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Limited-memory BFGS systems with diagonal updates

Authors
Journal
Linear Algebra and its Applications
0024-3795
Publisher
Elsevier
Publication Date
Volume
437
Issue
1
Identifiers
DOI: 10.1016/j.laa.2012.02.005
Keywords
  • L-Bfgs
  • Quasi-Newton Methods
  • Limited-Memory Methods
  • Inverses
  • Sherman–Morrison–Woodbury
  • Diagonal Updates

Abstract

Abstract In this paper, we investigate a formula to solve systems of the form (B+σI)x=y, where B is a limited-memory BFGS quasi-Newton matrix and σ is a positive constant. These types of systems arise naturally in large-scale optimization such as trust-region methods as well as doubly-augmented Lagrangian methods. We show that provided a simple condition holds on B0 and σ, the system (B+σI)x=y can be solved via a recursion formula that requires only vector inner products. This formula has complexity M2n, where M is the number of L-BFGS updates and n≫M is the dimension of x.

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