Polynomial root radius optimization with affine constraints

Julia Eaton, Sara Grundel, Mert Gürbüzbalaban, Michael L. Overton

Research output: Contribution to journalArticlepeer-review


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 linearly independent affine constraints on the coefficients. We show that there always exists an optimal polynomial with at most k- 1 inactive roots, that is, roots whose moduli are strictly less than the optimal root radius. We illustrate our results using some examples arising in feedback control.

Original languageEnglish (US)
Pages (from-to)509-528
Number of pages20
JournalMathematical Programming
Issue number2
StatePublished - Oct 1 2017


  • Frequency domain stabilization
  • Nonsmooth optimization
  • Polynomial optimization

ASJC Scopus subject areas

  • Software
  • General Mathematics


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