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On exact linesearch quasi-Newton methods for minimizing a quadratic function

Author

Listed:
  • Anders Forsgren

    (KTH Royal Institute of Technology)

  • Tove Odland

    (KTH Royal Institute of Technology)

Abstract

This paper concerns exact linesearch quasi-Newton methods for minimizing a quadratic function whose Hessian is positive definite. We show that by interpreting the method of conjugate gradients as a particular exact linesearch quasi-Newton method, necessary and sufficient conditions can be given for an exact linesearch quasi-Newton method to generate a search direction which is parallel to that of the method of conjugate gradients. We also analyze update matrices and give a complete description of the rank-one update matrices that give search direction parallel to those of the method of conjugate gradients. In particular, we characterize the family of such symmetric rank-one update matrices that preserve positive definiteness of the quasi-Newton matrix. This is in contrast to the classical symmetric-rank-one update where there is no freedom in choosing the matrix, and positive definiteness cannot be preserved. The analysis is extended to search directions that are parallel to those of the preconditioned method of conjugate gradients in a straightforward manner.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:coopap:v:69:y:2018:i:1:d:10.1007_s10589-017-9940-7
    DOI: 10.1007/s10589-017-9940-7
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    1. Anders Forsgren & Tove Odland, 2015. "On the connection between the conjugate gradient method and quasi-Newton methods on quadratic problems," Computational Optimization and Applications, Springer, vol. 60(2), pages 377-392, March.
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    Cited by:

    1. 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.

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