Non-polyhedral extensions of the Frank-and-Wolfe theorem

TL;DR

This paper explores broader classes of sets beyond polyhedra where quadratic functions bounded below attain their infimum, extending the classical Frank-and-Wolfe theorem through new characterizations and stability analysis.

Contribution

It introduces non-polyhedral Frank-and-Wolfe sets, provides internal asymptotic characterizations, and studies their stability under various operations.

Findings

Existence of non-polyhedral Frank-and-Wolfe sets

Internal asymptotic characterizations of these sets

Stability of the Frank-and-Wolfe property under operations

Abstract

In 1956 Marguerite Frank and Paul Wolfe proved that a quadratic function which is bounded below on a polyhedron $P$ attains its infimum on $P$. In this work we search for larger classes of sets $F$ with this Frank-and-Wolfe property. We establish the existence of non-polyhedral Frank-and-Wolfe sets, obtain internal characterizations by way of asymptotic properties, and investigate stability of the Frank-and-Wolfe class under various operations.