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Sufficient conditions for duality in homogeneous programming


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.

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  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.

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Communicated by O. L. Mangasarian

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Schechter, M. Sufficient conditions for duality in homogeneous programming. J Optim Theory Appl 23, 389–400 (1977).

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Key Words

  • Duality
  • mathematical programming
  • homogeneous functions
  • subgradients