Local Minimizers of the Crouzeix Ratio: A Nonsmooth Optimization Case Study

Given a square matrix $A$ and a polynomial $p$, the Crouzeix ratio is the norm of the polynomial on the field of values of $A$ divided by the 2-norm of the matrix $p(A)$. Crouzeix's conjecture states that the globally minimal value of the Crouzeix ratio is 0.5, regardless of the matrix order and polynomial degree, and it is known that 1 is a frequently occurring locally minimal value. Making use of a heavy-tailed distribution to initialize our optimization computations, we demonstrate for the first time that the Crouzeix ratio has many other locally minimal values between 0.5 and 1. Besides showing that the same function values are repeatedly obtained for many different starting points, we also verify that an approximate nonsmooth stationarity condition holds at computed candidate local minimizers. We also find that the same locally minimal values are often obtained both when optimizing over real matrices and polynomials, and over complex matrices and polynomials. We argue that minimization of the Crouzeix ratio makes a very interesting nonsmooth optimization case study, illustrating among other things how effective the BFGS method is for nonsmooth, nonconvex optimization. Our method for verifying approximate nonsmooth stationarity is based on what may be a novel approach to finding approximate subgradients of max functions on an interval. Our extensive computations strongly support Crouzeix's conjecture: in all cases, we find that the smallest locally minimal value is 0.5.