Saddle representations of positively homogeneous functions by linear functions

TL;DR

This paper introduces a method to represent continuous positively homogeneous functions using saddle representations via two-index families of linear functions, providing new insights into their structure.

Contribution

It establishes that every continuous positively homogeneous function can be characterized by a specific two-index family of linear functions for saddle representation.

Findings

Every continuous positively homogeneous function admits a saddle representation.

Characterization of two-index families for Lipschitz continuous, difference sublinear, and piecewise linear functions.

Provides a unified framework for representing various classes of positively homogeneous functions.

Abstract

We say that a positively homogeneous function admits a saddle representation by linear functions iff it admits both an inf-sup-representation and a sup-inf-representation with the same two-index family of linear functions. In the paper we show that each continuous positively homogeneous function can be associated with a two-index family of linear functions which provides its saddle representation. We also establish characteristic properties of those two-index families of linear functions which provides saddle representations of functions belonging to the subspace of Lipschitz continuous positively homogeneous functions as well as the subspaces of difference sublinear and piecewise linear functions.