### 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 language | English (US) |
---|---|

Pages (from-to) | 149-190 |

Number of pages | 42 |

Journal | Acta Numerica |

Volume | 5 |

DOIs | |

State | Published - 1996 |

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### ASJC Scopus subject areas

- Numerical Analysis
- Mathematics(all)

### Cite this

*Acta Numerica*,

*5*, 149-190. https://doi.org/10.1017/S0962492900002646

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

Research output: Contribution to journal › Article

*Acta Numerica*, vol. 5, pp. 149-190. https://doi.org/10.1017/S0962492900002646

}

TY - JOUR

T1 - Eigenvalue optimization

AU - Lewis, Adrian S.

AU - Overton, Michael

PY - 1996

Y1 - 1996

N2 - 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).

AB - 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).

UR - http://www.scopus.com/inward/record.url?scp=0042602157&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0042602157&partnerID=8YFLogxK

U2 - 10.1017/S0962492900002646

DO - 10.1017/S0962492900002646

M3 - Article

VL - 5

SP - 149

EP - 190

JO - Acta Numerica

JF - Acta Numerica

SN - 0962-4929

ER -