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Variational analysis of the Crouzeix ratio

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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 \(\Vert p(A)\Vert _2\). 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.

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Notes

  1. By this we mean, apart from making the following transformations: scaling \(p\), scaling \(A\), shifting the root of the monomial \(p\) and the diagonal of the matrix \(A\) by the same scalar, applying a unitary similarity transformation to \(A\), or replacing the zero block in \(A\) by any matrix whose field of values is contained in \(\mathcal {D}\). Note, however, that if the condition that p is a polynomial is relaxed to allow it to be analytic, there are many choices for (p, A) for which the ratio 0.5 is attained; for the case \(N=3\), see [8, Sec. 10].

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Authors and Affiliations

  1. Department of Applied Mathematics, University of Washington, Seattle, WA, USA

    Anne Greenbaum

  2. Department of Operations Research and Information Engineering, Cornell University, Ithaca, NY, USA

    Adrian S. Lewis

  3. Courant Institute of Mathematical Sciences, New York University, New York, NY, USA

    Michael L. Overton

Authors
  1. Anne Greenbaum
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  2. Adrian S. Lewis
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  3. Michael L. Overton
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Correspondence to Michael L. Overton.

Additional information

Dedicated to Terry Rockafellar on the Occasion of his 80th Birthday.

Anne Greenbaum: Supported in part by National Science Foundation Grant DMS-1210886. Adrian S. Lewis: Supported in part by National Science Foundation Grant DMS-1208338. Michael L. Overton: Supported in part by National Science Foundation Grant DMS-1317205.

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Greenbaum, A., Lewis, A.S. & Overton, M.L. Variational analysis of the Crouzeix ratio. Math. Program. 164, 229–243 (2017). https://doi.org/10.1007/s10107-016-1083-6

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