Convergence analysis of a regularized interior point algorithm for the barrier problems with singular solutions

Wächter and Biegler (2006) [25] presented and implemented an interior point filter line search method (called IPOPT) for large scale nonlinear programming. To make IPOPT efficient and robust, they add some safeguards in their implementation. One crucial safeguard is to apply the inertia correction technique to modify the KKT matrix. We focus on the degenerate problems in which the KKT matrix might be singular. To complement theoretical foundation of this technique in the degenerate case, we propose and analyze a regularized interior point method for the barrier problems with singular solutions which largely relies on IPOPT. Based on the inertia correction technique, the matrix arising in the KKT equation is corrected to be a nonsingular one so that the Newton step can be calculated at each iteration even in the degenerate case. We emphasize that the special inertia correction for the KKT matrix does not impose any negative impact on the fast convergence of our algorithm. Under some local error bound condition, we prove that the rate of quadratic convergence is achieved when the second order condition or some constraint qualification condition fails to hold (or both).