Australian Mathematical Society
Mathematics Department at Macquarie University

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50th Annual Meeting of the Australian Mathematical Society

Plenary Talk in Macquarie Theatre

Thursday 28 September 2006 at 08:45


Adrian Lewis (Cornell University)


Eigenvalues and Optimization


Eigenvalues of symmetric and nonsymmetric matrices are central in several areas of modern optimization, including semidefinite programming and robust control. For symmetric matrices, many elegant classical eigenvalue inequalities derive from convexity properties of the spectrum; singular values enjoy analogous inequalities and properties. I will outline the variational theory of spectral functions, and discuss unifying algebraic frameworks, such as semisimple Lie theory and hyperbolic polynomials. By contrast with the symmetric case, the spectrum of a nonsymmetric matrix is less robust and less predictive for modelling. In some contexts, such as transient dynamics, the pseudospectrum (the set of eigenvalues of all nearby matrices) may be a more pertinent and robust tool. I will discuss a variety of optimization problems involving pseudospectra.