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arXiv · 2011.00854

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

C. Cartis·N. I. M. Gould·Ph. L. Toint
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Abstract

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 $\mathcal{O}\big(\min_{j\in\{1,\ldots,q\}}\epsilon_j^{-(q+1)}\big)$ where the $\epsilon_j$ 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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C. Cartis, N. I. M. Gould, Ph. L. Toint. 2020-11-02. Strong Evaluation Complexity of An Inexact Trust-Region Algorithm for Arbitrary-Order Unconstrained Nonconvex Optimization. https://arxiv.org/abs/2011.00854

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