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

M. L. Overton, R. S. Womersley

Research output: Contribution to journalArticle

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.

Original languageEnglish (US)
Pages (from-to)321-357
Number of pages37
JournalMathematical Programming
Volume62
Issue number1-3
DOIs
StatePublished - Feb 1993

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Duality Theory
Largest Eigenvalue
Optimality Conditions
Symmetric matrix
Eigenvalue
Convex function
Multiple Eigenvalues
First-order Optimality Conditions
Composite function
Nonsmooth Function
Generalized Gradient
Subdifferential
Descent
Absolute value
Perturbation Theory
Optimality conditions
Duality theory
Eigenvalues
Verify
Composite materials

Keywords

  • convex composite optimization
  • generalized gradient
  • maximum eigenvalue
  • Nonsmooth optimization
  • sum of eigenvalues

ASJC Scopus subject areas

  • Applied Mathematics
  • Mathematics(all)
  • Safety, Risk, Reliability and Quality
  • Management Science and Operations Research
  • Software
  • Computer Graphics and Computer-Aided Design
  • Computer Science(all)

Cite this

Optimality conditions and duality theory for minimizing sums of the largest eigenvalues of symmetric matrices. / Overton, M. L.; Womersley, R. S.

In: Mathematical Programming, Vol. 62, No. 1-3, 02.1993, p. 321-357.

Research output: Contribution to journalArticle

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