Skip to main content
SpringerLink
Log in

Global Error Bounds for γ-paraconvex Multifunctions

  • Published:
Set-Valued and Variational Analysis Aims and scope Submit manuscript

Abstract

In this paper, error bounds for γ-paraconvex multifunctions are considered. A Robinson-Ursescu type Theorem is given in normed spaces. Some results on the existence of global error bounds are presented. Perturbation error bounds are also studied.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Azé, D.: A unified theory for metric regularity of multifunctions. J. Convex Anal. 13(2), 225–252 (2006)

    MathSciNet  MATH  Google Scholar 

  2. Azé, D., Corvellec, J.N.: On the sensitivity analysis of Hoffman constants for systems of linear inequalities. SIAM J. Optim. 12(4), 913–927 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Azé, D., Corvellec, J.N.: Characterizations of error bounds for lower semicontinuous functions on metric spaces. ESAIM, Controle Optim. Calc. Var. 10(3), 409–425 (2004)

    Article  MATH  Google Scholar 

  4. Bosch, P., Jourani, A., Henrion, R.: Sufficient conditions for error bounds and applications. Appl. Math. Optim. 50(2), 161–181 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Burke, J.V.: Calmness and exact penalization. SIAM J. Control Optim. 29(2), 493–497 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  6. Burke, J.V., Deng, S.: Weak sharp minima revisited. I. Basic theory. Control Cybern. 31(3), 439-469 (2002)

    MathSciNet  MATH  Google Scholar 

  7. Burke, J.V., Deng, S.: Weak sharp minima revisited. II. Application to linear regularity and error bounds. Math. Program., Ser. B 104(2–3), 235–261 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Burke, J.V., Ferris, M.C.: Weak sharp minima in mathematical programming. SIAM J. Control Optim. 31(5), 1340–1359 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  9. Clarke, F.H.: Optimization and Nonsmooth Analysis. Canadian Mathematical Society Series of Monographs and Advanced Texts. Wiley, New York, A Wiley-Interscience Publication (1983)

  10. Corvellec, J.N., Motreanu, V.V.: Nonlinear error bounds for lower semicontinuous functions on metric spaces. Math. Program., Ser. A 114(2), 291–319 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Deng, S.: Global error bounds for convex inequality systems in Banach spaces. SIAM J. Control Optim. 36(4), 1240-1249 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dontchev, A.L., Rockafellar, R.T.: Regularity and conditioning of solution mappings in variational analysis. Set-Valued Anal. 12(1–2), 79–109 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fabian, M.J., Henrion, R., Kruger, A.Y., Outrata, J.V.: Error bounds: Necessary and sufficient conditions. Set-Valued Anal. 18, 121–149 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Henrion, R., Jourani, A.: Subdifferential conditions for calmness of convex constraints. SIAM J. Optim. 13(2), 520–534 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. Henrion, R., Outrata, J.V.: A subdifferential condition for calmness of multifunctions. J. Math. Anal. Appl. 258(1), 110–130 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  16. Henrion, R., Outrata, J.V.: Calmness of constraint systems with applications. Math. Program., Ser. B 104(2), 437–464 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hoffman, A.J.: On approximate solutions of systems of linear inequalities. J. Res. Natl. Bur. Stand. 49, 263–265 (1952)

    Google Scholar 

  18. Huang, H.: Inversion theorem for nonconvex multifunctions. Math. Inequal. Appl. 13(4), 841–849 (2010)

    MathSciNet  Google Scholar 

  19. Ioffe, A.D.: Necessary and sufficient conditions for a local minimum. I. A reduction theorem and first order conditions. SIAM J. Control Optim. 17(2), 245–250 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ioffe, A.D.: Regular points of Lipschitz functions. Trans. Am. Math. Soc. 251, 61–69 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ioffe, A.D., Outrata, J.V.: On metric and calmness qualification conditions in subdifferential calculus. Set-Valued Anal. 16(2–3), 199–227 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Jourani, A.: Hoffman’s error bound, local controllability, and sensitivity analysis. SIAM J. Control Optim. 38(3), 947–970 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  23. Jourani, A.: Weak regularity of functions and sets in Asplund spaces. Nonlinear Anal. 65(3), 660–676 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Jourani, A.: Radiality and semismoothness. Control Cybern. 36(3), 669–680 (2007)

