The mathematics of eigenvalue optimization

 Optimization problems involving the eigenvalues of symmetric and nonsymmetric matrices present a fascinating mathematical challenge. Such problems arise often in theory and practice, particularly in engineering design, and are amenable to a rich blend of classical mathematical techniques and contemporary optimization theory. This essay presents a personal choice of some central mathematical ideas, outlined for the broad optimization community. I discuss the convex analysis of spectral functions and invariant matrix norms, touching briefly on semidefinite representability, and then outlining two broader algebraic viewpoints based on hyperbolic polynomials and Lie algebra. Analogous nonconvex notions lead into eigenvalue perturbation theory. The last third of the article concerns stability, for polynomials, matrices, and associated dynamical systems, ending with a section on robustness. The powerful and elegant language of nonsmooth analysis appears throughout, as a unifying narrative thread. Received: December 4, 2002 / Accepted: April 22, 2003 Published online: May 28, 2003 Key Words.  eigenvalue optimization – convexity – nonsmooth analysis – duality – semidefinite program – subdifferential – Clarke regular – chain rule – sensitivity – eigenvalue perturbation – partly smooth – spectral function – unitarily invariant norm – hyperbolic polynomial – stability – robust control – pseudospectrum – H _∞ norm Mathematics Subject Classification (2000): 90C30, 15A42, 65F15, 49K40