1 Convex and Non-convex Optimization Problems for the Field of Values of a Matrix

By Frank Uhlig, Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849–5310, USA; uhligfd@auburn.edu

We introduce and study numerical algorithms that compute the minimal and maximal distances between 0 ∈ ℂ and points in the field of values F (A) = {x^{∗}Ax | x ∈ ℂ^{n} , ∥x∥_{2} = 1} ⊂ ℂ for a complex matrix A_{n,n}. Finding the minimal distance from 0 ∈ ℂ to F (A) is a convex optimization problem if 0 ∉ F (A) and thus it has a unique solution, called the Crawford number whose magnitude relates information on the stability margin of the associated system. If 0 ∈ F (A), this is a non-convex optimization problem and consequently there can be multiple solutions or local minima that are not so globally. Non-convexity also holds for the maximal distance problem between points in F (A) and 0 ∈ ℂ. This maximal distance is commonly called the numerical radius numrad(A) for which the inequality ρ(A) ≤ numrad(A) ≤ ∥A∥ is well established. Both cases can be solved efficiently numerically by using ideas from geometric computing, eigenanalyses of linear combinations of the hermitean and skewhermitean parts of A and the rotation method introduced by C. R. Johnson in the 1970s to compute the boundary of the field of values.