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### Abstract

A stochastic adaptive regularization algorithm allowing random noise in
derivatives and inexact function values is proposed for computing strong
approximate minimizers of any order for inexpensively constrained smooth
optimization problems. For an objective function with Lipschitz continuous
p-th derivative in a convex neighbourhood of the feasible set and
given an arbitrary optimality order q, it is shown that this algorithm
will, in expectation, compute such a point in at most
O( ( \min_{1<=j<=q} \epsilon_j )^{-(p+1)/(p-q+1)} )
inexact evaluations of f and its derivatives whenever q=1 and the
feasible set is convex, or q=2 and the problem is unconstrained, where
\epsilon_j is the tolerance for j-th order accuracy. This bound becomes
at most
O( ( \min_{1<=j<=q} \epsilon_j )^{-q(p+1)/p} )
inexact evaluations in the other cases if all derivatives are Lipschitz
continuous. Moreover these bounds are sharp in the order of the accuracy
tolerances.

Original language | English |
---|---|

Publisher | Arxiv |

Number of pages | 22 |

Volume | 2005.04639 |

Publication status | Published - 13 May 2020 |

### Keywords

- evaluation complexity
- regularization methods
- lnexact functions and derivatives
- stochastic analysis

## Fingerprint Dive into the research topics of 'High-order Evaluation Complexity of a Stochastic Adaptive Regularization Algorithm for Nonconvex Optimization Using Inexact Function Evaluations and Randomly Perturbed Derivatives'. Together they form a unique fingerprint.

## Projects

- 2 Active

## Complexity in nonlinear optimization

TOINT, P., Gould, N. I. M. & Cartis, C.

1/11/08 → …

Project: Research

## Cite this

Bellavia, S., Gurioli, G., Morini, B., & TOINT, P. (2020).

*High-order Evaluation Complexity of a Stochastic Adaptive Regularization Algorithm for Nonconvex Optimization Using Inexact Function Evaluations and Randomly Perturbed Derivatives*. Arxiv.