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Diagonal Approximation of the Hessian by Finite Differences for Unconstrained Optimization

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  • Neculai Andrei

    (Academy of Romanian Scientists)

Abstract

A new quasi-Newton method with a diagonal updating matrix is suggested, where the diagonal elements are determined by forward or by central finite differences. The search direction is a direction of sufficient descent. The algorithm is equipped with an acceleration scheme. The convergence of the algorithm is linear. The preliminary computational experiments use a set of 75 unconstrained optimization test problems classified in five groups according to the structure of their Hessian: diagonal, block-diagonal, band (tri- or penta-diagonal), sparse and dense. Subject to the CPU time metric, intensive numerical experiments show that, for problems with Hessian in a diagonal, block-diagonal or band structure, the algorithm with diagonal approximation of the Hessian by finite differences is top performer versus the well-established algorithms: the steepest descent and the Broyden–Fletcher–Goldfarb–Shanno. On the other hand, as a by-product of this numerical study, we show that the Broyden–Fletcher–Goldfarb–Shanno algorithm is faster for problems with sparse Hessian, followed by problems with dense Hessian.

Suggested Citation

  • Neculai Andrei, 2020. "Diagonal Approximation of the Hessian by Finite Differences for Unconstrained Optimization," Journal of Optimization Theory and Applications, Springer, vol. 185(3), pages 859-879, June.
    • Handle: RePEc:spr:joptap:v:185:y:2020:i:3:d:10.1007_s10957-020-01676-z
    DOI: 10.1007/s10957-020-01676-z
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    References listed on IDEAS

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    1. Michael Bartholomew-Biggs, 2008. "Nonlinear Optimization with Engineering Applications," Springer Optimization and Its Applications, Springer, number 978-0-387-78723-7, September.
    2. Neculai Andrei, 2017. "Continuous Nonlinear Optimization for Engineering Applications in GAMS Technology," Springer Optimization and Its Applications, Springer, number 978-3-319-58356-3, September.
    3. David G. Luenberger & Yinyu Ye, 2016. "Linear and Nonlinear Programming," International Series in Operations Research and Management Science, Springer, edition 4, number 978-3-319-18842-3, September.
    4. Neculai Andrei, 2017. "Applications of Continuous Nonlinear Optimization," Springer Optimization and Its Applications, in: Continuous Nonlinear Optimization for Engineering Applications in GAMS Technology, chapter 0, pages 47-117, Springer.
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    2. Shi, Yanlin, 2022. "A closed-form estimator for the Markov switching in mean model," Finance Research Letters, Elsevier, vol. 44(C).

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