Properties of the complementarity set for the cone of copositive matrices
O. I. Kostyukova
TL;DR
This paper tackles the local description of the complementarity set for COP$(p)$ and CP$(p)$, dual non-symmetric cones central to copositive programming. It introduces a non-degenerate neighborhood framework based on a polyhedral decomposition of the zero-set of a COP-matrix and a corresponding cone decomposition of the CP-part, leading to a system of $m$ bilinear equations augmented by linear constraints that locally define ${\\mathbb C}({\\mathbb COP}(p))$. The authors prove that, under Assumptions $j)$-$jjj)$, these defining equations are independent (full rank), thereby yielding a robust, explicit characterization of the complementarity structure near a given point. This framework lays a foundation for sensitivity analysis and parametric studies in copositive optimization, and it points toward possible relaxations of the assumptions for broader applicability. The results advance understanding of complementarity beyond symmetric cones and provide a concrete tool for solving conic optimization problems over COP$(p)$ and CP$(p)$.
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.