Minimization of a Particular Singular Value
We consider the problem of minimizing a particular singular value of a matrix variable, neither the largest nor the smallest, 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 formulation of the problem as a polynomial optimization (PO) is considered, particularly for obtaining lower bounds on the value of the problem, along with the use of a Courant-Fischer characterization to sample smaller POs which also provide lower bounds. We lastly show how the other Courant-Fischer characterization can be used to formulate the problem as one with a bilinear matrix inequality (BMI) and a Stiefel manifold constraint.