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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 problems 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 \(3\times 3\) positive semidefinite cone does not admit any second-order cone representation. In fact we show that the slice consisting of \(3\times 3\) positive semidefinite Hankel matrices does not admit a second-order cone representation. Our proof relies on exhibiting a sequence of submatrices of the slack matrix of the \(3\times 3\) positive semidefinite cone whose “second-order cone rank” grows to infinity.

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