Advertisement

Positivity

, Volume 22, Issue 1, pp 219–244 | Cite as

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

  • Tiến Sơn Phạm
  • Xuân Ɖức Hà TrươngEmail author
  • Jen-Chih Yao
Article
  • 121 Downloads

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.

Keywords

Global weak sharp minima with explicit exponents Vector optimization Polynomials 

Mathematics Subject Classification

49J53 58C06 90C29 

Notes

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.

References

  1. 1.
    Bednarczuk, E.M.: Weak sharp efficiency and growth condition for vector-valued functions with applications. Optimization 53, 455–474 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Burke, J.V., Ferris, M.C.: Weak sharp minima in mathematical programming. SIAM J. Control Optim. 31, 1340–1359 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)zbMATHGoogle 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)CrossRefzbMATHGoogle Scholar
  5. 5.
    Deng, S., Yang, X.Q.: Weak sharp minima in multicriteria programming. SIAM J. Optim. 15, 456–460 (2004)MathSciNetCrossRefzbMATHGoogle 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 CrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)zbMATHGoogle Scholar
  10. 10.
    Ekeland, I.: Nonconvex minimization problems. Bull. Am. Math. Soc. 1, 443–474 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ferris, M.C.: Weak sharp minima and penalty functions in mathematical programming. Ph.D. Thesis, University of Cambridge, Cambridge (1988)Google Scholar
  12. 12.
    Hà, H.V., Phạm, T.S.: Genericity in Polynomial Optimization. World Scientific Publishing, Singapore (2017)CrossRefzbMATHGoogle Scholar
  13. 13.
    Hoffman, A.J.: On approximate solutions of linear inequalities. J. Res. Natl. Bur. Stand. 49, 263–265 (1952)MathSciNetCrossRefGoogle 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)MathSciNetzbMATHGoogle 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)Google Scholar
  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)CrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Luo, Z.Q., Pang, J.S.: Error bounds for analytic systems and their applications. Math. Program. 67, 1–28 (1994)MathSciNetCrossRefzbMATHGoogle 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)CrossRefGoogle Scholar
  22. 22.
    Mordukhovich, B.S.: Multiobjective optimization problems with equilibrium constraints. Math. Program. Ser. B 117, 331–354 (2009)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Phạm, T.S.: An explicit bound for the Łojasiewicz exponent of real polynomials. Kodai Math. J. 35(2), 311–319 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Pang, J.S.: Error bounds in mathematical programming. Math. Program. Ser. B 79, 299–332 (1997)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Polyak, B.T.: Introduction to Optimization. Optimization Software, New York (1987)zbMATHGoogle Scholar
  27. 27.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)CrossRefzbMATHGoogle Scholar
  28. 28.
    Studniarski, M.: Weak sharp minima in multiobjective optimization. Control Cybernet. 36, 925–937 (2007)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Studniarski, M., Ward, D.E.: Weak sharp minima: characterizations and sufficient conditions. SIAM J. Control Optim. 38, 219–236 (1999)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Ye, J.J., Zhu, D.L.: Optimality conditions for bilevel programming problems. Optimization 33(1), 9–27 (1995)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Tiến Sơn Phạm
    • 1
    • 2
  • Xuân Ɖức Hà Trương
    • 3
    Email author
  • Jen-Chih Yao
    • 4
  1. 1.Division of Computational Mathematics and Engineering, Institute for Computational ScienceTon Duc Thang UniversityHo Chi Minh CityVietnam
  2. 2.Faculty of Mathematics and StatisticsTon Duc Thang UniversityHo Chi Minh CityVietnam
  3. 3.Institute of MathematicsVietnam Academy of Science and TechnologyHanoiVietnam
  4. 4.Center for General Education, China Medical UniversityTaichungTaiwan

Personalised recommendations