A Computational View of Interior-Point Methods for Linear Programming

Many issues that are crucial for an e cient implementation of an interior point algorithm are addressed in this paper. To start with, a prototype primal{dual algorithm is presented. Next, many tricks that make it so e cient in practice are discussed in detail. Those include: the preprocessing techniques, the initialization approaches, the methods of computing search directions (and lying behind them linear algebra techniques), centering strategies and methods of stepsize selection. Several reasons for the manifestations of numerical di culties like e.g.: the primal degeneracy of optimal solutions or the lack of feasible solutions are explained in a comprehensive way. A motivation for obtaining an optimal basis is given and a practicable algorithm to perform this task is presented. Advantages of di erent methods to perform postoptimal analysis (applicable to interior point optimal solutions) are discussed. Important questions that still remain open in the implementations of interior point methods are also addressed, e.g.: performing correct postoptimal analysis, detecting infeasibility or resolving di culties arising in a presence of unbounded optimal faces. Challenging practical problem of warm start is recalled and two potentially attractive approaches to it are suggested. To facilitate the understanding of di erent implementation strategies, some illustrative numerical results on a subset of problems from the Netlib collection are presented.