eigenvalue perturbation – Optimization Online

SOME REGULARITY RESULTS FOR THE PSEUDOSPECTRAL ABSCISSA AND PSEUDOSPECTRAL RADIUS OF A MATRIX

Published: 2011/02/14

Global Optimization Theory, Nonlinear Optimization, Nonsmooth Optimization aubin property, eigenvalue perturbation, lipschitz multifunction, pseudospectral abscissa, pseudospectral radius, pseudospectrum

The $\epsilon$-pseudospectral abscissa $\alpha_\epsilon$ and radius $\rho_\epsilon$ of an n x n matrix are respectively the maximal real part and the maximal modulus of points in its $\epsilon$-pseudospectrum, defined using the spectral norm. It was proved in [A.S. Lewis and C.H.J. Pang. Variational analysis of pseudospectra. SIAM Journal on Optimization, 19:1048-1072, 2008] that for fixed … Read more

The mathematics of eigenvalue optimization

Published: 2003/04/18

Adrian Lewis

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