Skip to main content
SpringerLink
Log in

Relative Pareto minimizers for multiobjective problems: existence and optimality conditions

  • FULL LENGTH PAPER
  • Series A
  • Published:
Mathematical Programming Submit manuscript

Abstract

In this paper we introduce and study enhanced notions of relative Pareto minimizers for constrained multiobjective problems that are defined via several kinds of relative interiors of ordering cones and occupy intermediate positions between the classical notions of Pareto and weak Pareto efficiency/minimality. Using advanced tools of variational analysis and generalized differentiation, we establish the existence of relative Pareto minimizers for general multiobjective problems under a refined version of the subdifferential Palais-Smale condition for set-valued mappings with values in partially ordered spaces and then derive necessary optimality conditions for these minimizers (as well as for conventional efficient and weak efficient counterparts) that are new in both finite-dimensional and infinite-dimensional settings. Our proofs are based on variational and extremal principles of variational analysis; in particular, on new versions of the Ekeland variational principle and the subdifferential variational principle for set-valued and single-valued mappings in infinite-dimensional spaces.

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.: On Fréchet subdifferential sum rule and its applications (preprint) (2007)

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

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  4. Bao T.Q., Mordukhovich B.S.: Existence of minimizers and necessary conditions in set-valued optimization with equilibrium constraints. Appl. Math. 52, 453–472 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bednarczuk E.M., PrzybyPrzybyła M.J.: The vector-valued variational principle in Banach spaces ordered by cones with nonempty interiors. SIAM J. Optim. 18, 907–913 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  6. Borwein J.M., Goebel R.: Notions of relative interior in Banach spaces. J. Math. Sci. 115, 2542–2553 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  7. Borwein J.M., Lewis A.S.: Partially finite convex programming, Part I: quasi relative inteiors and duality theory. Math. Prog. 57, 15–48 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  8. Borwein J.M., Lewis A.S.: Partially finite convex programming, Part II: explicit lattice models. Math. Prog. 57, 49–83 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  9. Borwein J.M., Lucet Y., Mordukhovich B.S.: Compactly epi-Lipschitzian convex sets and functions in normed spaces. J. Convex Anal. 7, 375–393 (2000)

    MATH  MathSciNet  Google Scholar 

  10. Borwein J.M., Zhu Q.J.: Techniques of Variational Analysis, CMS Books in Mathematics, vol. 20. Springer, New York (2005)

    Google Scholar 

  11. Durea, M., Dutta, J., Tammer, C.: Lagrange multipliers for ε-Pareto solutions in vector optimization with non-solid cones in Banach spaces (preprint) (2007)

  12. Ekeland I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  13. Göpfert A., Riahi H., Tammer C., Zălinescu C.: Variational Methods in Partially Ordered Spaces, CMS Books in Mathematics, vol. 17. Springer, New York (2003)

    Google Scholar 

  14. Gorokhovik, V.V.: Convex and Nonsmooth Problems of Vector Optimization. Navuka i Tekhnika, Minsk (1990)

  15. Holmes R.B.: Geometric Functional Analysis and Its Applications. Graduate Texts in Mathematics, vol. 24. Springer, New York (1975)

    Google Scholar 

  16. Jahn, J.: Vector Optimization: Theory, Applications and Extensions. Series in Operations Research and Decision Theory. Springer, New York (2004)

  17. Luc, D.T.: Theory of vector optimization. In: Lecture Notes in Economics and Mathematics Systems, vol. 319. Springer, Berlin (1989)

  18. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory. Grundlehren Series in Fundamental Principles of Mathematical Sciences, vol. 330. Springer, Berlin (2006)

  19. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, II: Applications. Grundlehren Series in Fundamental Principles of Mathematical Sciences, vol. 331. Springer, Berlin (2006)

  20. Mordukhovich B.S., Nam N.M., Yen N.D.: échetsubdifferential calculus and optimality necessary conditions in nondifferentiable programming. Optimization 55, 685–708 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  21. Mordukhovich B.S., Wang B.: Necessary optimality and suboptimality conditions in nondifferentiable programming via variational principles. SIAM J. Control Optim. 41, 623–640 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  22. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Grundlehren Series in Fundamental Principles of Mathematical Sciences, vol. 317. Springer, Berlin (1998)

  23. Ye J.J., Zhu Q.J.: Multiobjective optimization problems with variational inequality constraints. Math. Prog. 96, 139–160 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  24. Zheng X.Y., Ng K.F.: The Fermat rule for multifunctions on Banach spaces. Math. Prog. 104, 69–90 (2005)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Truong Q. Bao & 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 Truong Q. Bao.

Additional information

This research was partly supported by the USA National Science Foundation under grants DMS-0304989 and DMS-0603846 and by the Australian Research Council under grant DP-9551168.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bao, T.Q., Mordukhovich, B.S. Relative Pareto minimizers for multiobjective problems: existence and optimality conditions. Math. Program. 122, 301–347 (2010). https://doi.org/10.1007/s10107-008-0249-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-008-0249-2

Keywords

Mathematics Subject Classification (2000)

Search

Navigation