Abstract
Linear programming is the workhorse of modern optimization, but despite its importance, we still face plenty of open mathematical questions about it. Dantzig’s simplex method is certainly one of the most popular algorithms for solving linear programs. Its intuitive geometry is quite simple. The simplex method searches the graph of the polyhedron, from a vertex of the one-skeleton to a better neighboring one according to some pivot rule, which selects an improving neighbor. Geometrically, the simplex method traces a path on the graph of the polytope. The diameter of the graph of a polytope is the length of the longest shortest path among all possible pairs of vertices.
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