Linear Programming Stories

The history of polyhedra, linear inequalities, and linear programming has many diverse origins. Polyhedra have been around since the beginning of mathematics in ancient times. It appears that Fourier was the first to consider linear inequalities seriously. This was in the first half of the 19 century. He invented a method, today often called Fourier-Motzkin elimination, with which linear programs can be solved, although this notion did not exist in his time. If you want to know anything about the history of linear programming, I strongly recommend consulting Schrijver’s book [5]. It covers all developments in deepest possible elaborateness. This section of the book contains some aspects that complement Schrijver’s historical notes. The origins of the interior point method for linear programming are explored as well as column generation, a methodology that has proved of considerable practical importance in linear and integer programming. The solution of the Hirsch conjecture is outlined, and a survey of the development of computer codes for the solution of linear (and mixed-integer) programs is given. And there are two articles related to the ellipsoid method to which I would like to add a few further details. According to the New York Times of November 7, 1979: “A surprise discovery by an obscure Soviet mathematician has rocked the world of mathematics . . . ”. This obscure person was L. G. Khachiyan who ingeniously modified an algorithm, the ellipsoid method, developed for nonlinear programming by N. Z. Shor, D. B. Yudin, and A. S. Nemirovskii and proved in a very short paper [3] that this method solves linear programs in polynomial time. This was indeed a sensation. The ellipsoid method is a failure in practical computation but turned out to be a powerful tool to show the polynomial time solvability of many optimization problems, see [2]. One step in the ellipsoid method is the computation of a least volume ellipsoid containing a given convex body. The story of the persons behind the result that this ellipsoid, the Löwner-John ellipsoid, is uniquely determined and has very interesting properties, is told in this section. A second important ingredient of Khachiyan’s modification is “clever rounding”. A best possible approximation of a real number by a rational number with a bounded denominator can be