A theorem of the alternative with an arbitrary number of inequalities and quadratic programming

TL;DR

This paper presents a generalized Gordan-type theorem for multiple inequalities, establishing its validity under weak convexity and demonstrating its optimality, with applications to quadratic optimization and Fenchel conjugates.

Contribution

It introduces an optimal Gordan-type theorem involving arbitrary inequalities and extends its application to nonlinear quadratic optimization and conjugate function formulas.

Findings

The theorem holds under weak convexity assumptions.

It generalizes recent results in nonlinear quadratic optimization.

Provides a formula for Fenchel conjugates of supremum functions.

Abstract

In this paper we are concerned with a Gordan-type theorem involving an arbitrary number of inequality functions. We not only state its validity under a weak convexity assumption on the functions, but also show it is an optimal result. We discuss generalizations of several recent results on nonlinear quadratic optimization, as well as a formula for the Fenchel conjugate of the supremum of a family of functions, in order to illustrate the applicability of that theorem of the alternative.