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Exploiting effective negative curvature directions via SYMMBK algorithm, in Newton–Krylov methods

Author

Listed:
  • Giovanni Fasano

    (University Ca’ Foscari)

  • Christian Piermarini

    (SAPIENZA University of Rome)

  • Massimo Roma

    (SAPIENZA University of Rome)

Abstract

In this paper we consider the issue of computing negative curvature directions, for nonconvex functions, within Newton–Krylov methods for large scale unconstrained optimization. In the last decades this issue has been widely investigated in the literature, and different approaches have been proposed. We focus on the well known SYMMBK method introduced for solving large scale symmetric possibly indefinite linear systems (Bunch and Kaufman in Math Comput 31:163–179, 2003; Chandra in Conjugate gradient methods for partial differential equations, Yale University, New Haven, 1978; Conn et al. Trust-region methods. MPS-SIAM Series on Optimization, Philadelphia, 2000; HSL 2013: A collection of Fortran codes for large scale scientific computation. http://www.hsl.rl.ac.uk/ ), and show how to exploit it to yield an effective negative curvature direction in optimization frameworks. The distinguishing feature of our proposal is that the computation of negative curvatures is basically carried out as by–product of SYMMBK procedure, without storing no more than two additional vectors. Hence, no explicit matrix factorization or matrix storage is required. An extensive numerical experimentation has been performed on CUTEst problems; the obtained results have been analyzed also through novel profiles (Quality Profiles) which highlighted the good capability of the algorithms which use negative curvature directions to determine better local minimizers.

Suggested Citation

  • Giovanni Fasano & Christian Piermarini & Massimo Roma, 2025. "Exploiting effective negative curvature directions via SYMMBK algorithm, in Newton–Krylov methods," Computational Optimization and Applications, Springer, vol. 91(2), pages 617-647, June.
    • Handle: RePEc:spr:coopap:v:91:y:2025:i:2:d:10.1007_s10589-025-00650-7
    DOI: 10.1007/s10589-025-00650-7
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    References listed on IDEAS

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    1. Hao Jiang & Daniel P. Robinson & René Vidal & Chong You, 2018. "A nonconvex formulation for low rank subspace clustering: algorithms and convergence analysis," Computational Optimization and Applications, Springer, vol. 70(2), pages 395-418, June.
    2. Andrea Caliciotti & Giovanni Fasano & Florian Potra & Massimo Roma, 2020. "Issues on the use of a modified Bunch and Kaufman decomposition for large scale Newton’s equation," Computational Optimization and Applications, Springer, vol. 77(3), pages 627-651, December.
    3. 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.
    4. Renato Leone & Giovanni Fasano & Massimo Roma & Yaroslav D. Sergeyev, 2020. "Iterative Grossone-Based Computation of Negative Curvature Directions in Large-Scale Optimization," Journal of Optimization Theory and Applications, Springer, vol. 186(2), pages 554-589, August.
    5. G. Fasano, 2005. "Planar Conjugate Gradient Algorithm for Large-Scale Unconstrained Optimization, Part 1: Theory," Journal of Optimization Theory and Applications, Springer, vol. 125(3), pages 523-541, June.
    6. Caliciotti, Andrea & Fasano, Giovanni & Roma, Massimo, 2018. "Preconditioned Nonlinear Conjugate Gradient methods based on a modified secant equation," Applied Mathematics and Computation, Elsevier, vol. 318(C), pages 196-214.
    7. G. Fasano, 2005. "Planar Conjugate Gradient Algorithm for Large-Scale Unconstrained Optimization, Part 2: Application," Journal of Optimization Theory and Applications, Springer, vol. 125(3), pages 543-558, June.
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    Cited by:

    1. Stefania Bellavia & Valentina Simone & Benedetta Morini, 2025. "Preface: New trends in large scale optimization," Computational Optimization and Applications, Springer, vol. 91(2), pages 351-356, June.

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