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Exact linesearch limited-memory quasi-Newton methods for minimizing a quadratic function

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  • David Ek

    (KTH Royal Institute of Technology)

  • Anders Forsgren

    (KTH Royal Institute of Technology)

Abstract

The main focus in this paper is exact linesearch methods for minimizing a quadratic function whose Hessian is positive definite. We give a class of limited-memory quasi-Newton Hessian approximations which generate search directions parallel to those of the BFGS method, or equivalently, to those of the method of preconditioned conjugate gradients. In the setting of reduced Hessians, the class provides a dynamical framework for the construction of limited-memory quasi-Newton methods. These methods attain finite termination on quadratic optimization problems in exact arithmetic. We show performance of the methods within this framework in finite precision arithmetic by numerical simulations on sequences of related systems of linear equations, which originate from the CUTEst test collection. In addition, we give a compact representation of the Hessian approximations in the full Broyden class for the general unconstrained optimization problem. This representation consists of explicit matrices and gradients only as vector components.

Suggested Citation

  • David Ek & Anders Forsgren, 2021. "Exact linesearch limited-memory quasi-Newton methods for minimizing a quadratic function," Computational Optimization and Applications, Springer, vol. 79(3), pages 789-816, July.
    • Handle: RePEc:spr:coopap:v:79:y:2021:i:3:d:10.1007_s10589-021-00277-4
    DOI: 10.1007/s10589-021-00277-4
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    References listed on IDEAS

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    1. Anders Forsgren & Tove Odland, 2018. "On exact linesearch quasi-Newton methods for minimizing a quadratic function," Computational Optimization and Applications, Springer, vol. 69(1), pages 225-241, January.
    2. Nicholas Gould & Dominique Orban & Philippe Toint, 2015. "CUTEst: a Constrained and Unconstrained Testing Environment with safe threads for mathematical optimization," Computational Optimization and Applications, Springer, vol. 60(3), pages 545-557, April.
    3. David F. Shanno, 1978. "Conjugate Gradient Methods with Inexact Searches," Mathematics of Operations Research, INFORMS, vol. 3(3), pages 244-256, August.
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