A robust primal-dual interior-point algorithm for nonlinear programs

Abstract

We present a primal-dual interior-point algorithm for solving optimization problems with nonlinear inequality constraints. The algorithm has some of the theoretical properties of trust region methods, but works entirely by line search. Global convergence properties are derived without assuming regularity conditions. The penalty parameter p in the merit function is updated adaptively and plays two roles in the algorithm. First, it guarantees that the search directions are descent directions of the updated merit function. Second, it helps to determine a suitable search direction in a decomposed SQP step. It is shown that if ρ is bounded for each barrier parameter μ, then every limit point of the sequence generated by the algorithm is a Karush Kuhn-Tucker point, whereas if ρ is unbounded for some μ, then the sequence has a limit point which is either a Fritz-John point or a stationary point of a function measuring the violation of the constraints. Numerical results confirm that the algorithm produces the correct results for some hard problems, including the example provided by Wächter and Biegler, for which many of the existing line search-based interior-point methods have failed to find the right answers.