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...
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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 vertical bar vertical bar x vertical bar vertical bar(B) = root x(T) B-x for all x is an element of 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 O(is an element of(-3/2)) We show the good potential of the energy norm on a set of numerical experiments.
adaptive regularized framework using cubics has emerged as an alternative to line-search and trust-region algorithms for smooth nonconvex optimization, with an optimal complexity among second-order methods. In this pa...
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adaptive regularized framework using cubics has emerged as an alternative to line-search and trust-region algorithms for smooth nonconvex optimization, with an optimal complexity among second-order methods. In this paper, we propose and analyze the use of an iteration dependent scaled norm in the adaptive regularized framework using cubics. Within such a scaled norm, the obtained method behaves as a line-search algorithm along the quasi-Newton direction with a special backtracking strategy. Under appropriate assumptions, the new algorithm enjoys the same convergence and complexity properties as adaptiveregularized algorithm usingcubics. The complexity for finding an approximate first-order stationary point can be improved to be optimal whenever a second-order version of the proposed algorithm is regarded. In a similar way, using the same scaled norm to define the trust-region neighborhood, we show that the trust-region algorithm behaves as a line-search algorithm. The good potential of the obtained algorithms is shown on a set of large-scale optimization problems.
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