The Clarke and Michel-Penot Subdifferentials of the Eigenvalues of a Symmetric Matrix

Authors:
J.-B. Hiriart-Urruty;A. S. Lewis
Affiliations:
LAO, U.F.R. Mathématiques, Informatique, Gestion, Université Paul Sabatier, 118, route de Narbonne, F-31062 Toulouse Cedex, France. jbhu@cict.fr;Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1. aslewis@orion.uwaterloo.ca URL: http://orion.uwaterloo.ca/∼aslewis
Venue:
Computational Optimization and Applications - Special issue on computational optimization—a tribute to Olvi Mangasarian, part II
Year:
1999

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Abstract

We calculate the Clarke and Michel-Penot subdifferentialsof the function which maps a symmetric matrix to its mth largesteigenvalue. We show these two subdifferentials coincide, and areidentical for all choices of index m corresponding to equaleigenvalues. Our approach is via the generalized directionalderivatives of the eigenvalue function, thereby completing earlierstudies on the classical directional derivative.