Title:
Minimization of a Particular Singular Value
Abstract:
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 counterintuitive 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 CourantFischer
characterization to sample smaller POs which also provide lower bounds. We lastly show how the other CourantFischer characterization can be
used to formulate the problem as one with a bilinear matrix inequality (BMI) and a Stiefel manifold constraint.
