arXiv subjects

N. I. M. Gould

Publications and source records attributed to N. I. M. Gould.

3 recordsLinked to original sources

arXiv2021

An adaptive regularization algorithm for unconstrained optimization with inexact function and derivatives values

An adaptive regularization algorithm for unconstrained nonconvex optimization is proposed that is capable of handling inexact objective-function and derivative values, and also of providing approximate minimizer of arbitrary order. In comparison with a similar algorithm proposed in Cartis, Gould, Toint (2021), its distinguishing feature is that it is based on controlling the relative error between the model and objective values. A sharp evaluation complexity complexity bound is derived for the new algorithm.

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arXiv2021

Strong Evaluation Complexity of An Inexact Trust-Region Algorithm for Arbitrary-Order Unconstrained Nonconvex Optimization

A trust-region algorithm using inexact function and derivatives values is introduced for solving unconstrained smooth optimization problems. This algorithm uses high-order Taylor models and allows the search of strong approximate minimizers of arbitrary order. The evaluation complexity of finding a $q$-th approximate minimizer using this algorithm is then shown, under standard conditions, to be $\mathcal{O}\big(\min_{j\in\{1,\ldots,q\}}ε_j^{-(q+1)}\big)$ where the $ε_j$ are the order-dependent requested accuracy thresholds. Remarkably, this order is identical to that of classical trust-region methods using exact information.

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arXiv2018

Optimality of orders one to three and beyond: characterization and evaluation complexity in constrained nonconvex optimization

Necessary conditions for high-order optimality in smooth nonlinear constrained optimization are explored and their inherent intricacy discussed. A two-phase minimization algorithm is proposed which can achieve approximate first-, second- and third-order criticality and its evaluation complexity is analyzed as a function of the choice (among existing methods) of an inner algorithm for solving subproblems in each of the two phases. The relation between high-order criticality and penalization techniques is finally considered, showing that standard algorithmic approaches will fail if approximate constrained high-order critical points are sought.

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