The global weak sharp minima with explicit exponents in polynomial vector optimization problems

Abstract

In this paper we discuss the global weak sharp minima property for vector optimization problems with polynomial data. Exploiting the imposed polynomial structure together with tools of variational analysis and a quantitative version of Łojasiewicz’s gradient inequality due to D’Acunto and Kurdyka, we establish the Hölder type global weak sharp minima with explicitly calculated exponents.

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Notes

  1. 1.

    By semialgebraic function, we mean any function whose graph can be described by finitely many intersections and unions of polynomial sublevel sets or level sets.

References

  1. 1.

    Bednarczuk, E.M.: Weak sharp efficiency and growth condition for vector-valued functions with applications. Optimization 53, 455–474 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    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 

  3. 3.

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

    Google Scholar 

  4. 4.

    D’Acunto, D., Kurdyka, K.: Explicit bounds for the Łojasiewicz exponent in the gradient inequality for polynomials. Ann. Pol. Math. 87, 51–61 (2005)

    Article  MATH  Google Scholar 

  5. 5.

    Deng, S., Yang, X.Q.: Weak sharp minima in multicriteria programming. SIAM J. Optim. 15, 456–460 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Dinh, S.T., Hà, H.V., Thao, N.T.: Łojasiewicz inequality for polynomial functions on non compact domains. Int. J. Math. 23(4), 1250033 (2012). doi:10.1142/S0129167X12500334

    Article  MATH  Google Scholar 

  7. 7.

    Dinh, S.T., Hà, H.V., Phạm, T.S., Thao, N.T.: Global Łojasiewicz-type inequality for non-degenerate polynomial maps. J. Math. Anal. Appl. 410(2), 541–560 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Dinh, S.T., Hà, H.V., Phạm, T.S.: Hölder-type global error bounds for non-degenerate polynomial systems (2014). arXiv:1411.0859 [math.OC]

  9. 9.

    Ehrgott, M.: Multicriteria Optimization, 2nd edn. Springer, Berlin (2005)

    Google Scholar 

  10. 10.

    Ekeland, I.: Nonconvex minimization problems. Bull. Am. Math. Soc. 1, 443–474 (1979)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Ferris, M.C.: Weak sharp minima and penalty functions in mathematical programming. Ph.D. Thesis, University of Cambridge, Cambridge (1988)

  12. 12.

    Hà, H.V., Phạm, T.S.: Genericity in Polynomial Optimization. World Scientific Publishing, Singapore (2017)

    Google Scholar 

  13. 13.

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

    MathSciNet  Article  Google Scholar 

  14. 14.

    Jongen, HTh, Rückmann, J.-J., Stein, O.: Generalized semi-infinite optimization: a first order optimality condition and examples. Math. Program. 83, 145–158 (1998)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Kurdyka, K., Spodzieja, S.: Separation of real algebraic sets and the Łojasiewicz exponent. Proc. Amer. Math. Soc., 142 (9) S 0002-993912061-2, 3089–3102 (2014)

  16. 16.

    Li, G., Mordukhovich, B.S., Phạm, T.S.: New fractional error bounds for polynomial systems with applications to Höderian stability in optimization and spectral theory of tensors. Math. Program. Ser. A 153(2), 333–362 (2015)

    Article  MATH  Google Scholar 

  17. 17.

    Li, G., Mordukhovich, B.S., Nghia, T.T.A., Phạm, T.S.: Error bounds for parametric polynomial systems with applications to higher-order stability analysis and convergence rates. Math. Program. doi:10.1007/s10107-016-1014-6

  18. 18.

    Liu, C.G., Ng, K.F., Yang, W.H.: Merit functions in vector optimization. Math. Program. Ser. A 119(2), 215–237 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Luo, Z.Q., Pang, J.S.: Error bounds for analytic systems and their applications. Math. Program. 67, 1–28 (1994)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Miettinen, K.: Nonlinear Multiobjective Optimization, International Series in Operations Research & Management Science, 12. Kluwer Academic Publishers, Boston (1999)

    Google Scholar 

  21. 21.

    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory; II: Applications. Springer, Berlin (2006)

    Google Scholar 

  22. 22.

    Mordukhovich, B.S.: Multiobjective optimization problems with equilibrium constraints. Math. Program. Ser. B 117, 331–354 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    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 

  24. 24.

    Phạm, T.S.: An explicit bound for the Łojasiewicz exponent of real polynomials. Kodai Math. J. 35(2), 311–319 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

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

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Polyak, B.T.: Introduction to Optimization. Optimization Software, New York (1987)

    Google Scholar 

  27. 27.

    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)

    Google Scholar 

  28. 28.

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

    MathSciNet  MATH  Google Scholar 

  29. 29.

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

    MathSciNet  Article  MATH  Google Scholar 

  30. 30.

    Wierzbicki, A.P.: On the completeness and constructiveness of parametric characterizations to vector optimization problems. OR Spektrum 8(2), 73–87 (1986)

    MathSciNet  Article  MATH  Google Scholar 

  31. 31.

    Yang, X.Q., Yen, N.D.: Structure and weak sharp minimum of the Pareto solution set for piecewise linear multiobjective optimization. J. Optim. Theory Appl. 147(1), 113–124 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  32. 32.

    Ye, J.J., Zhu, D.L.: Optimality conditions for bilevel programming problems. Optimization 33(1), 9–27 (1995)

    MathSciNet  Article  MATH  Google Scholar 

  33. 33.

    Zheng, X.Y., Yang, X.Q.: Weak sharp minima for piecewise linear multiobjective optimization in normed spaces. Nonlinear Anal. 68(12), 3771–3779 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  34. 34.

    Zheng, X.Y., Yang, X.Q.: Conic positive definiteness and sharp minima of fractional orders in vector optimization problems. J. Math. Anal. Appl. 391, 619–629 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  35. 35.

    Zheng, X.Y., Ng, K.F.: Metric regularity of piecewise linear multifunction and applications to piecewise linear multiobjective optimization. SIAM J. Optim. 24(1), 154–174 (2014)

    MathSciNet  Article  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the editor and the referees for useful remarks and comments which allow to improve the paper. This work was performed during research visits of the first and the second authors at the Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung, Taiwan and at the Vietnam Institute for Advanced Study of Mathematics. These authors wish to thank the mentioned organizations for hospitality and support.

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Correspondence to Xuân Ɖức Hà Trương.

Additional information

Tiến Sơn Phạm was partially supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED), Grant 101.04-2016.05. Xuân Ɖức Hà Trương was partially supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED). Jen-Chih Yao was partially supported by the Grant MOST 105-2115-M-039-002-MY3.

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Phạm, T.S., Trương, X.Ɖ.H. & Yao, J. The global weak sharp minima with explicit exponents in polynomial vector optimization problems. Positivity 22, 219–244 (2018). https://doi.org/10.1007/s11117-017-0509-6

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Keywords

  • Global weak sharp minima with explicit exponents
  • Vector optimization
  • Polynomials

Mathematics Subject Classification

  • 49J53
  • 58C06
  • 90C29