Second-order cone representable slices of the positive semidefinite cone of size three

TL;DR

This paper classifies which slices of the 3x3 positive semidefinite cone are second-order cone representable, revealing that low-dimensional slices or those orthogonal to singular matrices are socr.

Contribution

It provides a complete classification of socr slices of S_+^3 based on dimension and orthogonality to singular matrices.

Findings

Slices of S_+^3 with dimension ≤ 4 are socr.

Slices orthogonal to a non-zero singular matrix are socr.

Not all slices of S_+^3 are second-order cone representable.

Abstract

To demonstrate the discrepancy between second-order cone and semidefinite programming, Hamza Fawzi showed that the cone S_+^3 of symmetric positive semidefinite matrices of size 3 is not second-order cone representable (socr). A slice of S_+^3 is intersection of S_+^3 and a linear sub-space of the space S^3 of 3 \times 3 symmetric matrices. It is known that some slices of S_+^3 are socr, while some others are not. We classify socr slices of S_+^3 by showing that a slice of S_+^3 is socr if and if it has dimension at most 4 or is orthogonal to a non-zero singular matrix (where the orthogonality is considered with respect to the standard trace scalar product).