  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.
