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

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 $$\Vert x\Vert _B= \sqrt{x^TBx}$$ ‖ x ‖ B = x T B x for all $$x \in \mathbb {R}^n$$ x ∈ R 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 $$\mathcal {O}(\epsilon ^{-3/2})$$ O ( ϵ - 3 / 2 ) . We show the good potential of the energy norm on a set of numerical experiments.