On the classical logarithmic barrier function method for a class of smooth convex programming problems

In this paper, we describe a natural implementation of the classical logarithmic barrier function method for smooth convex programming. It is assumed that the objective and constraint functions fulfill the so-called relative Lipschitz condition, with Lipschitz constantM>0.In our method, we do line searches along the Newton direction with respect to the strictly convex logarithmic barrier function if we are far away from the central trajectory. If we are sufficiently close to this path, with respect to a certain metric, we reduce the barrier parameter. We prove that the number of iterations required by the algorithm to converge to an -optimal solution isO((1+M²)log) orO((1+M²)nlog), depending on the updating scheme for the lower bound.on leave from Eötvös University, Budapest, Hungary.