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

TL;DR

This paper introduces an adaptive regularization algorithm for unconstrained nonconvex optimization that effectively manages inexact function and derivative data, providing high-order approximate solutions with proven complexity bounds.

Contribution

It presents a novel algorithm controlling relative errors between model and objective, improving upon previous methods with new complexity analysis.

Findings

Achieves a sharp evaluation complexity bound.

Handles inexact function and derivative evaluations.

Provides approximate minimizers of arbitrary order.

Abstract

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.