Mathematical Properties of Optimization Problems Defined by Positively Homogeneous Functions

J. B. Lasserre and J. B. Hiriart-Urruty
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J. B. Lasserre: Directeur de Recherche, LAAS-CNRS
J. B. Hiriart-Urruty: Université Paul Sabatier

Journal of Optimization Theory and Applications, 2002, vol. 112, issue 1, No 2, 52 pages

Abstract: 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.

Keywords: Nonlinear programming; homogeneous programming; global optimization (search for similar items in EconPapers)
Date: 2002
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Citations: View citations in EconPapers (2)

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DOI: 10.1023/A:1013088311288

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