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A hybrid inexact regularized Newton and negative curvature method

Author

Listed:
  • Hong Zhu

    (Jiangsu University)

  • Yunhai Xiao

    (Henan University)

Abstract

In this paper, we propose a hybrid inexact regularized Newton and negative curvature method for solving unconstrained nonconvex problems. The descent direction is chosen based on different conditions, either the negative curvature or the inexact regularized direction. In addition, to minimize computational costs while obtaining the negative curvature, we employ a dimensionality reduction strategy to verify if the Hessian matrix exhibits negative curvatures within a three-dimensional subspace. We show that the proposed method can achieve the best-known global iteration complexity if the Hessian of the objective function is Lipschitz continuous on a certain compact set. Two simplified methods for nonconvex and strongly convex problems are analyzed as specific instances of the proposed method. We show that under the local error bound assumption with respect to the gradient, the distance between iterations generated by our proposed method and the local solution set converges to $$0$$ 0 at a superlinear rate. Additionally, for strongly convex problems, the quadratic convergence rate can be achieved. Extensive numerical experiments show the effectiveness of the proposed method.

Suggested Citation

  • Hong Zhu & Yunhai Xiao, 2024. "A hybrid inexact regularized Newton and negative curvature method," Computational Optimization and Applications, Springer, vol. 88(3), pages 849-870, July.
    • Handle: RePEc:spr:coopap:v:88:y:2024:i:3:d:10.1007_s10589-024-00576-6
    DOI: 10.1007/s10589-024-00576-6
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    References listed on IDEAS

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    1. Duan Li & Xiaoling Sun, 2006. "Nonlinear Integer Programming," International Series in Operations Research and Management Science, Springer, number 978-0-387-32995-6, December.
    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).
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