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Using Improved Directions of Negative Curvature for the Solution of Bound-Constrained Nonconvex Problems

Author

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  • Javier Cano

    (Rey Juan Carlos University)

  • Javier M. Moguerza

    (Rey Juan Carlos University)

  • Francisco J. Prieto

    (Carlos III University)

Abstract

In this work, we describe the efficient use of improved directions of negative curvature for the solution of bound-constrained nonconvex problems. We follow an interior-point framework, in which the key point is the inclusion of computational low-cost procedures to improve directions of negative curvature obtained from a factorisation of the KKT matrix. From a theoretical point of view, it is well known that these directions ensure convergence to second-order KKT points. As a novelty, we consider the convergence rate of the algorithm with exploitation of negative curvature information. Finally, we test the performance of our proposal on both CUTEr/st and simulated problems, showing empirically that the enhanced directions affect positively the practical performance of the procedure.

Suggested Citation

  • Javier Cano & Javier M. Moguerza & Francisco J. Prieto, 2017. "Using Improved Directions of Negative Curvature for the Solution of Bound-Constrained Nonconvex Problems," Journal of Optimization Theory and Applications, Springer, vol. 174(2), pages 474-499, August.
    • Handle: RePEc:spr:joptap:v:174:y:2017:i:2:d:10.1007_s10957-017-1137-9
    DOI: 10.1007/s10957-017-1137-9
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    References listed on IDEAS

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    1. S. Sanmatías & E. Vercher, 1998. "A Generalized Conjugate Gradient Algorithm," Journal of Optimization Theory and Applications, Springer, vol. 98(2), pages 489-502, August.
    2. Alberto Olivares & Javier Moguerza, 2009. "Improving directions of negative curvature in an efficient manner," Annals of Operations Research, Springer, vol. 166(1), pages 183-201, February.
    3. Sun, Jie & Yang, Xiaoqi & Chen, Xiongda, 2005. "Quadratic cost flow and the conjugate gradient method," European Journal of Operational Research, Elsevier, vol. 164(1), pages 104-114, July.
    4. Moguerza, Javier M. & Olivares, Alberto & Prieto, Francisco J., 2007. "A note on the use of vector barrier parameters for interior-point methods," European Journal of Operational Research, Elsevier, vol. 181(2), pages 571-585, September.
    5. Walter Murray & Kien-Ming Ng, 2010. "An algorithm for nonlinear optimization problems with binary variables," Computational Optimization and Applications, Springer, vol. 47(2), pages 257-288, October.
    6. Olivares, Alberto & Moguerza, Javier M. & Prieto, Francisco J., 2008. "Nonconvex optimization using negative curvature within a modified linesearch," European Journal of Operational Research, Elsevier, vol. 189(3), pages 706-722, September.
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    Cited by:

    1. Seonho Park & Seung Hyun Jung & Panos M. Pardalos, 2020. "Combining Stochastic Adaptive Cubic Regularization with Negative Curvature for Nonconvex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 184(3), pages 953-971, March.

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