Skip to main content
SpringerLink
Log in

Vector Optimization Problems via Improvement Sets

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

Abstract

Motivated by applications to the real world, various optimality criteria (also approximate ones) are developed for situations in vector optimization. We propose a new type of solution based on upper comprehensive sets and we discuss the existence of optimal points in multicriteria situations.

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. French, S.: Decision Theory: An Introduction to the Mathematics of Rationality. Ellis Horwood, New York (1993)

    Google Scholar 

  2. Stadler, W.: Initiations of multicriteria optimization. In: Jahn, J., Krabs, W. (eds.) Recent Advances and Historical Development of Vector Optimization, pp. 3–47. Springer, Berlin (1987)

    Google Scholar 

  3. Demyanov, V.F., Giannessi, F., Karelin, V.V.: Optimal control problems via exact penalty functions. J. Glob. Optim. 12, 215–324 (1998)

    Article  MathSciNet  Google Scholar 

  4. Deng, S.: Boundedness and nonemptiness of the efficient solution sets in multiobjective optimization. J. Optim. Theory Appl. 144(1), 29–42 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Sun, G.Z.: An economic approach to some classical theorems in optimization theory. Optim. Lett. 2(2), 281–286 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  6. Loridan, P.: Well posedness in vector optimization. In: Lucchetti, R., Revalski, J. (eds.) Recent Developments in Well Posed-Variational Problems, pp. 171–192. Kluwer Academic, Dordrecht (1995)

    Google Scholar 

  7. Gutierrez, C., Jiménez, B., Novo, V.: A unified approach and optimality conditions for approximate solutions of vector optimization problems. SIAM J. Optim. 17(3), 688–710 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gutierrez, C., Jiménez, B., Novo, V.: Optimality conditions via scalarization for a new ε-efficiency concept in vector optimization problems. Eur. J. Oper. Res. 201, 11–22 (2010)

    Article  MATH  Google Scholar 

  9. Gutierrez, C., Jiménez, B., Novo, V.: A generic approach to approximate efficiency and application to vector optimization with set-valued maps. J. Glob. Optim. 49, 313–342 (2011)

    Article  MATH  Google Scholar 

  10. Sawaragi, Y., Nakayama, H., Tanino, T.: Theory of Multiobjective Optimization. Academic Press, Orlando (1985)

    MATH  Google Scholar 

  11. Luc, D.T.: Theory of Vector Optimization. Lectures Notes in Economics and Mathematical Systems. Springer, Berlin (1989)

    Google Scholar 

  12. Di Pillo, G., Giannessi, F. (eds.): Nonlinear Optimization and Applications (1996)

    MATH  Google Scholar 

  13. Giannessi, F. (ed.): Vector Variational Inequalities and Vector Equilibria. Kluwer Academic, Dordrecht (2000)

    MATH  Google Scholar 

  14. Ehrgott, M.: Multicriteria Optimization, 2nd edn. Springer, Berlin (2005)

    MATH  Google Scholar 

  15. Chinchuluun, A., Pardalos, P.M.: A survey of recent developments in multiobjective optimization. Ann. Oper. Res. 154, 29–50 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Chinchuluun, A., Pardalos, P.M., Migdalas, A., Pitsoulis, L.: Pareto Optimality, Game Theory and Equilibria. Edward Elgar, Cheltenham Glos (2008)

    Book  MATH  Google Scholar 

  17. Pardalos, P.M., Rassias, T.M., Khan, A.A.: Nonlinear Analysis and Variational Problems. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  18. Bednarczuck, E., Penot, J.P.: Metrically well-set minimization problems. Appl. Math. Optim. 26, 273–285 (1992)

    Article  MathSciNet  Google Scholar 

  19. Bednarczuck, E.: An approach to well posedness in vector optimization: consequences to stability. Control Cybern. 23, 107–122 (1994)

    Google Scholar 

  20. Morgan, J.: Approximations and well-posedness in multicriteria games. Ann. Oper. Res. 137, 257–268 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  21. Patrone, F., Pusillo, L., Tijs, S.: Multicriteria games and potentials. Top 15, 138–145 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Klein, E., Thompson, A.: Theory of Correspondences. Wiley-Interscience, New York (1984)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  24. Peters, H.: Game Theory—A Multileveled Approach. Springer, Berlin (2008)

    Google Scholar 

  25. Miglierina, E., Molho, E., Patrone, F., Tijs, S.: Axiomatic approach to approximate solutions in multiobjective optimization. Decis. Econ. Finance 31(2), 95–115 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kuratowski, C.: Topologie. Monografie Matematyczne, vol. 20. Polska Akad. Nauk, Warszawa (1952)

    Google Scholar 

  27. Beer, C.: On convergence of closed sets in a metric space and distance functions. Bull. Aust. Math. Soc. 31, 421–431 (1985)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Diptem, Sezione Metodi e Modelli Matematici, Facoltà di Ingegneria, University of Genoa, P. le Kennedy, pad. D, 16129, Genoa, Italy

    M. Chicco

  2. Department of Mathematical Disciplines, Mathematical Finance and Econometrics, Catholic University Milan, Milan, Italy

    F. Mignanego

  3. Dima, Department of Mathematics, University of Genoa, via Dodecaneso 35, 16146, Genoa, Italy

    L. Pusillo

  4. CentER and Department of Econometrics and Operations Research, Tilburg University, P.O. Box 90153, 5000, LE Tilburg, The Netherlands

    S. Tijs

Authors
  1. M. Chicco
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. F. Mignanego
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. L. Pusillo
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. S. Tijs
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to L. Pusillo.

Additional information

Communicated by Panos M. Pardalos.

This research was partially supported by MUR (Ministero Università Ricerca—Italia) via a contract with S. Tijs. We thank Rodica Branzei for her helpful comments on a previous version of the paper. The valuable comments of the referees are gratefully acknowledged.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chicco, M., Mignanego, F., Pusillo, L. et al. Vector Optimization Problems via Improvement Sets. J Optim Theory Appl 150, 516–529 (2011). https://doi.org/10.1007/s10957-011-9851-1

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-011-9851-1

Keywords

Search

Navigation