On the Fast Local Convergence of Interior-point `2-penalty Methods for Nonlinear Programming

In [3], two interior-point `2-penalty methods with strong global convergence properties were proposed for solving nonlinear programming problems. In this paper we show that under standard assumptions, slight modifications of these methods lead to fast local convergence. Specifically, we show that for each fixed small barrier parameter μ, iterates in a small neighborhood (roughly within o(μ)) of the minimizer of the barrier subproblem converge Q-quadratically to the minimizer. The overall convergence rate of the iterates to the solution of the nonlinear program is Q-superlinear. Our modifications include refinements of the rule for updating the penalty parameter and the termination criteria used by the inner algorithms, and the computation at each iteration for two correction steps that incur only modest cost. We illustrate these convergence results by some examples.