Interior-point Methods for Nonconvex Nonlinear Programming: Convergence Analysis and Computational Performance

In this paper, we present global and local convergence results for an interior-point method for nonlinear programming and analyze the computational performance of its implementation. The algorithm uses an `1 penalty approach to relax all constraints, to provide regularization, and to bound the Lagrange multipliers. The penalty problems are solved using a simplified version of Chen and Goldfarb’s strictly feasible interior-point method [12]. The global convergence of the algorithm is proved under mild assumptions, and local analysis shows that it converges Q-quadratically for a large class of problems. The proposed approach is the first to simultaneously have all of the following properties while solving a general nonconvex nonlinear programming problem: (1) the convergence analysis does not assume boundedness of dual iterates, (2) local convergence does not require the Linear Independence Constraint Qualification, (3) the solution of the penalty problem is shown to locally converge to optima that may not satisfy the Karush-Kuhn-Tucker conditions, and (4) the algorithm is applicable to mathematical programs with equilibrium constraints. Numerical testing on a set of general nonlinear programming problems, including degenerate problems and infeasible problems, confirm the theoretical results. We also provide comparisons to a highly-efficient nonlinear solver and thoroughly analyze the effects of enforcing theoretical convergence guarantees on the computational performance of the algorithm.