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Journal of Optimization Theory and Applications

, Volume 23, Issue 3, pp 389–400 | Cite as

Sufficient conditions for duality in homogeneous programming

  • M. Schechter
Contributed Papers

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.

Key Words

Duality mathematical programming homogeneous functions subgradients 

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References

  1. 1.
    Eisenberg, E.,Duality in Homogeneous Programming, Proceedings of the American Mathematical Society, Vol. 12, pp. 783–787, 1961.Google Scholar
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    Mehendreta, S. L.,Symmetry and Self-Duality in Nonlinear Programming, Numerische Mathematik, Vol. 10, pp. 103–109, 1967.Google Scholar
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    Smiley, B. F.,Duality in Complex Homogeneous Programming, Journal of Mathematical Analysis and Applications, Vol. 40, pp. 153–158, 1972.Google Scholar
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    Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1969.Google Scholar
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    Schechter, M.,A Solvability Theorem for Homogeneous Functions, SIAM Journal on Mathematical Analysis (to appear).Google Scholar
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    Rockafellar, R. T.,Conjugate Duality and Optimization, Regional Conference Series in Applied Mathematics, No. 16, Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, 1974.Google Scholar

Copyright information

© Plenum Publishing Corporation 1977

Authors and Affiliations

  • M. Schechter
    • 1
    • 2
  1. 1.Department of MathematicsLehigh UniversityBethlehem
  2. 2.Department of Applied MathematicsTechnion—IIT, Technion CityHaifaIsrael

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