A linear programming primer: from Fourier to Karmarkar

The story of linear programming is one with all the elements of a grand historical drama. The original idea of testing if a polyhedron is non-empty by using a variable elimination to project down one dimension at a time until a tautology emerges dates back to a paper by Fourier in 1823. This gets re-invented in the 1930s by Motzkin. The real interest in linear programming happens during World War II when mathematicians ponder best ways of utilising resources at a time when they are constrained. The problem of optimising a linear function over a set of linear inequalities becomes the focus of the effort. Dantzig’s Simplex Method is announced and the Rand Corporation becomes a hot bed of computational mathematics. The range of applications of this modelling approach grows and the powerful machinery of numerical analysis and numerical linear algebra becomes a major driver for the advancement of computing machines. In the 1970s, constructs of theoretical computer science indicate that linear programming may in fact define the frontier of tractable problems that can be solved effectively on large instances. This raised a series of questions and answers: Is the Simplex Method a polynomial-time method and if not can we construct novel polynomial time methods, etc. And that is how the Ellipsoid Method from the Soviet Union and the Interior Point Method from Bell Labs make their way into this story as the heroics of Khachiyan and Karmarkar. We have called this paper a primer on linear programming since it only gives the reader a quick narrative of the grand historical drama. Hopefully it motivates a young reader to delve deeper and add another chapter.