numerical radius

On Properties of Univariate Max Functions at Local Maximizers

  • Tim Mitchell
  • Michael L. Overton
    • Categories Nonsmooth Optimization Tags , , , ,

    More than three decades ago, Boyd and Balakrishnan established a regularity result for the two-norm of a transfer function at maximizers. Their result extends easily to the statement that the maximum eigenvalue of a univariate real analytic Hermitian matrix family is twice continuously differentiable, with Lipschitz second derivative, at all local maximizers, a property that … Read more

    Disk matrices and the proximal mapping for the numerical radius

  • X.Y. Han
  • Adrian Lewis
    • Categories Convex Optimization, Semi-definite Programming Tags , , , ,

    Optimal matrices for problems involving the matrix numerical radius often have fields of values that are disks, a phenomenon associated with partial smoothness. Such matrices are highly structured: we experiment in particular with the proximal mapping for the radius, which often maps n-by-n random matrix inputs into a particular manifold of disk matrices that has … Read more