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Finding search directions in quasi-Newton methods for minimizing a quadratic function subject to uncertainty

Author

Listed:
  • Shen Peng

    (Xidian University
    KTH Royal Institute of Technology)

  • Gianpiero Canessa

    (KTH Royal Institute of Technology)

  • David Ek

    (KTH Royal Institute of Technology)

  • Anders Forsgren

    (KTH Royal Institute of Technology)

Abstract

We investigate quasi-Newton methods for minimizing a strongly convex quadratic function which is subject to errors in the evaluation of the gradients. In particular, we focus on computing search directions for quasi-Newton methods that all give identical behavior in exact arithmetic, generating minimizers of Krylov subspaces of increasing dimensions, thereby having finite termination. The BFGS quasi-Newton method may be seen as an ideal method in exact arithmetic and is empirically known to behave very well on a quadratic problem subject to small errors. We investigate large-error scenarios, in which the expected behavior is not so clear. We consider memoryless methods that are less expensive than the BFGS method, in that they generate low-rank quasi-Newton matrices that differ from the identity by a symmetric matrix of rank two. In addition, a more advanced model for generating the search directions is proposed, based on solving a chance-constrained optimization problem. Our numerical results indicate that for large errors, such a low-rank memoryless quasi-Newton method may perform better than a BFGS method. In addition, the results indicate a potential edge by including the chance-constrained model in the memoryless quasi-Newton method.

Suggested Citation

  • Shen Peng & Gianpiero Canessa & David Ek & Anders Forsgren, 2025. "Finding search directions in quasi-Newton methods for minimizing a quadratic function subject to uncertainty," Computational Optimization and Applications, Springer, vol. 91(1), pages 145-171, May.
  • Handle: RePEc:spr:coopap:v:91:y:2025:i:1:d:10.1007_s10589-025-00661-4
    DOI: 10.1007/s10589-025-00661-4
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    References listed on IDEAS

    as
    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. B. K. Pagnoncelli & S. Ahmed & A. Shapiro, 2009. "Sample Average Approximation Method for Chance Constrained Programming: Theory and Applications," Journal of Optimization Theory and Applications, Springer, vol. 142(2), pages 399-416, August.
    3. Brian Irwin & Eldad Haber, 2023. "Secant penalized BFGS: a noise robust quasi-Newton method via penalizing the secant condition," Computational Optimization and Applications, Springer, vol. 84(3), pages 651-702, April.
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