On the Symmetric Formulation of Interior-point Methods

We present a uni ed framework for studying interior point methods for linear programming. Within this framework, we compare three fundamental methods: (1) pathfollowing, (2) barrier, and (3) primal/dual a ne-scaling (our terminology di ers slightly from the commonly accepted terminology). The step directions for each of these methods are linear combinations of three fundamental directions: (1) step-toward-optimality, (2) step-toward-center, and (3) step-toward-feasibilty. By making very simple modi cations to LOQO, we have created e cient implementations of each of these methods, which we compared one to another. We shall present the results of this comparison. In addition, associated with each of the three fundamental algorithms, there are three choices for how to organize the system of equations that must be solved: (1) general ordering heuristics applied to the KKT system, (2) primal-scaling, and (3) dual-scaling. Each of these three choices produce identical step directions but the choice can have a dramatic e ect on the e ciency of the implementation. Again, by making minor modi cations, we have created variants of the path-following method corresponding to each of these approaches to computing the step directions. We shall also present the results of this comparison.