Advertisement

Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Sufficient conditions for duality in homogeneous programming

Abstract

A symmetric duality theory for programming problems with homogeneous objective functions was published in 1961 by Eisenberg and has been used by a number of authors since in establishing duality theorems for specific problems. In this paper, we study a generalization of Eisenberg's problem from the viewpoint of Rockafellar's very general perturbation theory of duality. The extension of Eisenberg's sufficient conditions appears as a special case of a much more general criterion for the existence of optimal vectors and lack of a duality gap. We give examples where Eisenberg's sufficient condition is not satisfied, yet optimal vectors exist, and primal and dual problems have the same value.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Eisenberg, E.,Duality in Homogeneous Programming, Proceedings of the American Mathematical Society, Vol. 12, pp. 783–787, 1961.

  2. 2.

    Mehendreta, S. L.,Symmetry and Self-Duality in Nonlinear Programming, Numerische Mathematik, Vol. 10, pp. 103–109, 1967.

  3. 3.

    Smiley, B. F.,Duality in Complex Homogeneous Programming, Journal of Mathematical Analysis and Applications, Vol. 40, pp. 153–158, 1972.

  4. 4.

    Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1969.

  5. 5.

    Schechter, M.,A Solvability Theorem for Homogeneous Functions, SIAM Journal on Mathematical Analysis (to appear).

  6. 6.

    Rockafellar, R. T.,Conjugate Duality and Optimization, Regional Conference Series in Applied Mathematics, No. 16, Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, 1974.

Download references

Author information

Additional information

Communicated by O. L. Mangasarian

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Schechter, M. Sufficient conditions for duality in homogeneous programming. J Optim Theory Appl 23, 389–400 (1977). https://doi.org/10.1007/BF00933448

Download citation

Key Words

  • Duality
  • mathematical programming
  • homogeneous functions
  • subgradients