In this paper we consider the problem of minimizing a smooth function by using the Adaptive Cubic Regularized framework (ARC). We focus on the computation of the trial step as a suitable approximate minimizer of the cubic model and discuss the use of matrix-free iterative methods. Our approach is alternative to the implementation proposed in the original version of ARC, involving a linear algebra phase, but preserves the same worst-case complexity count as ARC. Further we introduce a new stopping criterion in order to properly manage the ``over-solving'' issue arising whenever the cubic model is not an adequate model of the true objective function. Numerical experiments conducted by using a nonmonotone gradient method as inexact solver are presented. The obtained results clearly show the effectiveness of the new variant of ARC algorithm.
Citation
Computational Optimization and Applications, 60, 35–57 (2015). DOI:https://doi.org/10.1007/s10589-014-9672-x
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View On the use of iterative methods in cubic regularization for unconstrained optimization