Summary, conclusions and recommendations

Most of the papers on path-following methods are concerned with short-step methods. These methods are unattractive in practice since they use fixed short steps and small updates in the parameter, and therefore require many iterations. On the other hand, medium-and long-step path-following methods are much more flexible, since they allow to do large updates in the parameter and (approximate) line searches. In the literature such medium-and long-step methods are only analyzed for the logarithmic barrier method applied to linear programming.