The role of linear objective functions in barrier methods

We consider the asymptotic behavior of the Newton/log barrier method for inequality constrained optimization. We show that, when the objective function is linear, an effective step can be taken along the Newton direction after each reduction in the barrier parameter, leading to efficient performance during the final stages of the algorithm. This behavior contrasts with the case of a nonlinear objective, where Newton's method often performs more and more poorly as the barrier parameter is reduced to zero. We analyze the behavior and demonstrate our result on a simple example.