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The Clarke and Michel-Penot Subdifferentials of the Eigenvalues of a Symmetric Matrix |
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| Authors: | J.-B. Hiriart-Urruty A.S. Lewis |
| Affiliation: | (1) LAO, U.F.R. Mathématiques, Informatique, Gestion, Université, Paul Sabatier, 118, route de Narbonne, F-31062 Toulouse Cedex, France;(2) Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1 |
| Abstract: | We calculate the Clarke and Michel-Penot subdifferentials of the function which maps a symmetric matrix to its mth largest eigenvalue. We show these two subdifferentials coincide, and are identical for all choices of index m corresponding to equal eigenvalues. Our approach is via the generalized directional derivatives of the eigenvalue function, thereby completing earlier studies on the classical directional derivative. |
| Keywords: | eigenvalue optimization nonsmooth analysis Michel-Penot subdifferential Clarke subdifferential generalized derivative |
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