Slopes, Error Bounds and Weak Sharp Pareto Minima of a Vector-Valued Map

Abstract

In this paper, we provide a detailed study of the upper and lower slopes of a vector-valued map recently introduced by Bednarczuk and Kruger. We show that these slopes enjoy most properties of the strong slope of a scalar-valued function and can be explicitly computed or estimated in the convex, strictly differentiable, linear cases. As applications, we obtain error bounds for lower level sets (in particular, a Hoffman-type error bound for a system of linear inequalities in the infinite-dimensional space setting, existence of weak sharp Pareto minima) and sufficient conditions for Pareto minima.

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

References

  1. 1.

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

    MathSciNet  Article  Google Scholar 

  2. 2.

    Lojasiewicz, S.: Sur le problème de la division. Stud. Math. 18, 87–136 (1959)

    Article  MATH  Google Scholar 

  3. 3.

    Ferris, M. C.: Weak Sharp Minima and Penalty Functions in Mathematical Programming. PhD thesis, University of Cambridge, Cambridge (1988)

  4. 4.

    Polyak, B. T.: Sharp Minima. Institute of Control Sciences Lecture Notes, Moscow, USSR, 1979. Presented at the IIASA workshop on generalized Lagrangians and their applications, IIASA, Laxenburg, Austria (1979)

  5. 5.

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

  6. 6.

    Lewis, A. S., Pang, J.-S.: Error bounds for convex inequality systems. Generalized Convexity, Generalized Monotonicity: Recent Results (Luminy, 1996), 75–110. In: Nonconvex Optimization and Its Applications, vol. 27. Kluwer Acad. Publ., Dordrecht (1998)

  7. 7.

    Pang, J.S.: Error bounds in mathematical programming. Math. Program. 79, 299–332 (1997)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Bednarczuk, E.M., Kruger, A.Y.: Error bounds for vector-valued functions: necessary and sufficient conditions. Nonlinear Anal. 75, 1124–1140 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Bednarczuk, E.M., Kruger, A.Y.: Error bounds for vector-valued functions on metric spaces. Vietnam J. Math. 40(2–3), 165–180 (2012)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Burke, J.V., Deng, S.: Weak sharp minima revisited. Part I: basic theory. Control Cybern. 31, 439–469 (2002)

    MATH  Google Scholar 

  11. 11.

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

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Studniarski, M., Ward, D.E.: Weak sharp minima: characterizations and sufficient conditions. SIAM J. Control Optim. 38, 219–238 (1999)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Bednarczuk, E. M.: Stability analysis for parametric vector optimization problems. Dissertationes Mathematicae, 42, Warsaw (2007)

  14. 14.

    Bednarczuk, E.M.: On weak sharp minima in vector optimization with applications to parametric problems. Control Cybern. 36, 563–570 (2007)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Liu, C.G., Ng, K.F., Yang, W.H.: Merit functions in vector optimization. Math. Program. Ser. A 119, 215–237 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Ng, K.F., Zheng, X.Y.: Global weak sharp minima on Banach spaces. SIAM J. Control Optim. 41(6), 1868–1885 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Studniarski, M.: Weak sharp minima in multiobjective optimization. Control Cybern. 36, 925–937 (2007)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    De Giorgi, E., Marino, A., Tosques, M.: Problemi di evoluzione in spazi metrici e curve di massima pendenza (Evolution problems in metric spaces and curves of maximal slope). Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 68, 180–187 (1980)

  19. 19.

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

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Hiriart-Urruty, J.B.: New concepts in nondifferentiable programming. Bull. Soc. Math. France 60, 57–85 (1979)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Zaffaroni, A.: Degrees of efficiency and degrees of minimality. SIAM J. Control Optim. 42(3), 1071–1086 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Coulibaly, A., Crouzeix, J.-P.: Condition numbers and error bounds in convex programming. Math. Program. Ser. B 116, 79–113 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Jahn, J.: Vector Optimization—Theory, Applications, and Extensions, 2nd edn. Springer, Berlin (2011)

    Google Scholar 

  24. 24.

    Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    Google Scholar 

  25. 25.

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

    Google Scholar 

  26. 26.

    Zheng, X.Y., Ng, K.F.: Hoffman least error bounds for systems of linear inequalities. J. Glob. Optim. 30, 391–403 (2004)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Acknowledgements

The research was carried out during the author’s stays at the Vietnam Institute for Advanced Study in Mathematics and the Institute of Mathematics, University of Erlangen-Nuremberg, under the Georg Forster grant of the Alexander von Humboldt Foundation, and was partially supported by NAFOSTED, Grant 101.01-2017.20. The author thanks the editor and the referees for the helpful comments and suggestions, which allowed to improve the paper.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Xuan Duc Ha Truong.

Additional information

Communicated by Marcin Studniarski.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Truong, X.D.H. Slopes, Error Bounds and Weak Sharp Pareto Minima of a Vector-Valued Map. J Optim Theory Appl 176, 634–649 (2018). https://doi.org/10.1007/s10957-018-1240-6

Download citation

Keywords

  • Vector-valued map
  • Slope
  • Error bound
  • Weak sharp Pareto minima

Mathematics Subject Classification

  • 49J53
  • 58C06
  • 90C29