## 2006.176: Modifying the interia of matrices arising in optimization

2006.176:
Nicholas J. Higham and Sheung Hun Cheng
(1998)
*Modifying the interia of matrices arising in optimization.*
Linear Algebra and its Applications, 275-276.
pp. 261-279.
ISSN 0024-3795

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DOI: 10.1016/S0024-3795(97)10015-5

## Abstract

Applications in constrained optimization (and other areas) produce symmetric matrices with a natural block 2 × 2 structure. An optimality condition leads to the problem of perturbing the (1,1) block of the matrix to achieve a specific inertia. We derive a perturbation of minimal norm, for any unitarily invariant norm, that increases the number of nonnegative eigenvalues by a given amount, and we show how it can be computed efficiently given a factorization of the original matrix. We also consider an alternative way to satisfy the optimality condition based on a projection approach. Theoretical tools developed here include an extension of Ostrowski's theorem on congruences and some lemmas on inertias of block 2 × 2 symmetric matrices.

Item Type: | Article |
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Uncontrolled Keywords: | 65F15; 15A42Inertia; Optimization; Nonlinear programming; unitarily invariant norm |

Subjects: | MSC 2000 > 15 Linear and multilinear algebra; matrix theory MSC 2000 > 65 Numerical analysis |

MIMS number: | 2006.176 |

Deposited By: | Miss Louise Stait |

Deposited On: | 04 July 2006 |

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