Advertisement

Annals of Operations Research

, Volume 105, Issue 1–4, pp 155–184 | Cite as

Proving Strong Duality for Geometric Optimization Using a Conic Formulation

  • François Glineur
Article

Abstract

Geometric optimization1 is an important class of problems that has many applications, especially in engineering design. In this article, we provide new simplified proofs for the well-known associated duality theory, using conic optimization. After introducing suitable convex cones and studying their properties, we model geometric optimization problems with a conic formulation, which allows us to apply the powerful duality theory of conic optimization and derive the duality results valid for geometric optimization.

geometric optimization duality theory conic optimization 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    D. den Hertog, F. Jarre, C. Roos and T. Terlaky, A sufficient condition for self-concordance, with application to some classes of structured convex programming problems, Mathematical Programming 69 (1995) 75–88.Google Scholar
  2. [2]
    R.J. Duffin, E.L. Peterson and C. Zener, Geometric Programming (Wiley, New York, 1967).Google Scholar
  3. [3]
    A.J. Goldman and A.W. Tucker, Theory of linear programming, in: Linear Equalities and Related Systems, eds. H.W. Kuhn and A.W. Tucker, Annals of Mathematical Studies, Vol. 38 (Princeton University Press, Princeton, NJ, 1956) pp. 53–97.Google Scholar
  4. [4]
    E. Klafszky, Geometric programming and some applications, Ph.D. thesis, Tanulmányok, No. 8 (1974).Google Scholar
  5. [5]
    E. Klafszky, Geometric Programming, Seminar Notes, no. 11.976, Hungarian Committee for Systems Analysis, Budapest (1976).Google Scholar
  6. [6]
    Y.E. Nesterov and A.S. Nemirovsky, Interior-Point Polynomial Methods in Convex Programming, SIAM Studies in Applied Mathematics (SIAM Publications, Philadelphia, 1994).Google Scholar
  7. [7]
    R.T. Rockafellar, Convex Analysis (Princeton University Press, Princeton, NJ, 1970).Google Scholar
  8. [8]
    R.T. Rockafellar, Some convex programs whose duals are linearly constrained, in: Non-Linear Programming, ed. J.B. Rosen (Academic Press, 1970).Google Scholar
  9. [9]
    J. Stoer and Ch.Witzgall, Convexity and Optimization in Finite Dimensions I (Springer, Berlin, 1970).Google Scholar
  10. [10]
    J.F. Sturm, Primal-dual interior-point approach to semidefinite programming, Ph.D. thesis, Erasmus Universiteit Rotterdam, The Netherlands (1997) published in [11].Google Scholar
  11. [11]
    J.F. Sturm, Duality results, in: High Performance Optimization, eds. H. Frenk, C. Roos, T. Terlaky and S. Zhang (Kluwer Academic, 2000) pp. 21-60.Google Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • François Glineur
    • 1
  1. 1.Service de Mathématique et de Recherche Opérationnelle, Faculté Polytechnique de MonsMonsBelgium

Personalised recommendations