We prove a new local convergence property of a primal-dual method for solving nonlinear optimization problem. Following a standard interior point approach, the complementarity conditions of the original primal-dual system are perturbed by a parameter which is driven to zero during the iterations. The sequence of iterates is generated by a linearization of the perturbed system and by applying the fraction to the boundary rule to maintain strict feasibility of the iterates with respect to the nonnegativity constraints. The analysis of the rate of convergence is carried out by considering a linear or a superlinear arbitrary decreasing sequence of perturbation parameters. We show that, if the perturbation parameters converge to zero linearly or superlinearly and once an iterate belongs to a neighborhood of convergence of the Newton method applied to the original system, then the whole sequence of iterates converges and asymptotically follows the central trajectory in a natural way.
Report 2006-05, Département DMI, Laboratoire XLIM - UMR CNRS 6172, 123, avenue Albert Thomas, 87060 Limoges, FRANCE, January 2006