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Mathematical Properties of Optimization Problems Defined by Positively Homogeneous Functions

Author

Listed:
  • J. B. Lasserre

    (Directeur de Recherche, LAAS-CNRS)

  • J. B. Hiriart-Urruty

    (Université Paul Sabatier)

Abstract

We consider the nonlinear programming problem $$(\mathcal{P}) \mapsto \{ \min f(x)\left| {g_i } \right.(x) \leqslant b_i ,i = 1, \ldots ,m\} ,$$ with $$f$$ positively p-homogeneous and $$g_i $$ positively q-homogeneous functions. We show that $$(\mathcal{P})$$ admits a simple min–max formulation $$(\mathcal{D})$$ with the inner max-problem being a trivial linear program with a single constraint. This provides a new formulation of the linear programming problem and the linear-quadratic one as well. In particular, under some conditions, a global (nonconvex) optimization problem with quadratic data is shown to be equivalent to a convex minimization problem.

Suggested Citation

  • J. B. Lasserre & J. B. Hiriart-Urruty, 2002. "Mathematical Properties of Optimization Problems Defined by Positively Homogeneous Functions," Journal of Optimization Theory and Applications, Springer, vol. 112(1), pages 31-52, January.
    • Handle: RePEc:spr:joptap:v:112:y:2002:i:1:d:10.1023_a:1013088311288
    DOI: 10.1023/A:1013088311288
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    References listed on IDEAS

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    1. B. T. Polyak, 1998. "Convexity of Quadratic Transformations and Its Use in Control and Optimization," Journal of Optimization Theory and Applications, Springer, vol. 99(3), pages 553-583, December.
    2. J. R. Bar-On & K. A. Grasse, 1997. "Global Optimization of a Quadratic Functional with Quadratic Equality Constraints, Part 2," Journal of Optimization Theory and Applications, Springer, vol. 93(3), pages 547-556, June.
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    Cited by:

    1. Y. B. Zhao & D. Li, 2006. "On KKT Points of Homogeneous Programs," Journal of Optimization Theory and Applications, Springer, vol. 130(2), pages 369-376, August.
    2. Cheikh Toure & Armand Gissler & Anne Auger & Nikolaus Hansen, 2021. "Scaling-invariant Functions versus Positively Homogeneous Functions," Journal of Optimization Theory and Applications, Springer, vol. 191(1), pages 363-383, October.

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