A numerically exact implementation of the simplex method

Abstract

The numerical performance of the state-of-the-art simplex based optimizers is good. At the same time, a newly arising LP problem can cause troubles still. This is exactly what happened in the Summer of 1992. The appearance of a hard LP problem motivated the development of the idea of a numerically exact implementation of the simplex method. It is based on a super register (SR) capable of accumulating inner products with arbitrary accuracy. The necessary operations of SR that require assembly level programming are introduced. Vectors of super registers would require prohibitively much memory. Therefore, a single-SR technique is proposed that entails the reorganization of parts of the simplex method. The ideas have been implemented in the MILP LP optimizer. Experiences show that solution speed decreases by 30–50 percent but robustness increases which may be important in case of critical problems. A framework is outlined for a system where the advantages of the traditional and the SR technique can co-operate efficiently.