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Limited-memory BFGS with displacement aggregation


A displacement aggregation strategy is proposed for the curvature pairs stored in a limited-memory BFGS (a.k.a. L-BFGS) method such that the resulting (inverse) Hessian approximations are equal to those that would be derived from a full-memory BFGS method. This means that, if a sufficiently large number of pairs are stored, then an optimization algorithm employing the limited-memory method can achieve the same theoretical convergence properties as when full-memory (inverse) Hessian approximations are stored and employed, such as a local superlinear rate of convergence under assumptions that are common for attaining such guarantees. To the best of our knowledge, this is the first work in which a local superlinear convergence rate guarantee is offered by a quasi-Newton scheme that does not either store all curvature pairs throughout the entire run of the optimization algorithm or store an explicit (inverse) Hessian approximation. Numerical results are presented to show that displacement aggregation within an adaptive L-BFGS scheme can lead to better performance than standard L-BFGS.

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  1. Quasi-Newton methods offer the ability to update Hessian and/or inverse Hessian approximations, which is why we state inverse parenthetically here. For ease of exposition throughout the remainder of the paper, we often drop mention of the inverse, although in many cases it is the approximation of the inverse, not the Hessian approximation, that is used in practice.

  2. By limited-memory-type BFGS algorithm, we mean one that stores and employs a finite set of curvature pairs rather than an explicit Hessian approximation.

  3. This provides evidence for the belief, held by some optimization researchers, that when solving certain large-scale problems one often observes that consecutive steps lie approximately in low-dimensional subspaces.


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Correspondence to Frank E. Curtis.

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This material is based upon work supported by the National Science Foundation under grant numbers CCF–1618717 and CCF–1740796.

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Berahas, A.S., Curtis, F.E. & Zhou, B. Limited-memory BFGS with displacement aggregation. Math. Program. 194, 121–157 (2022).

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  • Nonlinear optimization
  • Quasi-Newton algorithms
  • Broyden–Fletcher–Goldfarb–Shanno (BFGS)
  • Limited-memory BFGS
  • Superlinear convergence

Mathematics Subject Classification

  • 49M37
  • 65K05
  • 65K10
  • 90C30
  • 90C53