The Geometry of the Newton Method on Non-Compact Lie Groups

Authors:
Robert Mahony;Jonathan H. Manton
Affiliations:
Department of Engineering, Australian National University, A.C.T., 0200, Australia E-mail: Robert.Mahony@anu.edu.au;Department of Electrical and Electronic Engineering, The University of Melbourne, Parkville, Victoria, 3010, Australia. E-mail: j.manton@ee.mu.oz.au
Venue:
Journal of Global Optimization
Year:
2002

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Abstract

An important class of optimization problems involve minimizing a cost function on a Lie group. In the case where the Lie group is non-compact there is no natural choice of a Riemannian metric and it is not possible to apply recent results on the optimization of functions on Riemannian manifolds. In this paper the invariant structure of a Lie group is exploited to provide a strong interpretation of a Newton iteration on a general Lie group. The paper unifies several previous algorithms proposed in the literature in a single theoretical framework. Local asymptotic quadratic convergence is proved for the algorithms considered.