Optimal step length for the Newton method: case of self-concordant functions

Abstract

The theoretical foundation of path-following methods is the performance analysis of the (damped) Newton step on the class of self-concordant functions. However, the bounds available in the literature and used in the design of path-following methods are not optimal. In this contribution we use methods of optimal control theory to compute the optimal step length of the Newton method on the class of self-concordant functions, as a function of the initial Newton decrement, and the resulting worst-case decrease of the decrement. The exact bounds are expressed in terms of solutions of ordinary differential equations which cannot be integrated explicitly. We provide approximate numerical and analytic expressions which are accurate enough for use in optimization methods. Consequently, the neighbourhood of the central path in which the iterates of path-following methods are required to stay can be enlarged, enabling faster progress along the central path during each iteration and hence fewer iterations to achieve a given accuracy.