Eigenvalue optimization

Adrian S. Lewis, Michael Overton

Research output: Contribution to journalArticle

Abstract

Optimization problems involving eigenvalues arise in many different mathematical disciplines. This article is divided into two parts. Part I gives a historical account of the development of the field. We discuss various applications that have been especially influential, from structural analysis to combinatorial optimization, and we survey algorithmic developments, including the recent advance of interior-point methods for a specific problem class: semidefinite programming. In Part II we primarily address optimization of convex functions of eigenvalues of symmetric matrices subject to linear constraints. We derive a fairly complete mathematical theory, some of it classical and some of it new. Using the elegant language of conjugate duality theory, we highlight the parallels between the analysis of invariant matrix norms and weakly invariant convex matrix functions. We then restrict our attention further to linear and semidefinite programming, emphasizing the parallel duality theory and comparing primal-dual interior-point methods for the two problem classes. The final section presents some apparently new variational results about eigenvalues of nonsymmetric matrices, unifying known characterizations of the spectral abscissa (related to Lyapunov theory) and the spectral radius (as an infimum of matrix norms).

Original languageEnglish (US)
Pages (from-to)149-190
Number of pages42
JournalActa Numerica
Volume5
DOIs
StatePublished - 1996

Fingerprint

Eigenvalue Optimization
Matrix Norm
Duality Theory
Semidefinite Programming
Eigenvalue
Convex function
Conjugate Duality
Abscissa
Primal-dual Interior Point Method
Nonsymmetric Matrix
Lyapunov Theory
Invariant
Interior Point Method
Combinatorial Optimization
Matrix Function
Spectral Radius
Structural Analysis
Linear Constraints
Symmetric matrix
Linear programming

ASJC Scopus subject areas

  • Numerical Analysis
  • Mathematics(all)

Cite this

Eigenvalue optimization. / Lewis, Adrian S.; Overton, Michael.

In: Acta Numerica, Vol. 5, 1996, p. 149-190.

Research output: Contribution to journalArticle

Lewis, Adrian S. ; Overton, Michael. / Eigenvalue optimization. In: Acta Numerica. 1996 ; Vol. 5. pp. 149-190.
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