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In path-following methods for conic programming knowledge of the performanceof the (damped) Newton method at finite distances from the minimizer of aself-concordant function is crucial for the tuning of the parameters of themethod. The available bounds on the progress and the step length to be used arebased on conservative relations between the Hessians at different points andare hence sub-optimal. In this contribution we use methods of optimal controltheory to compute the optimal step length of the Newton method on the class ofself-concordant functions, as a function of the initial Newton decrement, andthe resulting worst-case decrease of the decrement. The exact bounds areexpressed in terms of solutions of ordinary differential equations which cannotbe integrated explicitly. We provide approximate numerical and analyticexpressions which are accurate enough for use in optimization methods. As anapplication, the neighbourhood of the central path in which the iterates ofpath-following methods are required to stay can be enlarged, enabling fasterprogress along the central path during each iteration and hence feweriterations to achieve a given accuracy.

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