Summary

The development of the simplex method by George B Dantzig in the mid-1940s for solving linear programming problems is a great achievement in the theory of optimization. An optimal solution of a linear programming problem always lies at a vertex of the feasible region which is a polyhedron. The simplex method moves along the edges of the polyhedron from one vertex to the next in an orderly fashion until it arrives at an optimal vertex. Since the number of vertices of the polyhedron associated with the m x n coefficient matrix of the problem, is quite large for large values of m and n, there was apprehension that the simplex method would not prove to be efficient. However, in practice, it was found to perform exceedingly well. It was observed that for most of the practical problems, the number of iterations the method needs, is only between m and 3m. However, the fact remains that for certain problems, the simplex method may require to examine all the vertices to arrive at an optimal solution and therefore the number of iterations can grow exponentially An example given by Klee and Minty [275] indeed shows that in the worst case, the simplex method needs 2ⁿ iterations to find an optimal solution.