- Variational Analysis of the Crouzeix Ratio Anne Greenbaum(greenbauuw.edu) Adrian Lewis(adrian.lewiscornell.edu) Michael Overton(mo1nyu.edu) Abstract: Let $W(A)$ denote the field of values (numerical range) of a matrix $A$. For any polynomial $p$ and matrix $A$, define the Crouzeix ratio to have numerator $\max\left\{|p(\zeta)|:\zeta\in W(A)\right\}$ and denominator $\|p(A)\|_2$. M.~Crouzeix's 2004 conjecture postulates that the globally minimal value of the Crouzeix ratio is $1/2$, over all polynomials $p$ of any degree and matrices $A$ of any order. We derive the subdifferential of this ratio at pairs $(p,A)$ for which the largest singular value of $p(A)$ is simple. In particular, we show that at certain candidate minimizers $(p,A)$, the Crouzeix ratio is (Clarke) regular and satisfies a first-order nonsmooth optimality condition, and hence that its directional derivative is nonnegative there in every direction in polynomial-matrix space. We also show that pairs $(p,A)$ exist at which the Crouzeix ratio is not regular. Keywords: numerical range, field of values, nonsmooth analysis, Crouzeix's conjecture Category 1: Convex and Nonsmooth Optimization (Nonsmooth Optimization ) Citation: Submitted to Math Programming Download: [PDF]Entry Submitted: 01/22/2016Entry Accepted: 01/23/2016Entry Last Modified: 01/22/2016Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Optmization Society.