  In path-following methods for conic programming knowledge of the performance
of the (damped) Newton method at finite distances from the minimizer of a
self-concordant function is crucial for the tuning of the parameters of the
method. The available bounds on the progress and the step length to be used are
based on conservative relations between the Hessians at different points and
are hence sub-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. As an
application, 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.
