Minimization of a particular singular value

We consider the problem of minimizing a particular singular value of a matrix variable, which is then subject to some convex constraints. Convex heuristics for this problem are discussed, including some counter-intuitive results regarding which is best, which then provide upper bounds on the value of the problem. The use of polynomial optimization formulations is considered, particularly for obtaining lower bounds on the value of the problem. We show that the main problem can also be formulated as an optimization problem with a bilinear matrix inequality (BMI), and discuss the use of this formulation.