An interior-point piecewise linear penalty method for nonlinear programming

We present an interior-point penalty method for nonlinear programming (NLP), where the merit function consists of a piecewise linear penalty function and an ℓ2-penalty function. The piecewise linear penalty function is defined by a set of break points that correspond to pairs of values of the barrier function and the infeasibility measure at a subset of previous iterates and this set is updated at every iteration. The ℓ2-penalty function is a traditional penalty function defined by a single penalty parameter. At every iteration the step direction is computed from a regularized Newton system of the first-order equations of the barrier problem proposed in Chen and Goldfarb (Math Program 108:1–36, 2006). Iterates are updated using a line search. In particular, a trial point is accepted if it provides a sufficient reduction in either of the penalty functions. We show that the proposed method has the same strong global convergence properties as those established in Chen and Goldfarb (Math Program 108:1–36, 2006). Moreover, our method enjoys fast local convergence. Specifically, for each fixed small barrier parameter μ, iterates in a small neighborhood (roughly within o(μ)) of the minimizer of the barrier problem converge Q-quadratically to the minimizer. The overall convergence rate of the iterates to the solution of the nonlinear program is Q-superlinear.