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Adrian S. Lewis(adrian.lewiscornell.edu) Abstract: Solutions to optimization problems involving the numerical radius often belong to a special class: the set of matrices having field of values a disk centered at the origin. After illustrating this phenomenon with some examples, we illuminate it by studying matrices around which this set of "disk matrices" is a manifold with respect to which the numerical radius is partly smooth. We then apply our results to matrices whose nonzeros consist of a single superdiagonal, such as Jordan blocks and the Crabb matrix related to a wellknown conjecture of Crouzeix. Finally, we consider arbitrary complex threebythree matrices; even in this case, the details are surprisingly intricate. One of our results is that in this real vector space with dimension 18, the set of disk matrices is a semialgebraic manifold with dimension 12. Keywords: Partial smoothness Category 1: Convex and Nonsmooth Optimization (Nonsmooth Optimization ) Citation: arXiv/submit/2525964 Download: [PDF] Entry Submitted: 12/31/2018 Modify/Update this entry  
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