Copositive and completely positive cones over symmetric cones of rank at least 5

Mitsuhiro Nishijima

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arXiv · 2602.23725

Copositive and completely positive cones over symmetric cones of rank at least 5

Mitsuhiro Nishijima

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Abstract

We study copositive and completely positive cones over symmetric cones of rank at least $5$, with particular emphasis on whether these cones are spectrahedral shadows and on the behavior of a sum-of-squares inner-approximation hierarchy. We examine to what extent known results for nonnegative orthants of dimension at least $5$ carry over to general symmetric cones of rank at least $5$. We first prove that neither the copositive nor the completely positive cone over such a symmetric cone is a spectrahedral shadow. We then generalize the Horn matrix to this setting by introducing Horn transformations and analyzing their geometric and algebraic properties. We show that Horn transformations generate exposed rays of copositive cones over symmetric cones and that they evade the zeroth level of the sum-of-squares inner-approximation hierarchy. Finally, we examine the asymptotic exactness of this hierarchy over positive semidefinite cones. In contrast to the $5$-dimensional nonnegative orthant, where the hierarchy is known to recover the entire copositive cone in the limit, we construct instances over positive semidefinite cones of order at least $5$ certifying that the union of all levels remains strictly included in the copositive cone.

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Mitsuhiro Nishijima. 2026-02-27. Copositive and completely positive cones over symmetric cones of rank at least 5. https://arxiv.org/abs/2602.23725

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