Variational analysis of the Crouzeix ratio

• 成果类型: Article
• 署名作者: Greenbaum, Anne; Lewis, Adrian S.; Overton, Michael L.
• 署名单位: University of Washington; University of Washington Seattle; Cornell University; New York University
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-016-1083-6
• 发表日期: 2017
• 页码: 229-243
• 关键词: conjecture; PROOF

摘要

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 and denominator . 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.