Analysis of a Non-interior Continuation Method Based on Chen-mangasarian Smoothing Functions

Recently Chen and Mangasarian proposed a class of smoothing functions for linear/nonlinear programs and complementarity problems that uniies many previous proposals. Here we study a non-interior continuation method based on these functions in which, like interior path-following methods, the iterates are maintained to lie in a neighborhood of some path and, at each iteration, one or two Newton-type steps are taken and then the smoothing parameter is decreased. We show that the method attains global convergence and linear convergence under conditions similar to those required for other methods. We also show that these conditions are in some sense necessary. By introducing an inexpensive active-set strategy in computing one of the Newton directions, we show that the method attains local superlinear convergence under conditions milder than those for other methods. The proof of this uses a local error bound on the distance from an iterate to a solution in terms of the smoothing parameter .