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A Framework of Conjugate Direction Methods for Symmetric Linear Systems in Optimization

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  • Giovanni Fasano

    (University Ca’Foscari of Venice
    S.Giobbe)

Abstract

In this paper, we introduce a parameter-dependent class of Krylov-based methods, namely Conjugate Directions $$(CD)$$ ( C D ) , for the solution of symmetric linear systems. We give evidence that, in our proposal, we generate sequences of conjugate directions, extending some properties of the standard conjugate gradient (CG) method, in order to preserve the conjugacy. For specific values of the parameters in our framework, we obtain schemes equivalent to both the CG and the scaled-CG. We also prove the finite convergence of the algorithms in $$CD$$ C D , and we provide some error analysis. Finally, preconditioning is introduced for $$CD$$ C D , and we show that standard error bounds for the preconditioned CG also hold for the preconditioned $$CD$$ C D .

Suggested Citation

  • Giovanni Fasano, 2015. "A Framework of Conjugate Direction Methods for Symmetric Linear Systems in Optimization," Journal of Optimization Theory and Applications, Springer, vol. 164(3), pages 883-914, March.
    • Handle: RePEc:spr:joptap:v:164:y:2015:i:3:d:10.1007_s10957-014-0600-0
    DOI: 10.1007/s10957-014-0600-0
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    References listed on IDEAS

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    1. 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.
    2. Giovanni Fasano & Massimo Roma, 2013. "Preconditioning Newton–Krylov methods in nonconvex large scale optimization," Computational Optimization and Applications, Springer, vol. 56(2), pages 253-290, October.
    3. G. Fasano, 2007. "Lanczos Conjugate-Gradient Method and Pseudoinverse Computation on Indefinite and Singular Systems," Journal of Optimization Theory and Applications, Springer, vol. 132(2), pages 267-285, February.
    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.
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    Cited by:

    1. Giovanni Fasano & Raffaele Pesenti, 2017. "Conjugate Direction Methods and Polarity for Quadratic Hypersurfaces," Journal of Optimization Theory and Applications, Springer, vol. 175(3), pages 764-794, December.

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    More about this item

    Keywords

    Krylov-based methods; Conjugate direction methods; Conjugacy loss and error analysis; Preconditioning;
    All these keywords.

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques

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