A robust Kantorovich's theorem on inexact Newton method with relative residual error tolerance

TL;DR

This paper establishes a semi-local convergence analysis for the inexact Newton method with fixed relative residual error tolerance, extending classical results to practical implementations with inexact computations.

Contribution

It provides a new convergence theorem for the inexact Newton method under semi-local assumptions, applicable to self-concordant and analytic functions, with fixed residual error tolerance.

Findings

Proves Q-linear convergence under semi-local conditions.

Shows applicability to minimizing self-concordant functions.

Retrieves classical Kantorovich theorem in the error-free case.

Abstract

We prove that under semi-local assumptions, the inexact Newton method with a fixed relative residual error tolerance converges Q-linearly to a zero of the non-linear operator under consideration. Using this result we show that Newton method for minimizing a self-concordant function or to find a zero of an analytic function can be implemented with a fixed relative residual error tolerance. In the absence of errors, our analysis retrieve the classical Kantorovich Theorem on Newton method.