Two–parameter scaled memoryless BFGS methods with a nonmonotone choice for the initial step length


A class of two–parameter scaled memoryless BFGS methods is developed for solving unconstrained optimization problems. Then, the scaling parameters are determined in a way to improve the condition number of the corresponding memoryless BFGS update. It is shown that for uniformly convex objective functions, search directions of the method satisfy the sufficient descent condition which leads to the global convergence. To achieve convergence for general functions, a revised version of the method is developed based on the Li–Fukushima modified secant equation. To enhance performance of the methods, a nonmonotone scheme for computing the initial value of the step length is suggested to be used in the line search procedure. Numerical experiments are done on a set of unconstrained optimization test problems of the CUTEr collection. They show efficiency of the proposed algorithms in the sense of the Dolan–Moré performance profile.

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This research was in part supported by the grant 96013024 from Iran National Science Foundation (INSF), and in part by the Research Council of Semnan University. The authors thank the anonymous reviewers for their valuable comments and suggestions helped to improve the quality of this work. They are also grateful to Professor Michael Navon for providing the line search code.

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Correspondence to Saman Babaie–Kafaki.

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Babaie–Kafaki, S., Aminifard, Z. Two–parameter scaled memoryless BFGS methods with a nonmonotone choice for the initial step length. Numer Algor 82, 1345–1357 (2019).

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  • Unconstrained optimization
  • Quasi–Newton method
  • Memoryless BFGS update
  • Global convergence
  • Line search

Mathematics Subject Classification (2010)

  • 90C53
  • 65K05