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Error Bounds: Necessary and Sufficient Conditions

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Abstract

The paper presents a general classification scheme of necessary and sufficient criteria for the error bound property incorporating the existing conditions. Several derivative-like objects both from the primal as well as from the dual space are used to characterize the error bound property of extended-real-valued functions on a Banach space.

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References

  1. Azé, D.: A survey on error bounds for lower semicontinuous functions. In: Proceedings of 2003 MODE-SMAI Conference, ESAIM Proc., vol. 13, pp. 1–17. EDP Sci., Les Ulis (2003)

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

    MATH  MathSciNet  Google Scholar 

  3. 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  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  7. Burke, J.V., Deng, S.: Weak sharp minima revisited. I. Basic theory. Control Cybernet. 31(3), 439–469 (2002). Well-Posedness in Optimization and Related Topics (Warsaw, 2001)

    MATH  MathSciNet  Google Scholar 

  8. 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  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  10. Clarke, F.H.: A new approach to Lagrange multipliers. Math. Oper. Res. 1(2), 165–174 (1976)

    Article  MATH  MathSciNet  Google Scholar 

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

  12. Cornejo, O., Jourani, A., Zălinescu, C.: Conditioning and upper-Lipschitz inverse subdifferentials in nonsmooth optimization problems. J. Optim. Theory Appl. 95(1), 127–148 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  13. 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  MATH  MathSciNet  Google Scholar 

  14. De Giorgi, E., Marino, A., Tosques, M.: Problems of evolution in metric spaces and maximal decreasing curve. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 68(3), 180–187 (1980)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  16. 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  MATH  MathSciNet  Google Scholar 

  17. Fabian, M., Habala, P., Hájek, P., Montesinos Santalucía, V., Pelant, J., Zizler, V.: Functional Analysis and Infinite-Dimensional Geometry. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 8. Springer-Verlag, New York (2001)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  20. Henrion, R., Outrata, J.V.: Calmness of constraint systems with applications. Math. Program. 104(2–3, Ser. B), 437–464 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  21. Hoffman, A.J.: On approximate solutions of systems of linear inequalities. J. Research Nat. Bur. Standards 49, 263–265 (1952)

    MathSciNet  Google Scholar 

  22. 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  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  24. Ioffe, A.D.: Metric regularity and subdifferential calculus. Russian Math. Surveys 55, 501–558 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  25. 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  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  29. Klatte, D., Kummer, B.: Nonsmooth Equations in Optimization, Nonconvex Optimization and its Applications, vol. 60. Kluwer Academic Publishers, Dordrecht (2002) Regularity, calculus, methods and applications

    Google Scholar 

  30. Klatte, D., Kummer, B.: Stability of inclusions: characterizations via suitable Lipschitz functions and algorithms. Optimization 55(5–6), 627–660 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  31. Kruger, A.Y.: Generalized differentials of nonsmooth functions. Deposited in VINITI no. 1332–81. Minsk (1981, in Russian)

  32. Kruger, A.Y.: ε-semidifferentials and ε-normal elements. Deposited in VINITI no. 1331-81. Minsk (1981, in Russian)

  33. Kruger, A.Y.: Strict (ε,δ)-subdifferentials and extremality conditions. Optimization 51(3), 539–554 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  34. Kruger, A.Y.: On Fréchet subdifferentials. J. Math. Sci. (N. Y.) 116(3), 3325–3358 (2003), Optimization and Related Topics, 3

    Article  MATH  MathSciNet  Google Scholar 

  35. 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)

    Google Scholar 

  36. Łojasiewicz, S.: Sur le problème de la division. Studia Math. 18, 87–136 (1959)

    MATH  MathSciNet  Google Scholar 

  37. Mifflin, R.: Semismooth and semiconvex functions in constrained optimization. SIAM J. Control Optim. 15(6), 959–972 (1977)

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  41. Ngai, H.V., Kruger, A.Y., Théra, M.: Stability of error bounds for semi-infinite convex constraint systems. SIAM J. Optim. 20 (2010)

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

    Article  MATH  MathSciNet  Google Scholar 

  43. Ngai, H.V., Théra, 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 

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

    Google Scholar 

  45. Penot, J.P.: Error bounds, calmness and their applications in nonsmooth analysis (2010, to be published)

  46. Polyak, B.T.: Introduction to Optimization. Translations Series in Mathematics and Engineering. Optimization Software Inc. Publications Division, New York (1987, translated from Russian)

  47. Rockafellar, R.T.: Directionally Lipschitzian functions and subdifferential calculus. Proc. London Math. Soc. (3) 39(2), 331–355 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  48. Rockafellar, R.T., Wets, R.J.B.: Variational Analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 317. Springer-Verlag, Berlin (1998)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  50. Wu, Z., Ye, J.J.: Sufficient conditions for error bounds. SIAM J. Optim. 12(2), 421–435 (2001/02)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  52. 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  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Mathematical Institute, Academy of Sciences of the Czech Republic, Žitná 25, 11567, Prague 1, Czech Republic

    Marian J. Fabian

  2. Weierstrass Institute for Applied Analysis and Stochastics, 10117, Berlin, Germany

    René Henrion

  3. School of Information Technology and Mathematical Sciences, Centre for Informatics and Applied Optimization, University of Ballarat, POB 663, Ballarat, Vic, 3350, Australia

    Alexander Y. Kruger

  4. Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, 18208, Prague, Czech Republic

    Jiří V. Outrata

Authors
  1. Marian J. Fabian
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  2. René Henrion
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  3. Alexander Y. Kruger
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  4. Jiří V. Outrata
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Corresponding author

Correspondence to Alexander Y. Kruger.

Additional information

The research of Marian J. Fabian was supported by Institutional Research Plan of the Academy of Sciences of Czech Republic AVOZ 101 905 03, and by GAČR 201/07/0394.

The research of René Henrion was supported by the DFG Research Center Matheon “Mathematics for key technologies” in Berlin.

The main structure of the article was developed during Alexander Y. Kruger’s stay at the Institute of Information Theory and Automation of the Academy of Sciences of the Czech Republic in July–August 2008; this author is grateful to the Institute for support and hospitality.

The research of Jiří V. Outrata was supported by IAA 100750802 of the Grant Agency of the Czech Academy of Sciences.

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Fabian, M.J., Henrion, R., Kruger, A.Y. et al. Error Bounds: Necessary and Sufficient Conditions. Set-Valued Anal 18, 121–149 (2010). https://doi.org/10.1007/s11228-010-0133-0

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  • DOI: https://doi.org/10.1007/s11228-010-0133-0

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