On representing the positive semidefinite cone using the second-order cone

TL;DR

This paper investigates whether the positive semidefinite cone can be represented using second-order cones, concluding that the 3x3 case cannot, due to the unbounded growth of a specific matrix rank.

Contribution

It provides a negative answer to the open question by demonstrating the impossibility of representing the 3x3 positive semidefinite cone with second-order cones.

Findings

3x3 positive semidefinite cone cannot be represented by second-order cones

The second-order cone rank of certain submatrices grows unbounded

Some slices of the 3x3 cone may be representable

Abstract

The second-order cone plays an important role in convex optimization and has strong expressive abilities despite its apparent simplicity. Second-order cone formulations can also be solved more efficiently than semidefinite programming in general. We consider the following question, posed by Lewis and Glineur, Parrilo, Saunderson: is it possible to express the general positive semidefinite cone using second-order cones? We provide a negative answer to this question and show that the 3x3 positive semidefinite cone does not admit any second-order cone representation. Our proof relies on exhibiting a sequence of submatrices of the slack matrix of the 3x3 positive semidefinite cone whose "second-order cone rank" grows to infinity. We also discuss the possibility of representing certain slices of the 3x3 positive semidefinite cone using the second-order cone.