Modifying the interia of matrices arising in optimization

Higham, Nicholas J. and Cheng, Sheung Hun (1998) Modifying the interia of matrices arising in optimization. Linear Algebra and its Applications, 275-27. pp. 261-279. ISSN 0024-3795

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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
Uncontrolled Keywords:	65F15; 15A42Inertia; Optimization; Nonlinear programming; unitarily invariant norm
Subjects:	MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User:	Ms Lucy van Russelt
Date Deposited:	04 Jul 2006
Last Modified:	20 Oct 2017 14:12
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/359

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