Properties of the complementarity set for the cone of copositive matrices

TL;DR

This paper investigates the properties of the complementarity set for the dual cones of copositive and completely positive matrices, aiming to deepen understanding for optimization applications.

Contribution

It provides new insights into the structure of the complementarity set for these non-symmetric cones, which have been underexplored in prior research.

Findings

Characterization of the complementarity set for copositive matrices.

Identification of linear identities in the complementarity set.

Potential implications for optimization over these cones.

Abstract

For a proper cone $K$ and its dual cone $K^*$ in $\mathbb R^n$, the complementarity set of $K$ is defined as ${\mathbb C}(K)=\{(x,y): x\in K,\; y\in K^*,\, x^\top y=0\}$. It is known that ${\mathbb C}(K)$ is an $n$-dimensional manifold in the space $\mathbb R^{2n}$. If $ K$ is a symmetric cone, points in ${\mathbb C}(K)$ must satisfy at least $n$ linearly independent bi-linear identities. Since this knowledge comes in handy when optimizing over such cones, it makes sense to search for similar relationships for non-symmetric cones. In this paper, we study properties of the complementarity set for the dual cones of copositive and completely positive matrices. Despite these cones are of great interest due to their applications in optimization, they have not yet been sufficiently studied.