Skip to main content
SpringerLink
Log in

Weak minimal elements and weak minimal solutions of a nonconvex set-valued optimization problem

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

In this paper, we characterize the nonemptiness of the set of weak minimal elements for a nonempty subset of a linear space. Moreover, we obtain some existence results for a nonconvex set-valued optimization problem under weaker topological conditions.

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. Adán, M., Novo, V.: Weak efficiency in vector optimization using a closure of algebraic type under some cone-convexlikeness. Eur. J. Oper. Res. 149, 641–653 (2003)

    Article  MathSciNet  Google Scholar 

  2. Adán, M., Novo, V.: Proper efficiency in vector optimization on real linear spaces. J. Optim. Theory Appl. 121, 515–540 (2004)

    Article  MathSciNet  Google Scholar 

  3. Chen, G., Huang, X., Yang, X.: Vector Optimization. Springer, Berlin (2005)

    Google Scholar 

  4. Conway, J.: A Course in Functional Analysis. Springer, New York (1985)

    Book  Google Scholar 

  5. Dhingra, M., Lalitha, C.S.: Approximate solutions and scalarization in set-valued optimization. Optimization 66, 1793–1805 (2017)

    Article  MathSciNet  Google Scholar 

  6. Gerstewitz, Ch., Iwanow, E.: Dualität für nichtkonvexe Vektoroptimierungsprobleme. Wiss. Z. Tech. Hochsch. Ilmenau. 31, 61–81 (1985)

    MathSciNet  MATH  Google Scholar 

  7. Gutiérrez, C., Jiménez, B., Miglierina, E., Molho, E.: Scalarization in set optimization with solid and nonsolid ordering cones. J. Glob. Optim. 61, 525–552 (2015)

    Article  MathSciNet  Google Scholar 

  8. Gutiérrez, C., Miglierina, E., Molho, E., Novo, V.: Pointwise well-posedness in set optimization with cone proper sets. Nonlinear Anal. 75, 1822–1833 (2012)

    Article  MathSciNet  Google Scholar 

  9. Gutiérrez, C., Novo, V., Ródenas-Pedregosa, J.L., Tanaka, T.: Nonconvex separation functional in linear spaces with applications to vector equilibria. SIAM J. Optim. 26, 2677–2695 (2016)

    Article  MathSciNet  Google Scholar 

  10. Gutiérrez, C., Kassay, G., Novo, V., Ródenas-Pedregosa, J.L.: Ekeland variational principles in vector equilibrium problems. SIAM J. Optim. 27, 2405–2425 (2017)

    Article  MathSciNet  Google Scholar 

  11. Göpfert, A., Riahi, H., Tammer, Ch., Zălinescu, C.: Variational Methods in Partially Ordered Spaces. Springer, New York (2003)

    MATH  Google Scholar 

  12. Hernández, E., Rodríguez-Marín, L.: Nonconvex scalarization in set optimization with set-valued maps. J. Math. Anal. Appl. 325, 1–18 (2007)

    Article  MathSciNet  Google Scholar 

  13. Holmes, R.B.: Geometric Functional Analysis and Its Applications. Springer, New York-Heidelberg (1975)

    Book  Google Scholar 

  14. Jahn, J.: Vector Optimization: Theory, Applications and Extensions. Springer, Berlin (2011)

    Book  Google Scholar 

  15. Khan, A.A., Tammer, C., Zălinescu, C.: Set-Valued Optimization: An Introduction with Applications. Vector Optimization. Springer, Heidelberg (2015)

    Book  Google Scholar 

  16. Kuroiwa, D.: The natural criteria in set-valued optimization. Surikaisekikenkyusho Kokyuroku 1031, 85–90 (1998)

    MathSciNet  MATH  Google Scholar 

  17. Luc, D.T.: Theory of Vector Optimization. Springer, Berlin (1989)

    Book  Google Scholar 

  18. Pascoletti, A., Serafini, P.: Scalarizing vector optimization problems. J. Optim. Theory Appl. 42, 499–524 (1984)

    Article  MathSciNet  Google Scholar 

  19. Qiu, J.H., He, F.: A general vectorial Ekeland’s variational principle with a \(p\)-distance. Acta Math. Sin. 29, 1655–1678 (2013)

    Article  MathSciNet  Google Scholar 

  20. Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)

    Book  Google Scholar 

  21. Weidner, P.: Gerstewitz functionals on linear spaces and functionals with uniform sublevel sets. J. Optim. Theory Appl. 173, 812–827 (2017)

    Article  MathSciNet  Google Scholar 

  22. Zhang, W.Y., Li, S.J., Teo, K.L.: Well-posedness for set optimization problems. Nonlinear Anal. 71, 3769–3778 (2009)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the associate editor and reviewers for their constructive comments, which helped us to improve the paper. We also thank Professor Nicolas Hadjisavvas and Professor Constantin Zalinescu who read our manuscript and provided us with valuable comments. The third and fourth authors were partially supported by a Grant from IPM (Nos. 96550414, 98460038).

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Isfahan, Isfahan, 81745-163, Iran

    M. Chinaie, F. Fakhar & M. Fakhar

  2. Department of Mathematics, University of Khansar, P. O. Box 87916854163, Khansar, Iran

    H. R. Hajisharifi

  3. School of Mathematics, Institute for Research in Fundamental Sciences, P. O. Box 19395-5746, Tehran, Iran

    M. Fakhar & H. R. Hajisharifi

Authors
  1. M. Chinaie
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. F. Fakhar
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Fakhar
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. H. R. Hajisharifi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Fakhar.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chinaie, M., Fakhar, F., Fakhar, M. et al. Weak minimal elements and weak minimal solutions of a nonconvex set-valued optimization problem. J Glob Optim 75, 131–141 (2019). https://doi.org/10.1007/s10898-019-00810-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-019-00810-0

Keywords

Search

Navigation