Skip to main content
SpringerLink
Log in

Sufficient conditions for global weak Pareto solutions in multiobjective optimization

  • Published:
Positivity Aims and scope Submit manuscript

Abstract

In this paper we derive new sufficient conditions for global weak Pareto solutions to set-valued optimization problems with general geometric constraints of the type

$$\begin{aligned} \text{ maximize}\quad F(x) \quad \text{ subject} \text{ to}\quad x\in \Omega , \end{aligned}$$

where \(F: X\rightrightarrows Z\) is a set-valued mapping between Banach spaces with a partial order on \(Z\). Our main results are established by using advanced tools of variational analysis and generalized differentiation; in particular, the extremal principle and full generalized differential calculus for the subdifferential/coderivative constructions involved. Various consequences and refined versions are also considered for special classes of problems in vector optimization including those with Lipschitzian data, with convex data, with finitely many objectives, and with no constraints.

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. Ansari, Q.H., Yao, J.-C. (eds.): Recent Advances in Vector Optimization. Springer, Berlin (2011)

    Google Scholar 

  2. Bao, T.Q.: Subdifferential necessary conditions in set-valued optimization problems with equilibrium constraints. Optimization (2012) (to appear)

  3. Bao, T.Q., Mordukhovich, B.S.: Variational principles for set-valued mappings with applications to multiobjective optimization. Control Cybern. 36, 531–562 (2007)

    MathSciNet  Google Scholar 

  4. Bao, T.Q., Mordukhovich, B.S.: Necessary conditions for super minimizers in constrained multiobjective optimization. J. Global Optim. 43, 533–552 (2009)

    Article  MathSciNet  Google Scholar 

  5. Bao, T.Q., Mordukhovich, B.S.: Relative Pareto minimizers in multiobjective optimization: existence and optimality conditions. Math. Program. 122, 301–347 (2010)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  7. Bao, T.Q., Mordukhovich, B.S.: Refined necessary conditions in multiobjective optimization with applications to microeconomic modeling. Disc. Cont. Dyn. Syst. Ser. A 31, 1069–1096 (2011)

    Article  MathSciNet  Google Scholar 

  8. Bao, T.Q., Mordukhovich, B.S.: Extended Pareto optimality in multiobjective problems. In: Ansari, Q.H., Yao, J.-C. (eds.) Recent Advances in Vector Optimization, Chapter 13, pp. 467–516. Springer, Berlin (2011)

  9. Borwein, J.M., Zhu, Q.J.: Techniques of Variational Analysis. Springer, Berlin (2005)

    Google Scholar 

  10. Durea, M., Strugariu, R.: On some Fermat principles for set-valued optimization problems. Optimization 60, 575–591 (2011)

    Article  MathSciNet  Google Scholar 

  11. Dutta, J.: Optimality conditions for maximizing a locally Lipschitz function. Optimization 54, 377–389 (2005)

    Article  MathSciNet  Google Scholar 

  12. Ekeland, I., Turnbull, T.: Infinite-Dimensional Optimization and Convexity. University of Chicago Press, Chicago (1983)

    Google Scholar 

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

    Google Scholar 

  14. Jahn, J.: Vector Optimization: Theory, Applications and Extensions. Springer, Berlin (2004)

    Google Scholar 

  15. Jeyakumar, V., Luc, D.T.: Nonsmooth Vector Functions and Continuous Optimization, Springer Optimization and Its Applications, vol. 10. Springer, New York (2008)

    Google Scholar 

  16. Ha, T.X.D.: The Fermat rule and Lagrange multiplier rule for various efficient solutions of set-valued optimization problems expressed in terms of coderivatives. In: Ansari, Q.H., Yao, J.-C. (eds.) Recent Advances in Vector Optimization, Chapter 12, pp. 417–466. Springer, Berlin (2011)

  17. Hiriart-Urruty, J.B., Ledyaev, Y.S.: A note on the characterization of the global maxima of a (tangentially) convex function over a convex set. J. Convex Anal. 3, 55–61 (1996)

    MathSciNet  Google Scholar 

  18. Mishra, S.K., Giorgi, G.: Invexity and Optimization. Springer, Berlin (2008)

    Book  Google Scholar 

  19. Mordukhovich, B.S.: Complete characterizations of covering, metric regularity, and Lipschitzian properties of multifunctions. Trans. Am. Math. Soc. 340, 1–35 (1993)

    Article  MathSciNet  Google Scholar 

  20. Mordukhovich, B.S.: Necessary conditions in nonsmooth minimization via lower and upper subgradients. Set-Valued Anal. 12, 163–193 (2004)

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Book  Google Scholar 

  25. Schirotzek, W.: Nonsmooth Analysis. Springer, Berlin (2007)

    Book  Google Scholar 

  26. Tammer, C., Zălinescu, C.: Vector variational principles for set-valued functions. In: Ansari, Q.H., Yao, J.-C. (eds.) Recent Advances in Vector Optimization, Chapter 11, pp. 367–416. Springer, Berlin (2011)

  27. Zheng, X.Y., Ng, K.F.: The Lagrange multiplier rule for multifunctions in Banach spaces. SIAM J. Optim. 17, 1154–1175 (2006)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Northern Michigan University, Marquette, MI, 49855, USA

    Truong Q. Bao

  2. Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA

    Boris S. Mordukhovich

Authors
  1. Truong Q. Bao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Boris S. Mordukhovich
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Boris S. Mordukhovich.

Additional information

Research of this author was partly supported by the National Science Foundation under grant DMS-1007132, by the Australian Research Council under grant DP-12092508, and by the Portuguese Foundation of Science and Technologies under grant MAT/11109.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bao, T.Q., Mordukhovich, B.S. Sufficient conditions for global weak Pareto solutions in multiobjective optimization. Positivity 16, 579–602 (2012). https://doi.org/10.1007/s11117-012-0194-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11117-012-0194-4

Keywords

Mathematics Subject Classification (2000)

Search

Navigation