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

In this paper we propose a large{step analytic center method for smooth convex programming. The method is a natural implementation of the classical method of centers. It is assumed that the objective and constraint functions ful l the socalled Relative Lipschitz Condition, with Lipschitz constant M > 0. A great advantage of the method, above the existing path{following methods, is that the steps can be made long by performing linesearches. In our method we do linesearches along the Newton direction with respect to a strictly convex potential function if we are far away from the central path. If we are su ciently close to this path we update a lower bound for the optimal value. We prove that the number of iterations required by the algorithm to converge to an {optimal solution is O((1+M2)pnj ln j) or O((1 +M2)nj ln j), dependent on the updating scheme for the lower bound.