A robust Kantorovich's theorem on inexact Newton method with relative residual error tolerance
Provides a theoretical foundation for using fixed relative residual tolerances in inexact Newton methods, which is important for practitioners in optimization and nonlinear equations.
The paper proves that the inexact Newton method with a fixed relative residual error tolerance converges Q-linearly under semi-local assumptions, extending classical Kantorovich theory to practical implementations with controlled error. The result is applied to self-concordant minimization and analytic function zero-finding.
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.