# Preprint No. MPIMD/14-24

#### Author(s): Julie Eaton, Sara Grundel, Mert Gürbüzbalaban, Michael Overton

#### Email: jreaton@uw.edu

#### Date: 2014-12-16

#### Revised: 2015-03-05

#### Abstract:

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.

#### BibTeX:

@TECHREPORT{MPIMD14-24,

author = {Julie Eaton and Sara Grundel and Mert Gürbüzbalaban and Michael Overton},

title = {Polynomial Root Radius Optimization with Affine Constraints},

number = {MPIMD/14-24},

month = dec,

year = 2014,

institution = {Max Planck Institute Magdeburg},

type = {Preprint},

note = {Available from \url{http://www.mpi-magdeburg.mpg.de/preprints/}},

}