Optimality conditions and duality theory for minimizing sums of the largest eigenvalues of symmetric matrices

Optimality conditions and duality theory for minimizing sums of the largest eigenvalues of symmetric matrices

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Article ID: iaor19951841
Country: Netherlands
Volume: 62
Issue: 2
Start Page Number: 321
End Page Number: 357
Publication Date: Nov 1993
Journal: Mathematical Programming
Authors: ,
Keywords: programming: mathematical
Abstract:

The sum of the largest eigenvalues of a symmetric matrix is a nonsmooth convex function of the matrix elements. Max characterizations for this sum are established, giving a concise characterization of the subdifferential in terms of a dual matrix. This leads to a very useful characterization of the generalized gradient of the following convex composite function: the sum of the largest eigenvalues of a smooth symmetric matrix-valued function of a set of real parameters. The dual matrix provides the information required to either verify first-order optimality conditions at a point or to generate a descent direction for the eigenvalue sum from that point, splitting a multiple eigenvalue if necessary. Connections with the classical literature on sums of eigenvalues and eigenvalue perturbation theory are discussed. Sums of the largest eigenvalues in the absolute value sense are also addressed.

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