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Improved second-order evaluation complexity for unconstrained nonlinear optimization using high-order regularized models

Coralia Cartis(cartis***at***maths.ox.ac.uk)
Nicholas I M Gould(nimg***at***stfc.ac.uk)
Philippe L Toint(philippe.toint***at***unamur.be)

Abstract: The unconstrained minimization of a sufficiently smooth objective function $f(x)$ is considered, for which derivatives up to order $p$, $p\geq 2$, are assumed to be available. An adaptive regularization algorithm is proposed that uses Taylor models of the objective of order $p$ and that is guaranteed to find a first- and second-order critical point in at most $O \left(\max\left( \epsilon_1^{-\frac{p+1}{p}}, \epsilon_2^{-\frac{p+1}{p-1}} \right) \right)$ function and derivatives evaluations, where $\epsilon_1$ and $\epsilon_2 >0$ are prescribed first- and second-order optimality tolerances. Our approach extends the method in Birgin et al. (2016) to finding second-order critical points, and establishes the novel complexity bound for second-order criticality under identical problem assumptions as for first-order, namely, that the $p$-th derivative tensor is Lipschitz continuous and that $f(x)$ is bounded from below. The evaluation-complexity bound for second-order criticality improves on all such known existing results.

Keywords: complexity analysis, regularisation methods

Category 1: Nonlinear Optimization

Citation: Technical Report, University of Oxford, Mathematical Institute, 2017.

Download: [PDF]

Entry Submitted: 08/13/2017
Entry Accepted: 08/13/2017
Entry Last Modified: 08/13/2017

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