Maximize the minimum singular value

Nonconvex
1

Hello CVXers,

How can I maximize the minimum singular value of a matrix in cvx? The minimum singular value is a concave function of the underlying matrix so cvx should be able to handle this. Unfortunately there is no function that can do this.

Thank you very much in advance for your help

Best,
curalph

2

Do you have a reference demonstrating the concavity of the minimum singular value? I am not convinced that is the case.

3

Unlike the minimum eigenvalue, which is concave, the minimum singular value is the difference of two convex functions, and is neither concave nor convex for matrix dimension >= 2. See section 5 of “On Extreme Singular Values of Matrix Valued Functions”, L. Qi and R.S. Womersley, Journal of Convex Analysis, volume 3 (1996), available at http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.43.299&rep=rep1&type=pdf . A simple 2 by 2 counter-example to the concavity of the minimum singular value is presented in Remark 5.2.