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Interior-point methods for nonconvex nonlinear programming: cubic regularization

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  • Hande Benson
  • David Shanno

Abstract

In this paper, we present a primal-dual interior-point method for solving nonlinear programming problems. It employs a Levenberg-Marquardt (LM) perturbation to the Karush-Kuhn-Tucker (KKT) matrix to handle indefinite Hessians and a line search to obtain sufficient descent at each iteration. We show that the LM perturbation is equivalent to replacing the Newton step by a cubic regularization step with an appropriately chosen regularization parameter. This equivalence allows us to use the favorable theoretical results of Griewank (The modification of Newton’s method for unconstrained optimization by bounding cubic terms, 1981 ), Nesterov and Polyak (Math. Program., Ser. A 108:177–205, 2006 ), Cartis et al. (Math. Program., Ser. A 127:245–295, 2011 ; Math. Program., Ser. A 130:295–319, 2011 ), but its application at every iteration of the algorithm, as proposed by these papers, is computationally expensive. We propose a hybrid method: use a Newton direction with a line search on iterations with positive definite Hessians and a cubic step, found using a sufficiently large LM perturbation to guarantee a steplength of 1, otherwise. Numerical results are provided on a large library of problems to illustrate the robustness and efficiency of the proposed approach on both unconstrained and constrained problems. Copyright Springer Science+Business Media New York 2014

Suggested Citation

  • Hande Benson & David Shanno, 2014. "Interior-point methods for nonconvex nonlinear programming: cubic regularization," Computational Optimization and Applications, Springer, vol. 58(2), pages 323-346, June.
    • Handle: RePEc:spr:coopap:v:58:y:2014:i:2:p:323-346
    DOI: 10.1007/s10589-013-9626-8
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    References listed on IDEAS

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    1. NESTEROV, Yu., 2006. "Cubic regularization of Newton’s method for convex problems with constraints," LIDAM Discussion Papers CORE 2006039, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. NESTEROV, Yurii & POLYAK, B.T., 2006. "Cubic regularization of Newton method and its global performance," LIDAM Reprints CORE 1927, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. N. Gould & M. Porcelli & P. Toint, 2012. "Updating the regularization parameter in the adaptive cubic regularization algorithm," Computational Optimization and Applications, Springer, vol. 53(1), pages 1-22, September.
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    Cited by:

    1. Tommaso Bianconcini & Giampaolo Liuzzi & Benedetta Morini & Marco Sciandrone, 2015. "On the use of iterative methods in cubic regularization for unconstrained optimization," Computational Optimization and Applications, Springer, vol. 60(1), pages 35-57, January.
    2. Paul Armand & Isaï Lankoandé, 2017. "An inexact proximal regularization method for unconstrained optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 85(1), pages 43-59, February.
    3. Yonggang Pei & Shaofang Song & Detong Zhu, 2023. "A sequential adaptive regularisation using cubics algorithm for solving nonlinear equality constrained optimization," Computational Optimization and Applications, Springer, vol. 84(3), pages 1005-1033, April.
    4. 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.

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