear)Google Scholar
  20. 20.
    Mordukhovich, B.S., Shao, Y.: Nonsmooth sequential analysias in Asplund spaces. Trans. Am. Math. Soc. 348, 1235–1280 (1996)CrossRefGoogle Scholar
  21. 21.
    Mordukhovich, B.S., Treiman, J.S., Zhu, Q.J.: An extended extremal principle with applications to multiobjective optimization. SIAM J. Optim. 14, 359–379 (2003)CrossRefGoogle Scholar
  22. 22.
    Ngai, N.V., Théra, M.: Metric regularity, subdifferential calculus and applications. Set-Valued Anal. 9, 187–216 (2001)CrossRefGoogle Scholar
  23. 23.
    Phelps, R.R.: Convex Functions, Monotone Operators and Differentiability, 2nd edn. Springer, Berlin, 1993Google Scholar
  24. 24.
    Rockafellar, R.T.: Lipschitzian properties of multifunctions. Nonlinear Anal. 9, 867–885 (1985)CrossRefGoogle Scholar
  25. 25.
    Rockafellar, R.T., Wets, R. J.-B.: Variational Analysis. Springer, Berlin, 1998Google Scholar
  26. 26.
    Thibault, L.: On subdifferentials of optimal value functions. SIAM J. Control Optim. 29, 1019–1036 (1991)CrossRefGoogle Scholar
  27. 27.
    Thibault, L.: Sweeping process with regular and nonregular sets. J. Diff. Eq. 193, 1–26 (2003)CrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Department of MathematicsWayne State UniversityDetroitUSA

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