Optimal step length for the maximal decrease of a self-concordant function by the Newton method

TL;DR

This paper investigates the optimal step length in Newton's method for self-concordant functions, formulating it as an optimal control problem and applying control theory to determine the maximal decrease in function value.

Contribution

It introduces a novel optimal control framework to identify the best step length for Newton's method on self-concordant functions, enhancing convergence analysis.

Findings

Derived explicit formulas for optimal step length

Demonstrated improved convergence rates

Provided theoretical insights into Newton's method behavior

Abstract

In this paper we consider the problem of finding the optimal step length for the Newton method on the class of self-concordant functions, with the decrease in function value as criterion. We formulate this problem as an optimal control problem and use optimal control theory to solve it.