Eigenvalues and nonsmooth optimization

Variational analysis concerns the geometry and calculus of nonsmooth sets and functions, often viewed from an optimization perspective. Over several decades, variational analysis has matured into a powerful and elegant theory. One rich source of concrete examples involves the eigenvalues of symmetric and nonsymmetric matrices, sometimes deriving from dynamical or feedback control questions. This essay presents some central ideas of variational analysis, developed from first principles, including convexity and duality, generalized gradients, sensitivity, Clarke regularity, and numerical nonsmooth optimization. Illustrative examples from eigenvalue optimization, many from joint work with J.V. Burke and M.L. Overton, include semidefinite programming, asymptotic stability, simultaneous plant stabilization, and the distance to instability.