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A limited memory steepest descent method

Abstract

The possibilities inherent in steepest descent methods have been considerably amplified by the introduction of the Barzilai–Borwein choice of step-size, and other related ideas. These methods have proved to be competitive with conjugate gradient methods for the minimization of large dimension unconstrained minimization problems. This paper suggests a method which is able to take advantage of the availability of a few additional ‘long’ vectors of storage to achieve a significant improvement in performance, both for quadratic and non-quadratic objective functions. It makes use of certain Ritz values related to the Lanczos process (Lanczos in J Res Nat Bur Stand 45:255–282, 1950). Some underlying theory is provided, and numerical evidence is set out showing that the new method provides a competitive and more simple alternative to the state of the art l-BFGS limited memory method.

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Correspondence to Roger Fletcher.

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Fletcher, R. A limited memory steepest descent method. Math. Program. 135, 413–436 (2012). https://doi.org/10.1007/s10107-011-0479-6

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  • DOI: https://doi.org/10.1007/s10107-011-0479-6

Mathematics Subject Classification (2000)

  • 90C06
  • 90C26
  • 65K05