L-Broyden methods: a generalization of the L-BFGS method to the limited-memory Broyden family

Authors:
M. B. Reed
Affiliations:
Department of Mathematical Sciences, University of Bath, Bath, UK
Venue:
International Journal of Computer Mathematics
Year:
2009

Citing 7
Cited 0

Remark on algorithm 630

ACM Transactions on Mathematical Software (TOMS)
On the limited memory BFGS method for large scale optimization

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Convergence of quasi-Newton matrices generated by the symmetric rank one update

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Representations of quasi-Newton matrices and their use in limited memory methods

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Algorithm 630: BBVSCG–a variable-storage algorithm for function minimization

ACM Transactions on Mathematical Software (TOMS)
BFGS with Update Skipping and Varying Memory

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Updating the Self-Scaling Symmetric Rank One Algorithm with Limited Memory for Large-Scale Unconstrained Optimization

Computational Optimization and Applications

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Abstract

The paper derives a multiplicative update equation for the convex Broyden family of quasi-Newton (QN) updates. The well-known multiplicative Broyden-fletcher-Goldfarb-Shanno (BFGS) update is a special case of this. Using a self-scaling parameter, the formula is extended to the SR1 update. It is shown that for each Broyden update, a pair of multiplicative update formulae can be defined (coincident in the case of Davidon-fletcher-Powell (DFP)). The multiplicative updates are used to define limited-memory QN algorithms, denoted the L-Broyden methods, of which L-BFGS is a special case. Numerical results show that the L-Broyden methods are competitive with extensions of the variable storage conjugate gradients limited memory QN method to other Broyden updates, but that L-BFGS with Shanno scaling remains the most efficient and reliable method in the L-Broyden family.