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2016 | OriginalPaper | Chapter

A Riemannian BFGS Method for Nonconvex Optimization Problems

Authors : Wen Huang, P.-A. Absil, Kyle A. Gallivan

Published in: Numerical Mathematics and Advanced Applications ENUMATH 2015

Publisher: Springer International Publishing

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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 removes completely the need to consider 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.

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previous chapter On the Stability of a Weighted Finite Difference Scheme for Hyperbolic Equation with Integral Boundary Conditions

next chapter Discrete Lie Derivative

This mapping is not even required to be continuous in the definition. The smoothness is imposed in the convergence analyses.

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Title: A Riemannian BFGS Method for Nonconvex Optimization Problems
Authors: Wen Huang
P.-A. Absil
Kyle A. Gallivan
Publisher: Springer International Publishing
Book: Numerical Mathematics and Advanced Applications ENUMATH 2015
Print ISBN: 978-3-319-39927-0

Electronic ISBN: 978-3-319-39929-4

Copyright Year: 2016
DOI: https://doi.org/10.1007/978-3-319-39929-4_60

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