Skip to main content
SpringerLink
Log in

Simplified Optimality Conditions for Minimizing the Difference of Vector-Valued Functions

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

Abstract

This paper deals with an axiomatic approach to certain optimality conditions for the vector nonconvex minimization problem min{g(x)−h(x): x∈X}, where X is an arbitrary set and g, h are functions defined on X with values in an ordered topological vector space Z.

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. Flores-BazÁn, F., On Minima of the Difference of Functions, Journal of Optimization Theory and Applications, Vol. 93, pp. 525–531, 1997.

    Google Scholar 

  2. Flores-BÁzan, F., and Martinez-Legaz, J. E., Simplified Global Optimality Conditions in Generalized Theory, Generalized Convexity, Generalized Monotonicity: Recent Results, Edited by J. P. Crouzeix et al., Kluwer Academic Publishers, Dordrecht, Holland, pp. 305–329, 1998.

    Google Scholar 

  3. Tuy, H., D.C. Optimization: Theory, Methods, and Algorithms, Handbook of Global Optimization, Edited by R. Horst and P. M. Pardalos, Kluwer Academic Publishers, Dordrecht, Holland, pp. 149–216, 1995.

    Google Scholar 

  4. Peressini, A. L., Ordered Topological Vector Spaces, Harper and Row, New York, NY, 1967.

    Google Scholar 

  5. Hiriart-Urruty, J. B., From Convex Optimization to Nonconvex Optimization: Necessary and Sufficient Conditions for Global Optimality, Nonsmooth Optimization and Related Topics, Edited by F. H. Clarke, V. F. Dem'yanov, and F. Giannessi, Plenum, New York, NY, pp. 219–239, 1989.

    Google Scholar 

  6. Tuy, H., and Oettli, W., On Necessary and Sufficient Conditions for Global Optimality, Revista de Matemáticas Aplicadas, Vol. 15, pp. 39–41, 1994.

    Google Scholar 

  7. Giannessi, F., Mastroeni, G., and Pellegrini, L., On the Theory of Vector Optimization and Variational Inequalities: Image Space Analysis and Separation, Vector Variational Inequalities and Vector Equilibria: Mathematical Theories, Edited by F. Giannessi, Kluwer Academic Publishers, Dordrecht, Holland, pp. 153–215, 1999.

    Google Scholar 

  8. Ekeland, I., and Temam, R., Convex Analysis and Variational Problems, North-Holland, Amsterdam, Holland, 1976.

    Google Scholar 

  9. Stromberg, K. R., An Introduction to Classical Real Analysis, Wadsworth International, Belmont, California, 1981.

    Google Scholar 

  10. Zowe, J., Convex Functions and Convex Duality Theory in Ordered Vector Spaces, Habilitationsschrift, Universität Würzburg, 1976 (in German).

  11. Borwein, J. M., and ThÉra, M., Sandwich Theorems for Semicontinuous Operators, Canadian Mathematical Bulletin, Vol. 35, pp. 463–474, 1992.

    Google Scholar 

  12. Martinez-Legaz, J. E., and Seeger, A., A Formula on the Approximate Subdifferential of the Difference of Convex Functions, Bulletin of the Australian Mathematical Society, Vol. 45, pp. 37–41, 1992.

    Google Scholar 

  13. Dolecki, S., and Kurcyusz, S., On Φ-Convexity in Extremal Problems, SIAM Journal on Control and Optimization, Vol. 16, pp. 277–300, 1978.

    Google Scholar 

  14. Oettli, W., Kolmogorov Conditions for Vectorial Optimization Problems, OR Spektrum, Vol. 17, pp. 227–229, 1995.

    Google Scholar 

  15. DÜr, M., Global Optimality Conditions for Minimizing Differences of Functions, Forschungsbericht 57, Institut für Statistik, Wirtschaftsuniversität Wien, 1998.

  16. Singer, I., A Fenchel–Rockafellar Type Duality Theorem for Maximization, Bulletin of the Australian Mathematical Society, Vol. 20, pp. 193–198, 1979.

    Google Scholar 

  17. Toland, J. F., A Duality Principle for Nonconvex Optimization and the Calculus of Variations, Archive for Rational Mechanics and Analysis, Vol. 71, pp. 41–61, 1979.

    Google Scholar 

  18. Kutateladze, S. S., and Rubinov, A. M., Minkowsky Duality and Its Applications, Russian Mathematical Surveys, Vol. 27, pp. 137–192, 1972.

    Google Scholar 

  19. Oettli, W., and SchlÄger, D., Conjugate Functions for Convex and Nonconvex Duality, Journal of Global Optimization, Vol. 13, pp. 337–347, 1998.

    Google Scholar 

  20. Flores-BazÁn, F., On a Class of Phi-Convex Functions, Technical Report 98–04, Departamento de Ingeniería Matemática, Universidad de Concepció n, Concepción, Chile, 1998.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Ingeniería Matemática, Universidad de Concepción, Concepción, Chile

    F. Flores-BAZÁN (Professor)

  2. Fakultät für Mathematik und Informatik, Universität Mannheim, Mannheim, Germany

    W. Oettli (Professor)

Authors
  1. F. Flores-BAZÁN
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. W. Oettli
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Flores-BAZÁN, F., Oettli, W. Simplified Optimality Conditions for Minimizing the Difference of Vector-Valued Functions. Journal of Optimization Theory and Applications 108, 571–586 (2001). https://doi.org/10.1023/A:1017535424813

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1017535424813

Search

Navigation