# Adaptive Regularization for Nonconvex Optimization sing Inexact Function Values and Randomly Perturbed Derivatives

Stefania Bellavia, Gianmarco Gurioli, Benedetta Morini, Philippe TOINT

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## Résumé

A regularization algorithm allowing random noise in derivatives and inexact
function values is proposed for computing approximate local critical points
of any order for smooth unconstrained optimization problems. For an
objective function with Lipschitz continuous p-th derivative and given an
arbitrary optimality order q <= p, it is shown that this algorithm will, in
expectation, compute such a point in at most
O((min_{j=1,...,q}epsilon_j)^{-(p+1)/(p-q+1)})
inexact evaluations of f and its derivatives whenever q in {1,2}, where
\epsilon_j is the tolerance for j-th order accuracy.  This bound becomes at most
O((min_{j=1,...,q}epsilon_j)^{-(q(p+1))/(p)})
inexact evaluations if q>2 and all derivatives are Lipschitz
continuous. Moreover these bounds are sharp in the order of the accuracy
tolerances. An extension to convexly constrained problems is also outlined.
langue originale Anglais Arxiv 22 2005.04639 Publié - 13 mai 2020

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• ### Complexity in nonlinear optimization

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

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