Speaker
Description
We propose an extremely versatile approach to address a large family of matrix nearness problems, possibly with additional linear constraints. Our method is based on splitting a matrix nearness problem into two nested optimization problems, of which the inner one can be solved either exactly or cheaply, while the outer one can be recast as an unconstrained optimization task over a smooth real Riemannian manifold. We observe that this paradigm applies to many matrix nearness problems of practical interest appearing in the literature, thus revealing that they are equivalent in this sense to a Riemannian optimization problem. We also show that the objective function to be minimized on the Riemannian manifold can be discontinuous, thus requiring regularization techniques, and we give conditions for this to happen. Time permitting, we demonstrate the practical applicability of our method by implementing it for a number of matrix nearness problems that are relevant for applications and are currently considered very demanding in practice. Among them, we recall the problem of finding the distance to singularity for matrix polynomials, the computation of the approximate gcd between to scalar polynomials and the nearest matrix with at least one eigenvalue
in a prescribed closed set, also known as distance to instability.
