A Riemannian BFGS Method for Nonconvex Optimization Problems

Authors

Wen Huang, P.-A. Absil, K. A. Gallivan

Abstract

In this paper, a Riemannian BFGS method is defined for minimizing a smooth function on a Riemannian manifold endowed with a retraction and a vector transport. The method is based on a Riemannian generalization of a cautious update and a weak line search condition. It is shown that the Riemannian BFGS method converges (i) globally to a stationary point without assuming that the objective function is convex and (ii) superlinearly to a nondegenerate minimizer. The weak line search condition completely exempts us from resorting to the differentiated retraction. The joint diagonalization problem is used to demonstrate the performance of the algorithm with various parameters, line search conditions, and pairs of retraction and vector transport.

Key words

Riemannian optimization; manifold optimization; Quasi-Newton methods; BFGS method; Stiefel manifold, Soft ICA;

Status

Lecture Notes in Computational Science and Engineering, DOI:10.1007/978-3-319-39929-4_60, 2016.

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BibTex entry

Technical Report
@TECHREPORT{HAG2015,
author = "Wen Huang and P.-A. Absil and K. A. Gallivan",
title = "A Riemannian BFGS Method for Nonconvex Optimization Problems",
institution = "U.C.Louvain",
number = "UCL-INMA-2015.11",
year = 2015,
}