Skip to main content
SpringerLink
Log in

Existence Theorems in Vector Optimization with Generalized Order

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

In the present paper, we establish some results for the existence of optimal solutions in vector optimization in infinite-dimensional spaces, where the optimality notion is understood in the sense of generalized order (may not be convex and/or conical). This notion is induced by the concept of set extremality and covers many of the conventional notions of optimality in vector optimization. Some sufficient optimality conditions for optimal solutions of a class of vector optimization problems, which satisfies the free disposal hypothesis, are also examined.

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

Fig. 1

Similar content being viewed by others

References

  1. Yu, P.L.: Cone convexity, cone extreme points, and nondominated solutions in decision problems with multiobjectives. J. Optim. Theory Appl. 14, 319–337 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  2. Borwein, J.M.: On the existence of Pareto efficient points. Math. Oper. Res. 8, 64–73 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  3. Jahn, J.: Existence theorems in vector optimization. J. Optim. Theory Appl. 50, 397–406 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  4. Luc, D.T.: An existence theorem in vector optimization. Math. Oper. Res. 14, 693–699 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  5. Sterna-Karwat, A.: On existence of cone-maximal points in real topological linear spaces. Isr. J. Math. 54, 33–41 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  6. Sonntag, Y., Zălinescu, C.: Comparison of existence results for efficient points. J. Optim. Theory Appl. 105, 161–198 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Al-Homidan, S., Ansari, Q.H.: Equilibrium constrained vector optimization problems. J. Nonlinear Convex Anal. 17, 891–898 (2016)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Oppezzi, P., Rossi, A.: Existence and convergence of optimal points with respect to improvement sets. SIAM J. Optim. 26, 1293–1311 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  10. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, II: Applications. Springer, Berlin (2006)

    Google Scholar 

  11. Kruger, A.Y., Mordukhovich, B.S.: Extremal points and the Euler equation in nonsmooth optimization problems. Dokl. Akad. Nauk BSSR 24, 684–687 (1980). (In Russian)

    MathSciNet  MATH  Google Scholar 

  12. Tuyen, N.V., Yen, N.D.: On the concept of generalized order optimality. Nonlinear Anal. 75, 1592–1601 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bao, T.Q., Mordukhovich, B.S.: Set-valued optimization in welfare economics. Adv. Math. Econ. 13, 113–153 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Arrow, K.J.: An extension of the basic theorems of classical welfare economics. In: Neyman, J. (ed.) Proceedings of the Second Berkeley Symposium, pp. 507–532. University of California Press, Berkeley and Los Angeles (1950)

  15. Bao, T.Q.: Subdifferential necessary conditions in set-valued optimization problems with equilibrium constraints. Optimization 63, 181–205 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Bonnisseau, J.M., Cornet, B.: On the characterization of efficient production vectors. Econom. Theory 31, 213–223 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jofré, A., Jourani, A.: Characterizations of the free disposal condition for nonconvex economies on infinite dimensional commodity spaces. SIAM J. Optim. 25, 699–712 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Bao, T.Q., Mordukhovich, B.S., Gupta, P.: Necessary conditions for multiobjective optimization with equilibrium constraints. J. Optim. Theory Appl. 135, 179–203 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Huy, N.Q., Mordukhovich, B.S., Yao, J.-C.: Coderivatives of frontier and solution maps in parametric multiobjective optimization. Taiwan. J. Math. 12, 2083–2111 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Tuyen, N.V.: Some characterizations of solution sets of vector optimization problems with generalized order. Acta Math. Vietnam. 41, 677–694 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  22. Bao, T.Q., Mordukhovich, B.S.: Relative Pareto minimizers for multiobjective problems: existence and optimality conditions. Math. Program. 122, 101–138 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Brecker, W.W., Kassay, G.: A systematization of convexity concepts for sets and functions. J. Convex Anal. 4, 109–127 (1997)

    MathSciNet  MATH  Google Scholar 

  24. Zheng, X.Y., Yang, X.Q.: The structure of weak Pareto solution sets in piecewise linear multiobjective optimization in normed spaces. Sci. China Ser. A. 51, 1243–1256 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (1990)

    MATH  Google Scholar 

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

    Book  Google Scholar 

  27. Berge, C.: Topological Spaces. Oliver and Boyd, London (1963)

Download references

Acknowledgements

The authors would like to thank the anonymous referees and the handling Associate Editor for their valuable remarks and detailed suggestions that allowed us to improve the original version. Research of the first author is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 101.02-2012.03. Research of the second and the third author was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2016R1A2B4011589).

Author information

Authors and Affiliations

  1. Department of Mathematics, Hanoi Pedagogical University 2, Xuan Hoa, Phuc Yen, Vinh Phuc, Vietnam

    Nguyen Quang Huy & Nguyen Van Tuyen

  2. Department of Applied Mathematics, Pukyong National University, Busan, 48513, Korea

    Do Sang Kim

Authors
  1. Nguyen Quang Huy
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Do Sang Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Nguyen Van Tuyen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Do Sang Kim.

Additional information

Communicated by Qamrul Hasan Ansari.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huy, N.Q., Kim, D.S. & Tuyen, N.V. Existence Theorems in Vector Optimization with Generalized Order. J Optim Theory Appl 174, 728–745 (2017). https://doi.org/10.1007/s10957-017-1146-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-017-1146-8

Keywords

Mathematics Subject Classification

Search

Navigation