Eigenvalue problems in optimization
Claude Lemarechal(Inria, Grenoble)
Tuesday December 8, 11.00 a.m. CERFACS Conference Room
Abstract : The last decade has seen a tremendous increase of interest for the following problem: A(x) is a symmetric n by n matrix depending on x in R^m (usually, A(.) is affine). Then one wants to maximize a function f(x) (usually linear) under the constraint that A(x) is positive semi-definite. A slightly simpler variant of this problem is to minimize the maximal eigenvalue of A(x).
In this talk, we review some of the applications underlying this problem. We then describe briefly some of the known methods of solution (interior points, nonsmooth optimization). This gives us the opportunity to demonstrate the need for fast and reliable computation of information on the spectrum of possibly large symmetric matrices. Depending on the situation considered, such information can be:
the full spectrum,
just the largest eigenvualue and one associated eigenvector,
a few largest eigenvalues and associated eigenvectors.
Furthermore, the computation must be performed at points where the largest eigenvalue is multiple, with a possibly large associated eigenspace.
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