Mathematical Properties of Optimization Problems Defined by Positively Homogeneous Functions

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.