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IMA Journal of Numerical Analysis Advance Access originally published online on June 5, 2008
IMA Journal of Numerical Analysis 2008 28(4):665-689; doi:10.1093/imanum/drn029
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© The author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

An implicit trust-region method on Riemannian manifolds

C. G. Baker{dagger}

Computational Mathematics and Algorithms, Sandia National Laboratories, Albuquerque, NM 87185-1320, USA and School of Computational Science, Florida State University, Tallahassee, FL 32306-4120, USA

P.-A. Absil

Département d'ingénierie mathématique, Université catholique de Louvain, Avenue Georges Lemaître 4, 1348 Louvain-la-Neuve, Belgium

K. A. Gallivan

School of Computational Science, Florida State University, Tallahassee, FL 32306-4120, USA

1 Email: cgbaker{at}gmail.com

Received on 30 March 2007. Revised on 29 December 2007.


  Abstract

We propose and analyse an implicit trust-region method in the general setting of Riemannian manifolds. The method is implicit in that the trust region is defined as a superlevel set of the {rho} ratio of the actual over-predicted decrease in the objective function. Since this method potentially requires the evaluation of the objective function at each step of the inner iteration, we do not recommend it for problems where the objective function is expensive to evaluate. However, we show that on some instances of a very structured problem—the extreme symmetric eigenvalue problem or equivalently the optimization of the Rayleigh quotient on the unit sphere—the resulting numerical method outperforms state-of-the-art algorithms. Moreover, the new method inherits the detailed convergence analysis of the generic Riemannian trust-region method.

Key Words: optimization on manifolds; trust-region methods; Newton's method; symmetric generalized eigenvalue problem

Dedicated to Prof. M. J. D. Powell on the occasion of his 70th birthday.


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