Commentary—Interior-Point Methods: Algorithms and Formulations

The paper by Lustig, Marsten, and Shanno (LMS) gives an excellent presentation of the current state of the art for interior-point methods (as represented by their OB1 code) as it compares to the current state of the art for the simplex method (represented by OSL). The paper is well organized and thoughtful. The results of their experiments clearly indicate that for large problems interior-point methods offer a serious alternative to the simplex method. In these comments, we will try to clarify the relation among some of the algorithms discussed in LMS. In addition, we will compare the implementation strategy described in LMS, which is based on solving the so-called normal equations, to an alternative implementation strategy based on solving the so-called reduced Karush-Kuhn-Tucker (KKT) system. We will show that for many problems the two approaches yield virtually identical results but, for certain classes of challenging problems, the reduced KKT approach yields a much more efficient code. Also, we will argue that the reduced KKT system approach is more easily extended to quadratic and convex optimization problems. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.