A potential reduction method for a class of smooth convex programming problems

In this paper we propose a potential reduction method for smooth convex programming. It is assumed that the objective and constraint functions ful l the socalled Relative Lipschitz Condition, with Lipschitz constant M > 0. The great advantage of this method, above the existing path{following methods, is that it allows 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.