Interior Point Algorithms: Polyhedral Transformations

Abstract

The simplex algorithm is not a polynomial algorithm for solving the linear programming problem. The first polynomial algorithm for linear programming, the ellipsoid algorithm, is due to L. G. Khachiyan [262, 263]. This was met with great excitement from the mathematical programming community and stimulated considerable research. See Bland, Goldfarb, and Todd [63] for a good survey. There are also many important consequences of the ellipsoid algorithm in combinatorial optimization. See, for example, the book by Grötschel, Lovász, and Schrijver [212]. Unfortunately, the ellipsoid performed very poorly from a computational standpoint. Then in 1984, N. K. Karmarkar [260] created new excitement with claims of an algorithm that not only was polynomial in complexity, but also outperformed the simplex algorithm on large sparse problems. This created an incredible amount of research on interior point methods