The root radius of a polynomial is the maximum of the moduli of its roots (zeros). We consider the following optimization problem: minimize the root radius over monic polynomials of degree $n$, with either real or complex coefficients, subject to $k$ consistent affine constraints on the coefficients. We show that there always exists an optimal polynomial with at most $k-1$ inactive roots, that is, whose modulus is strictly less than the optimal root radius. We illustrate our results using some examples arising in feedback control.
2015-03-4808, Eaton: Interdisciplinary Arts & Sciences, University of Washington Tacoma, Tacoma, WA 98402; Grundel: Max Planck Institute for Dynamics of Complex Technical Systems, 39106 Magdeburg, Germany; Gurbuzbalaban: Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Boston, MA 02139; Overton: Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, March/2015