We present a null-space primal-dual interior-point algorithm for solving nonlinear optimization problems with general inequality and equality constraints. The algorithm approximately solves a sequence of equality constrained barrier subproblems by computing a predictor step and a null space step in every iteration. The $\ell_2$ penalty function is taken as the merit function. Under very mild conditions on predictor steps and approximate Hessians, without assuming any regularity, it is proved that the limit point of the iterate sequence is either a Karush-Kuhn-Tucker point of the barrier subproblem, or a point that is strictly feasible for inequality constraints of the original problem and stationary for minimizing the $\ell_2$ norm of violations of equality constraints, provided the penalty parameter remains bounded; if the penalty parameter tends to infinity, there exists a limit point that is either an infeasible stationary point of minimizing the $\ell_2$ norm of violations of constraints of the original problem, or a Fritz-John point of the original problem. In addition, we analyze the local convergence properties of the algorithm, and prove that by suitably controlling the exactness of predictor steps, selecting the barrier parameter and Hessian approximation, the algorithm generates a superlinearly or quadratically convergent step. The conditions on guaranteeing the positiveness of slack variable vector for a full step are presented.
Citation
Research report, Department of Applied Mathematics, Hebei University of Technology, Tianjin, China, March 2007