Advertisement

Journal of Optimization Theory and Applications

, Volume 122, Issue 2, pp 285–307 | Cite as

Conic Formulation for l p -Norm Optimization

  • F. Glineur
  • T. Terlaky
Article

Abstract

In this paper, we formulate the l p -norm optimization problem as a conic optimization problem, derive its duality properties (weak duality, zero duality gap, and primal attainment) using standard conic duality and show how it can be solved in polynomial time applying the framework of interior-point algorithms based on self-concordant barriers.

Duality theory lp-norm optimization conic optimization interior-point methods self-concordant barrier 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Peterson, E.L., and Ecker, J.G., Geometric Programming:Duality in Quadratic Programming and l p -Approximation, I, Proceedings of the International Symposium on Mathematical Programming, Edited by H. W. Kuhn and A. W. Tucker, Princeton University Press, Princeton, New Jersey, pp. 445-480, 1970.Google Scholar
  2. 2.
    Peterson, E.L., and Ecker, J.G., Geometric Programming: Duality in Quadratic Programming and l p -Approximation, II, SIAM Journal on Applied Mathematics, Vol. 13, pp. 317-340, 1967.Google Scholar
  3. 3.
    Peterson, E.L., and Ecker, J.G., Geometric Programming: Duality in Quadratic Programming and l p -Approximation, III, Journal of Mathematical Analysis and Applications, Vol. 29, pp. 365-383, 1970.Google Scholar
  4. 4.
    Terlaky, T., On l p -Programming, European Journal of Operational Research, Vol. 22, pp. 70-100, 1985.Google Scholar
  5. 5.
    Nesterov, Y.E., and Nemirovski, A.S., Interior-Point Polynomial Methods in Convex Programming, SIAM Studies in Applied Mathematics, SIAM Publications, Philadelphia, Pennsylvania, 1994.Google Scholar
  6. 6.
    Glineur, F., Topics in Convex Optimization: Interior-Point Methods, Conic Duality, and Approximations, PhD Thesis, Faculté Polytechnique de Mons, Mons, Belgium, 2001.Google Scholar
  7. 7.
    Stoer, J., and Witzgall, C., Convexity and Optimization in Finite Dimensions, I, Springer Verlag, Berlin, Germany, 1970.Google Scholar
  8. 8.
    Sturm, J.F., Primal-Dual Interior-Point Approach to Semidefinite Programming, PhD Thesis, Erasmus University Rotterdam, Rotterdam, Netherlands, 1997.Google Scholar
  9. 9.
    Rockafellar, R.T., Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.Google Scholar
  10. 10.
    Boyd, S., Lobo, M.S., Vandenberghe, L., and Lebret, H., Applications of Second-Order Cone Programming, Linear Algebra and Applications, Vol. 284, pp.193-228, 1998.Google Scholar
  11. 11.
    Goldman, A.J., and Tucker, A.W., Theory of Linear Programming, Linear Equalities and Related Systems, Edited by H.W. Kuhn and A.W. Tucker, Annals of Mathematical Studies, Princeton University Press, Princeton, New Jersey, Vol. 38, pp. 53-97, 1956.Google Scholar
  12. 12.
    Xue, G., and Ye, Y., An Efficient Algorithm for Minimizing a Sum of p-Norms, SIAM Journal on Optimization, Vol. 10, pp. 551-579, 2000.Google Scholar
  13. 13.
    Roos, C., den hertog, D., Jarre, F., and Terlaky, T., A Sufficient Condition for Self-Concordance, with Application to Some Classes of Structured Convex Programming Problems, Mathematical Programming, Vol. 69, pp. 75-88, 1995.Google Scholar
  14. 14.
    Glineur, F., Proving Strong Duality for Geometric Optimization Using a Conic Formulation, Annals of Operations Research, Vol. 105, pp. 155-184, 2001.Google Scholar
  15. 15.
    Glineur, F., An Extended Conic Formulation for Geometric Optimization, Foundations of Computing and Decision Sciences, Vol. 25, pp. 161-174, 2000.Google Scholar

Copyright information

© Plenum Publishing Corporation 2004

Authors and Affiliations

  • F. Glineur
    • 1
  • T. Terlaky
    • 2
  1. 1.Service de Mathématique et de Recherche Opérationnelle, Faculté Polytechnique de MonsMonsBelgium
  2. 2.Canadian Research Chair in Optimization, Department of Computing and SoftwareMcMaster UniversityHamilton, OntarioCanada

Personalised recommendations