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On the use of iterative methods in cubic regularization for unconstrained optimization

Tommaso Bianconcini (tommaso.bianconcini***at***unifi.it)
Giampaolo Liuzzi (giampaolo.liuzzi***at***iasi.cnr.it)
Benedetta Morini (benedetta.morini***at***unifi.it)
Marco Sciandrone (marco.sciandrone***at***unifi.it)

Abstract: 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.

Keywords: Unconstrained optimization, cubic regularization, worst-case complexity, matrix-free subproblem solvers.

Category 1: Nonlinear Optimization

Citation: Rapporto Tecnico n. 6/2013, Dipartimento di Ingegneria Industriale, Universita degli Studi di Firenze

Download: [PDF]

Entry Submitted: 09/11/2013
Entry Accepted: 09/11/2013
Entry Last Modified: 09/24/2013

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