Large-Scale Optimization of Eigenvalues (1991)
| Venue: | SIAM J. Optimization |
| Citations: | 71 - 3 self |
BibTeX
@ARTICLE{Overton91large-scaleoptimization,
author = {Michael L. Overton},
title = {Large-Scale Optimization of Eigenvalues},
journal = {SIAM J. Optimization},
year = {1991},
volume = {2},
pages = {88--120}
}
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Abstract
Optimization problems involving eigenvalues arise in many applications. Let x be a vector of real parameters and let A(x) be a continuously differentiable symmetric matrix function of x. We consider a particular problem which occurs frequently: the minimization of the maximum eigenvalue of A(x), subject to linear constraints and bounds on x. The eigenvalues of A(x) are not differentiable at points x where they coalesce, so the optimization problem is said to be nonsmooth. Furthermore, it is typically the case that the optimization objective tends to make eigenvalues coalesce at a solution point. There are three main purposes of the paper. The first is to present a clear and self-contained derivation of the Clarke generalized gradient of the max eigenvalue function in terms of a "dual matrix". The second purpose is to describe a new algorithm, based on the ideas of a previous paper by the author (SIAM J. Matrix Anal. Appl. 9 (1988) 256-268), which is suitable for solving l...
