On Recent Developments in Linear Programming

Abstract

Linear programming remains a central subject of Mathematical Programming although already in 1947, Dantzig developed the Simplex method for solving Linear Programming problems. That method has been implemented in quite powerful codes which are very efficient for solving real world problems. With the growing interest in the complexity of algorithms, however, it turned out that the Simplex method is exponential in the worst case (Klee-Minty [1972]). Until now the existence of pivoting rules resulting in a polynomial variant of the Simplex Method is an open question. For most known rules conterexamples to polynomiality have been constructed. The worst case approach itself is due to much criticizm. The analysis of the average case is usually much harder, but seems to explain the typically efficient performance of the Simplex method much better. For a broad discussion of average case analysis of the Simplex method we refer the reader to Borgwardt [1984].