On the connection of facially exposed and nice cones

TL;DR

This paper explores the properties of nice convex cones, establishing their connection with facial exposedness, providing characterizations, and conjecturing their equivalence, which has implications for convex analysis and optimization algorithms.

Contribution

It offers new characterizations of nice cones, links them to facial exposedness, and conjectures their equivalence, advancing understanding in convex cone theory.

Findings

Nice cones are facially exposed.

Facial exposedness with additional conditions implies niceness.

Conjecture that nice and facially exposed cones are equivalent.

Abstract

A closed convex cone K is called nice, if the set K^* + F^\perp is closed for all F faces of K, where K^* is the dual cone of K, and F^\perp is the orthogonal complement of the linear span of F. The niceness property is important for two reasons: it plays a role in the facial reduction algorithm of Borwein and Wolkowicz, and the question whether the linear image of a nice cone is closed also has a simple answer. We prove several characterizations of nice cones and show a strong connection with facial exposedness. We prove that a nice cone must be facially exposed; in reverse, facial exposedness with an added condition implies niceness. We conjecture that nice, and facially exposed cones are actually the same, and give supporting evidence.