A projected-search interior-point method for nonlinearly constrained optimization

This paper concerns the formulation and analysis of a new interior-point method for constrained optimization that combines a shifted primal-dual interior-point method with a projected-search method for bound-constrained optimization. The method involves the computation of an approximate Newton direction for a primal-dual penalty-barrier function that incorporates shifts on both the primal and dual variables. Shifts on the dual variables allow the method to be safely "warm started" from a good approximate solution and avoids the possibility of very large solutions of the associated path-following equations. The approximate Newton direction is used in conjunction with a new projected-search line-search algorithm that employs a flexible non-monotone quasi-Armijo line search for the minimization of each penalty-barrier function. Numerical results are presented for a large set of constrained optimization problems. For comparison purposes, results are also given for two primal-dual interior-point methods that do not use projection. The first is a method that shifts both the primal and dual variables. The second is a method that involves shifts on the primal variables only. The results show that the use of both primal and dual shifts in conjunction with projection gives a method that is more robust and requires significantly fewer iterations. In particular, the number of times that the search direction must be computed is substantially reduced. Results from a set of quadratic programming test problems indicate that the method is particularly well-suited to solving the quadratic programming subproblem in a sequential quadratic programming method for nonlinear optimization.