Terracini Convexity

TL;DR

This paper introduces a flexible generalization of neighborliness called Terracini convexity for non-polyhedral convex cones, including positive-semidefinite cones, and demonstrates its application in characterizing semidefinite relaxation tightness.

Contribution

It proposes a new, more inclusive notion of convex cone neighborliness, extending to non-polyhedral cases, and applies it to analyze semidefinite relaxation tightness in inverse problems.

Findings

Terracini convexity characterizes non-polyhedral cones including positive-semidefinite cones.

Polyhedral cones are exactly the Terracini convex cones if and only if they are neighborly.

The framework helps identify when semidefinite relaxations are tight in inverse problems.

Abstract

We present a generalization of the notion of neighborliness to non-polyhedral convex cones. Although a definition of neighborliness is available in the non-polyhedral case in the literature, it is fairly restrictive as it requires all the low-dimensional faces to be polyhedral. Our approach is more flexible and includes, for example, the cone of positive-semidefinite matrices as a special case (this cone is not neighborly in general). We term our generalization Terracini convexity due to its conceptual similarity with the conclusion of Terracini's lemma from algebraic geometry. Polyhedral cones are Terracini convex if and only if they are neighborly. More broadly, we derive many families of non-polyhedral Terracini convex cones based on neighborly cones, linear images of cones of positive semidefinite matrices, and derivative relaxations of Terracini convex hyperbolicity cones. As a demonstration of the utility of our framework in the non-polyhedral case, we give a characterization based on Terracini convexity of the tightness of semidefinite relaxations for certain inverse problems.