Skip to main content
SpringerLink
Log in

Generalized Fenchel’s Conjugation Formulas and Duality for Abstract Convex Functions

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

Abstract

In this paper, we present a generalization of Fenchel’s conjugation and derive infimal convolution formulas, duality and subdifferential (and ε-subdifferential) sum formulas for abstract convex functions. The class of abstract convex functions covers very broad classes of nonconvex functions. A nonaffine global support function technique and an extended sum-epiconjugate technique of convex functions play a crucial role in deriving the results for abstract convex functions. An additivity condition involving global support sets serves as a constraint qualification for the duality.

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. Moreau, J.J.: Inf-convolution, sous-additivité, convexité des fonctions numériques. J. Math. Pures Appl. 49, 109–154 (1970)

    MATH  Google Scholar 

  2. Pallaschke, D., Rolewicz, S.: Foundations of Mathematical Optimization. Kluwer Academic, Dordrecht (1997)

    MATH  Google Scholar 

  3. Rubinov, A.M.: Abstract Convexity and Global Optimization. Kluwer Academic, Dordrecht (2000)

    MATH  Google Scholar 

  4. Singer, I.: Abstract Convex Analysis. Wiley-Interscience, New York (1997)

    MATH  Google Scholar 

  5. Burachik, R.S., Jeyakumar, V.: A simple closure condition for the normal cone intersection formula. Proc. Am. Math. Soc. 133, 1741–1748 (2005)

    Article  MATH  Google Scholar 

  6. Burachik, R.S., Jeyakumar, V.: A new geometric condition for Fenchel’s duality in infinite dimensions. Math. Program. 104B, 229–233 (2005)

    Article  Google Scholar 

  7. Burachik, R.S., Jeyakumar, V.: A dual condition for the subdifferential sum formula and applications. J. Convex Anal. 12, 279–290 (2005)

    MATH  Google Scholar 

  8. Rubinov, A.M., Glover, B.M., Jeyakumar, V.: A general approach to dual characterizations of solvability of inequality systems with applications. J. Convex Anal. 2, 309–344 (1995)

    MATH  Google Scholar 

  9. Stromberg, T.: The operation of infimal convolution. Diss. Math. 352, 1–61 (1996)

    Google Scholar 

  10. Rubinov, A.M., Wu, Z.Y., Duan, L.: Hidden abstract convex functions. J. Nonlinear Convex Anal. 6, 203–216 (2005)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, University of New South Wales, Sydney, Australia

    V. Jeyakumar

  2. CIAO, School of Information Technology and Mathematical Sciences, University of Ballarat, Ballarat, Victoria, Australia

    A. M. Rubinov

  3. School of Mathematics and Computer Science, Chongqing Normal University, Chongqing, China

    Z. Y. Wu

Authors
  1. V. Jeyakumar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. M. Rubinov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Z. Y. Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. Jeyakumar.

Additional information

Communicated by X.Q. Yang.

Work of Z.Y. Wu was carried out while the author was at the Department of Applied Mathematics, University of New South Wales, Sydney, Australia.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jeyakumar, V., Rubinov, A.M. & Wu, Z.Y. Generalized Fenchel’s Conjugation Formulas and Duality for Abstract Convex Functions. J Optim Theory Appl 132, 441–458 (2007). https://doi.org/10.1007/s10957-007-9185-1

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-007-9185-1

Keywords

Search

Navigation