Smooth hyperbolicity cones are second-order cone representable

TL;DR

The paper shows that every Nash-smooth hyperbolicity cone is second-order cone representable (socr) and that every compact convex semialgebraic set with Nash-smooth boundary and strict positive curvature is socr. It achieves this by employing tensor evaluation to obtain lifted linear matrix inequalities with only $2\times2$ blocks, thereby giving SOC program representations and improving the understanding of geometric complexity via the sxdeg hierarchy. The results generalize and sharpen prior work on smooth hyperbolicity cones, and connect real-zero polynomials, the Lax conjecture, and deterministic curvature conditions to concrete semidefinite and second-order cone representations. Practically, these findings imply faster, SOC-based optimization for the considered convex sets and deepen the link between boundary regularity, curvature, and semidefinite representability.

Abstract

Netzer and Sanyal proved that every smooth hyperbolicity cone is a spectrahedral shadow. We generalize and sharpen this result at the same time, by showing that every Nash-smooth hyperbolicity cone is even second-order cone representable (socr). The result is proved as a consequence of our second theorem, according to which every compact convex semialgebraic set with Nash-smooth boundary of strict positive curvature is socr. The proof uses the technique of tensor evaluation.

Figures (4)

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• Figure : Fig. 1

Theorems & Definitions (16)

• Theorem 1.1
• Theorem 1.2
• Definition 2.4
• Theorem 3.3
• Lemma 3.5
• proof
• Lemma 3.7
• proof
• Proposition 3.11
• Lemma 3.12