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Spectral Projected Gradient Methods: Review and Perspectives

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  • Birgin, Ernesto G.
  • Martínez, Jose Mario
  • Raydan, Marcos

Abstract

Over the last two decades, it has been observed that using the gradient vector as a search direction in large-scale optimization may lead to efficient algorithms. The effectiveness relies on choosing the step lengths according to novel ideas that are related to the spectrum of the underlying local Hessian rather than related to the standard decrease in the objective function. A review of these so-called spectral projected gradient methods for convex constrained optimization is presented. To illustrate the performance of these low-cost schemes, an optimization problem on the set of positive definite matrices is described.

Suggested Citation

  • Birgin, Ernesto G. & Martínez, Jose Mario & Raydan, Marcos, 2014. "Spectral Projected Gradient Methods: Review and Perspectives," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 60(i03).
    • Handle: RePEc:jss:jstsof:v:060:i03
    DOI: http://hdl.handle.net/10.18637/jss.v060.i03
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    References listed on IDEAS

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    1. M. A. Diniz-Ehrhardt & M. A. Gomes-Ruggiero & J. M. Martínez & S. A. Santos, 2004. "Augmented Lagrangian Algorithms Based on the Spectral Projected Gradient Method for Solving Nonlinear Programming Problems," Journal of Optimization Theory and Applications, Springer, vol. 123(3), pages 497-517, December.
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    Cited by:

    1. Wolfgang Schadner, 2021. "Feasible Implied Correlation Matrices from Factor Structures," Papers 2107.00427, arXiv.org.
    2. J. M. Martínez & M. Raydan, 2017. "Cubic-regularization counterpart of a variable-norm trust-region method for unconstrained minimization," Journal of Global Optimization, Springer, vol. 68(2), pages 367-385, June.
    3. Yu-Hong Dai & Yakui Huang & Xin-Wei Liu, 2019. "A family of spectral gradient methods for optimization," Computational Optimization and Applications, Springer, vol. 74(1), pages 43-65, September.
    4. Crisci, Serena & Ruggiero, Valeria & Zanni, Luca, 2019. "Steplength selection in gradient projection methods for box-constrained quadratic programs," Applied Mathematics and Computation, Elsevier, vol. 356(C), pages 312-327.
    5. Nataša Krejić & Nataša Krklec Jerinkić, 2019. "Spectral projected gradient method for stochastic optimization," Journal of Global Optimization, Springer, vol. 73(1), pages 59-81, January.
    6. Marco Viola & Mara Sangiovanni & Gerardo Toraldo & Mario R. Guarracino, 2019. "Semi-supervised generalized eigenvalues classification," Annals of Operations Research, Springer, vol. 276(1), pages 249-266, May.
    7. Na Zhao & Qingzhi Yang & Yajun Liu, 2017. "Computing the generalized eigenvalues of weakly symmetric tensors," Computational Optimization and Applications, Springer, vol. 66(2), pages 285-307, March.
    8. N. Krejić & E. H. M. Krulikovski & M. Raydan, 2023. "A Low-Cost Alternating Projection Approach for a Continuous Formulation of Convex and Cardinality Constrained Optimization," SN Operations Research Forum, Springer, vol. 4(4), pages 1-24, December.
    9. Roberto Andreani & Marcos Raydan, 2021. "Properties of the delayed weighted gradient method," Computational Optimization and Applications, Springer, vol. 78(1), pages 167-180, January.
    10. di Serafino, Daniela & Ruggiero, Valeria & Toraldo, Gerardo & Zanni, Luca, 2018. "On the steplength selection in gradient methods for unconstrained optimization," Applied Mathematics and Computation, Elsevier, vol. 318(C), pages 176-195.
    11. J. Martínez & M. Raydan, 2015. "Separable cubic modeling and a trust-region strategy for unconstrained minimization with impact in global optimization," Journal of Global Optimization, Springer, vol. 63(2), pages 319-342, October.
    12. Milagros Loreto & Hugo Aponte & Debora Cores & Marcos Raydan, 2017. "Nonsmooth spectral gradient methods for unconstrained optimization," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 5(4), pages 529-553, December.
    13. Pospíšil, Lukáš & Dostál, Zdeněk, 2018. "The projected Barzilai–Borwein method with fall-back for strictly convex QCQP problems with separable constraints," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 145(C), pages 79-89.
    14. V. S. Amaral & R. Andreani & E. G. Birgin & D. S. Marcondes & J. M. Martínez, 2022. "On complexity and convergence of high-order coordinate descent algorithms for smooth nonconvex box-constrained minimization," Journal of Global Optimization, Springer, vol. 84(3), pages 527-561, November.
    15. Geovani N. Grapiglia & Ekkehard W. Sachs, 2017. "On the worst-case evaluation complexity of non-monotone line search algorithms," Computational Optimization and Applications, Springer, vol. 68(3), pages 555-577, December.
    16. Andrej Čopar & Blaž Zupan & Marinka Zitnik, 2019. "Fast optimization of non-negative matrix tri-factorization," PLOS ONE, Public Library of Science, vol. 14(6), pages 1-15, June.
    17. Filippozzi, Rafaela & Gonçalves, Douglas S. & Santos, Luiz-Rafael, 2023. "First-order methods for the convex hull membership problem," European Journal of Operational Research, Elsevier, vol. 306(1), pages 17-33.
    18. Harry Fernando Oviedo Leon, 2019. "A delayed weighted gradient method for strictly convex quadratic minimization," Computational Optimization and Applications, Springer, vol. 74(3), pages 729-746, December.
    19. O. P. Ferreira & M. Lemes & L. F. Prudente, 2022. "On the inexact scaled gradient projection method," Computational Optimization and Applications, Springer, vol. 81(1), pages 91-125, January.
    20. Wolfgang Schadner & Joshua Traut, 2022. "Estimating Forward-Looking Stock Correlations from Risk Factors," Mathematics, MDPI, vol. 10(10), pages 1-19, May.
    21. Varadhan, Ravi, 2014. "Numerical Optimization in R: Beyond optim," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 60(i01).

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