Construction of exceptional copositive matrices
Tea Štrekelj, Aljaž Zalar
TL;DR
This work advances the understanding of copositive versus completely positive matrix cones by (i) providing a free probability–inspired bootstrap to construct exceptional doubly nonnegative (e-DNN) matrices for all sizes $\ge 5$ via a cosine-basis multiplication operator and SOS certificates, (ii) deriving exceptional copositive (e-COP) matrices from these e-DNN matrices through SDP feasibility with a negative Frobenius inner product against a COP witness, and (iii) establishing asymptotic bounds on ball-truncated volume radii that prove a positive gap between COP and CP cones as dimension grows. It also clarifies the geometry of these cones by comparing their sectional volumes under Frobenius norms and by leveraging hyperplane sections and duality, yielding concrete numerical bounds such as $\tfrac{1}{8\sqrt{2}}\leq \mathrm{vrad}(\mathrm{CP}^{(B_n)})\leq \tfrac{1}{2}\leq \mathrm{vrad}(\mathrm{COP}^{(B_n)})\leq 1$. Overall, the paper provides explicit constructions of exceptional matrices and a rigorous quantitative gap result, with potential implications for optimization problems formulated over COP and CP cones. All mathematical expressions are presented with $...$ delimiters for clarity in computational and search contexts.
Abstract
An $n\times n$ symmetric matrix $A$ is copositive if the quadratic form $x^TAx$ is nonnegative on the nonnegative orthant $\mathbb{R}^{n}_{\geq 0}$. The cone of copositive matrices contains the cone of matrices which are the sum of a positive semidefinite matrix and a nonnegative one and the latter contains the cone of completely positive matrices. These are the matrices of the form $BB^T$ for some $n\times r$ matrix $B$ with nonnegative entries. The above inclusions are strict for $n\geq5.$ The first main result of this article is a free probability inspired construction of exceptional copositive matrices of all sizes $\geq 5$, i.e., copositive matrices that are not the sum of a positive semidefinite matrix and a nonnegative one. The second contribution of this paper addresses the asymptotic ratio of the volume radii of compact sections of the cones of copositive and completely positive matrices. In a previous work by the authors, it was shown that, by identifying symmetric matrices naturally with quartic even forms, and equipping them with the $L^2$ inner product and the Lebesgue measure, the ratio of the volume radii of sections with a suitably chosen hyperplane is bounded below by a constant independent of $n$ as $n$ tends to infinity. In this paper, we extend this result by establishing an analogous bound when the sections of the cones are unit balls in the Frobenius inner product.