clarke regular – Optimization Online

The mathematics of eigenvalue optimization

Published: 2003/04/18

Control Applications, Nonsmooth Optimization, Semi-definite Programming chain rule, clarke regular, convexity, duality, eigenvalue optimization, eigenvalue perturbation, hyperbolic polynomial, nonsmooth analysis, partly smooth, pseudospectrum, robust control, semidefinite programming, sensitivity analysis, spectral function, stability, subdifferential, unitarily invariant norm

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 … Read more

Variational Analysis of the Abscissa Mapping for Polynomials

Published: 2001/06/01

James V. Burke

Michael L. Overton

Nonsmooth Optimization clarke regular, nonsmooth analysis, stability

The abscissa mapping on the affine variety $M_n$ of monic polynomials of degree $n$ is the mapping that takes a monic polynomial to the maximum of the real parts of its roots. This mapping plays a central role in the stability theory of matrices and dynamical systems. It is well known that the abscissa mapping … Read more