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

Martin B Reed

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)

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.
Original languageEnglish
Pages (from-to)606-615
Number of pages10
JournalInternational Journal of Computer Mathematics
Volume86
Issue number4
DOIs
Publication statusPublished - 2009

Keywords

  • limited memory
  • quasi-Newton methods
  • unconstrained optimisation

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