Q-superlinear convergence of primal–dual interior point quasi-Newton methods for constrained optimization

This paper analyzes local convergence rates of primal–dual interior point methods for general nonlinearly constrained optimization problems. For this purpose, we first discuss modified Newton methods and modified quasi-Newton methods for solving a nonlinear system of equations, and show local and Q-quadratic/Q-superlinear convergence of these methods. These methods are characterized by a perturbation of the right-hand side of the Newton equation applied to the system, an approximation of the Jacobian matrix by some matrix, and component-wise dampings of the step. By applying these convergence results for the nonlinear system of equations to the primal–dual interior point methods for nonlinear optimization, we obtain convergence results of the primal–dual interior point Newton and quasi-Newton methods. A necessary and sufficient condition for Q-superlinear convergence of the latter methods corresponds to the Dennis–Moré condition. Furthermore, we present some quasi-Newton updating formulae. Finally, we give an analysis of the Q-rate in a part of variables for the primal–dual interior point quasi-Newton methods, and obtain a necessary and sufficient conditon for the Q-rate. This condition is a generalization of the result given by Martinez et al., which was done independently.