    MathSciNet  MATH  Google Scholar 

  25. Jourani, A.: Open mapping theorem and inversion theorem for γ-paraconvex multivalued mappings and applications. Studia Mathematica. 117(2), 123–136 (1996)

    MathSciNet  MATH  Google Scholar 

  26. Lewis, A.S., Pang, J.S.: Error bounds for convex inequality systems. In: Generalized Convexity, Generalized Monotonicity: Recent Results (Luminy, 1996), Nonconvex Optim. Appl., vol. 27, pp. 75–110. Kluwer Acad. Publ., Dordrecht (1998)

  27. Li, W., Singer, I.: Global error bounds for convex multifunctions and applications. Math. Oper. Res. 23(2), 443–462 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  28. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. I: Basic Theory, Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 330. Springer, Berlin (2006)

    Google Scholar 

  29. Ng, K.F., Yang, W.H.: Regularities and their relations to error bounds. Math. Program., Ser. A 99, 521–538 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  30. Ng, K.F., Zheng, X.Y.: Characterizations of error bounds for convex multifunctions on Banach spaces. Math. Oper. Res. 29(1), 45–63 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  31. Ng, K.F., Zheng, X.Y.: Error bounds for lower semicontinuous functions in normed spaces. SIAM J. Optim. 12(1), 1–17 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  32. Ngai, H.V., Kruger, A.Y., Thera, M.: Stability of error bounds for semi-infinite convex constraint systems. SIAM J. Optim. 20(4), 2080–2096 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  33. Ngai, H.V., Thera, M.: Error bounds in metric spaces and application to the perturbation stability of metric regularity. SIAM J. Optim. 19(1), 1–20 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  34. Ngai, H.V., Thera, M.: Error bounds for systems of lower semicontinuous functions in Asplund spaces. Math. Program., Ser. B 116(1–2), 397–427 (2009)

    Article  MATH  Google Scholar 

  35. Pang, J.S.: Error bounds in mathematical programming. Math. Program., Ser. B 79(1–3), 299–332 (1997) (Lectures on Mathematical Programming (ISMP97) (Lausanne, 1997))

    MATH  Google Scholar 

  36. Robinson, S.M.: Regularity and stability for convex multivalued functions. Math. Oper. Res. 1(2), 130–143 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  37. Rolewicz, S.: On γ-paraconvex multifunctions. Math. Jpn. 24(3), 293–300 (1979)

    MathSciNet  MATH  Google Scholar 

  38. Studniarski, M., Ward, D.E.: Weak sharp minima: characterizations and sufficient conditions. SIAM J. Control Optim. 38(1), 219–236 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  39. Ursescu C.: Multifunctions with convex closed graph. Czechoslov. Math. J. 25(100), 438–441 (1975)

    MathSciNet  Google Scholar 

  40. Wu, Z., Ye, J.J.: Sufficient conditions for error bounds. SIAM J. Optim. 12(2), 421–435 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  41. Wu, Z., Ye, J.J.: On error bounds for lower semicontinuous functions. Math. Program., Ser. A 92(2), 301–314 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  42. Ye, J.J., Ye, X.Y.: Necessary optimality conditions for optimization problems with variational inequality constraints. Math. Oper. Res. 22(4), 977–997 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  43. Zălinescu, C.: A nonlinear extension of Hoffman’s error bounds for linear inequalities. Math. Oper. Res. 28(3), 524–532 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  44. Zheng, X.Y.: Error bounds for set inclusion. Sci. China, Ser. A 46(6), 750–763 (2003)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Yunnan University, Kunming, 650091, People’s Republic of China

    Hui Huang & Runxin Li

Authors
  1. Hui Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Runxin Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hui Huang.

Additional information

The research was supported by the Natural Science Foundation of Yunnan Province (2009CD011) and the Foundation of Yunnan University (21132014).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Huang, H., Li, R. Global Error Bounds for γ-paraconvex Multifunctions. Set-Valued Anal 19, 487–504 (2011). https://doi.org/10.1007/s11228-010-0172-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11228-010-0172-6

Keywords

Mathematics Subject Classifications (2010)

Search

Navigation