Convergence rate of primal-dual reciprocal barrier Newton interior-point methods

Primal-dual interior-point methods for linear programming are often motivated by a certain nonlinear transformation of the Karush-Kuhn-Tucker conditions of the logarithmic barrier formulation. Recently, Nassar [5] studied the reciprocal barrier function formulation of the problem. Using a similar nonlinear transformation, he proved local convergence for Newton interior-point method on the resulting perturbed Karush-Kuhn-Tucker system. This result poses the question whether this method can exhibit fast convergence rate for linear programming. In this paper we prove that, for linear programming, Newton's method on the reciprocal barrier formulation exhibits at best Q-linear convergence rate. Moreover, an exact Q(1) factor is established which precludes fast linear convergence.