An improvement of adaptive cubic regularization method for unconstrained optimization problems

Abstract

In this paper, we present two nonmonotone versions of adaptive cubic regularized (ARC) method for unconstrained optimization problems. The proposed methods are a combination of the ARC algorithm with the nonmonotone line search methods introduced by Zhang and Hager [A nonmonotone line search technique and its application to unconstrained optimization, SIAM J. Optim. 14 (2004), pp. 1043–1056] and Ahookhosh et al. [A nonmonotone trust-region line search method for large-scale unconstrained optimization, Appl. Math. Model. 36 (2012), pp. 478–487]. The global convergence analysis for these iterative algorithms is established under suitable conditions. Several numerical examples are given to illustrate the efficiency and robustness of the newly suggested methods. The obtained results show the satisfactory performance of the proposed algorithms when compared to the basic ARC algorithm.

Keywords:

• Unconstrained optimization
• cubic regularization
• ARC algorithm
• nonmonotone line search
• global convergence

Mathematics Subject Classification (2000):

• 49M37
• 65K05
• 90C30

Acknowledgments

We would like to thank Professor Stefano Lucidi for their valuable comments and suggestions, which helped us to considerably improve the manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).