Skip to main content
SpringerLink
Log in

Convergence of relaxed minimizers in set optimization

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

In this paper we investigate, in a unified way, the stability of several relaxed minimizers of set optimization problems. To this end, we introduce a topology on vector ordered spaces from which we derive a concept of convergence that allows us to study both the upper and the lower stability of the sets of relaxed minimizers we consider.

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. Bao, T.Q., Mordukhovich, B.S.: Relative Pareto minimizers for multiobjective problems: existence and optimality conditions. Math. Program. Ser. A 122(2), 301–347 (2010)

  2. Borwein, J., Goebel, R.: Notions of relative interior in Banach spaces, optimization and related topics, 1. J. Math. Sci. (N. Y.) 115(4), 2542–2553 (2003)

  3. Borwein, J., Lewis, A.S.: Partially finite convex programming. I. Quasi relative interiors and duality theory. Math. Program. Ser. B 57(1), 15–48 (1992) II. Explicit lattice models. Math. Program. Ser. B 57(1) 49–83 (1992)

  4. Braides, A.: A handbook of \(\Gamma \)-convergence. In: Chipot, M., Quittner, P. (Eds.) Handbook of Differential Equations. Stationary Partial Differential Equations, vol. 3, pp. 101–213. Elsevier, Amsterdam (2006)

  5. Chicco, M., Mignanego, F., Pusillo, L., Tijs, S.: Vector optimization problems via improvement sets. J. Optim. Theory Appl. 150(3), 516–529 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  6. De Giorgi, E., Franzoni, T.: Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 58(6), 842–850 (1975)

  7. Dudley, R.M.: On sequential convergence. Trans. Am. Math. Soc. 112, 483–507 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  8. Flores-Bazán, F., Hernández, E.: A unified vector optimization problem: complete scalarizations and applications. Optimization 60(12), 1399–1419 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  9. Gaydu, M., Geoffroy, M.H., Jean-Alexis, C., Nedelcheva, D.: Stability of minimizers of set optimization problems. Positivity (2016). doi:10.1007/s11117-016-0412-6

    MATH  Google Scholar 

  10. Ha, T.X.D.: Optimality conditions for several types of efficient solutions of set-valued optimization problems. Nonlinear analysis and variational problems, vol. 35, pp. 305–324, Springer Optim. Appl., Springer, New York (2010)

  11. Holmes, R.B.: Geometric functional analysis and its applications. Graduate Texts in Mathematics, No. 24, pp. x+246. Springer-Verlag, New-York (1975)

  12. Khan, A.A., Tammer, C., Zălinescu, C.: Set-valued optimization. An introduction with applications. Vector Optimization. pp. xxii+765. Springer, Heidelberg (2015)

  13. Lemaire, B.: Approximation in multiobjective optimization. J. Glob. Optim. 2(2), 117–132 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  14. Li, X.-B., Wang, Q.-L., Lin, Z.: Stability of set-optimization problems with naturally quasi-functions. J. Optim. Theory Appl. (2015). doi:10.1007/s10957-015-0802-0

    MATH  Google Scholar 

  15. López, R.: Approximations of equilibrium problems. SIAM J. Control Optim. 50(2), 1038–1070 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  16. López, R.: Variational convergence for vector-valued functions and its applications to convex multiobjective optimization. Math. Meth. Oper. Res. 78(1), 1–34 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  17. Loridan, P.: \(\epsilon \)-solutions in vector minimization problems. J. Optim. Theory Appl. 43(2), 265–276 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  18. Nieuwenhuis, J.W.: Separating sets with relative interior in Fréchet spaces. Appl. Math. Optim. 3(4), 373–376 (1976/77)

  19. Oppezzi, P., Rossi, A.M.: A convergence for infinite dimensional vector valued functions. J. Glob. Optim. 42(4), 577–586 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  20. Oppezzi, P., Rossi, A.M.: Improvement sets and convergence of optimal points. J. Optim. Theory Appl. 165(2), 405–419 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  21. Peressini, A.L.: Ordered topological vector spaces. pp. x+228. Harper & Row, Publishers, New York (1967)

  22. Ponstein, J.: Approaches to the theory of optimization. Cambridge Tracts in Mathematics, vol. 77. pp. xii+205. Cambridge University Press, Cambridge (1980)

  23. Proinov, P.D.: A unified theory of cone metric spaces and its applications to the fixed point theory. Fixed Point Theory Appl. 103, 38 (2013)

  24. Rockafellar, R.T., Wets, R.: Variational analysis, 317. Springer-Verlag, Berlin (1998)

    Book  MATH  Google Scholar 

  25. White, D.J.: Epsilon efficiency. J. Optim. Theory Appl. 49(2), 319–337 (1986)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

We would like to thank the anonymous referee for his/her careful reading of our manuscript and for providing accurate comments and, in particular, for bringing some references and works to our attention.

Author information

Authors and Affiliations

  1. LAMIA, Department of Mathematics, Université des Antilles, Guadeloupe, Pointe-à-Pitre, France

    Michel H. Geoffroy & Yvesner Marcelin

  2. Department of Mathematics, Technical University, Varna, Bulgaria

    Diana Nedelcheva

Authors
  1. Michel H. Geoffroy
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yvesner Marcelin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Diana Nedelcheva
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michel H. Geoffroy.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Geoffroy, M.H., Marcelin, Y. & Nedelcheva, D. Convergence of relaxed minimizers in set optimization. Optim Lett 11, 1677–1690 (2017). https://doi.org/10.1007/s11590-016-1079-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-016-1079-4

Keywords

Search

Navigation