# On the use of the energy norm in trust-region and adaptive cubic regularization subproblems

Bergou, El Houcine and Diouane, Youssef and Gratton, Serge On the use of the energy norm in trust-region and adaptive cubic regularization subproblems. (2017) Computational Optimization and Applications, 68 (3). 533-554. ISSN 0926-6003

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Official URL: http://dx.doi.org/10.1007/s10589-017-9929-2

## Abstract

We consider solving unconstrained optimization problems by means of two popular globalization techniques: trust-region (TR) algorithms and adaptive regularized framework using cubics (ARC). Both techniques require the solution of a so-called subproblem'' in which a trial step is computed by solving an optimization problem involving an approximation of the objective function, called the model". The latter is supposed to be adequate in a neighborhood of the current iterate. In this paper, we address an important practical question related with the choice of the norm for defining the neighborhood. More precisely, assuming here that the Hessian $B$ of the model is symmetric positive definite, we propose the use of the so-called energy norm'' -- defined by $\|x\|_B= \sqrt{x^TBx}$ for all $x \in \real^n$ -- in both TR and ARC techniques. We show that the use of this norm induces remarkable relations between the trial step of both methods that can be used to obtain efficient practical algorithms. We furthermore consider the use of truncated Krylov subspace methods to obtain an approximate trial step for large scale optimization. Within the energy norm, we obtain line search algorithms along the Newton direction, with a special backtracking strategy and an acceptability condition in the spirit of TR/ARC methods. The new line search algorithm, derived by ARC, enjoys a worst-case iteration complexity of $\mathcal{O}(\epsilon^{-3/2})$. We show the good potential of the energy norm on a set of numerical experiments.

Item Type: Article Thanks to Elsevier editor. The definitive version is available at https://link.springer.com/article/10.1007/s10589-017-9929-2 hal-01654736 International peer-reviewed journal French research institutions > Centre National de la Recherche Scientifique - CNRS (FRANCE)Université de Toulouse > Institut National Polytechnique de Toulouse - Toulouse INP (FRANCE)Université de Toulouse > Institut Supérieur de l'Aéronautique et de l'Espace - ISAE-SUPAERO (FRANCE)Université de Toulouse > Université Toulouse III - Paul Sabatier - UT3 (FRANCE)Université de Toulouse > Université Toulouse - Jean Jaurès - UT2J (FRANCE)Université de Toulouse > Université Toulouse 1 Capitole - UT1 (FRANCE) download 27 Nov 2017 11:07

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