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# Smooth convex approximation to the maximum eigenvalue function

Chen, Xin, Qi, Houduo, Qi, Liqun and Teo, Kok-Lay (2004) Smooth convex approximation to the maximum eigenvalue function Journal of Global Optimization, 30, (2-3), pp. 253-270.

Record type: Article

## Abstract

In this paper, we consider smooth convex approximations to the maximum eigenvalue function. To make it applicable to a wide class of applications, the study is conducted on the composite function of the maximum eigenvalue function and a linear operator mapping Rm to Sn, the space of n-by-n symmetric matrices. The composite function in turn is the natural objective function of minimizing the maximum eigenvalue function over an affine space in Sn. This leads to a sequence of smooth convex minimization problems governed by a smoothing parameter. As the parameter goes to zero, the original problem is recovered. We then develop a computable Hessian formula of the smooth convex functions, matrix representation of the Hessian, and study the regularity conditions which guarantee the nonsingularity of the Hessian matrices. The study on the well-posedness of the smooth convex function leads to a regularization method which is globally convergent.

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Published date: November 2004
Keywords: matrix representation, spectral function, symmetric function, tikhonov regularization
Organisations: Operational Research

## Identifiers

Local EPrints ID: 29650
URI: http://eprints.soton.ac.uk/id/eprint/29650
ISSN: 0925-5001
PURE UUID: 127e8404-9ed7-4405-a783-b30caeda3f4a

## Catalogue record

Date deposited: 11 May 2006

## Contributors

Author: Xin Chen
Author: Houduo Qi
Author: Liqun Qi
Author: Kok-Lay Teo