Optimal step length for the Newton method near the minimum of a self-concordant function

6 figuresThe quadratic convergence region of the exact Newton method around the minimum of a self-concordant function makes up a fraction of the Dikin ellipsoid. Outside of this region, the Newton method has to be damped in order to ensure convergence. However, the available estimates of both the size of the convergence region and the step length to be used outside of it 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 Newton decrement. With this step length quadratic convergence can be achieved on the whole Dikin ellipsoid. The exact bounds are expressed in terms of solutions of ordinary differential equations which cannot be integrated explicitly. As an application, the neighbourhood of the central path in which the iterates of path-following methods for conic programming 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