The (Dantzig) simplex method for linear programming

G eorge Dantzig, in describing the " Origins of the Simplex Method, " 1 noted that it was the availability of early digital computers that supported and invited the development of LP models to solve real-world problems. He agreed that invention is sometimes the mother of necessity. Moreover, he commented that he initially rejected the simplex method because it seemed intuitively more attractive to pursue the objective function downhill—as in the currently popular interior-point methods—rather than search along the constraint set's edges. It is this latter approach that the simplex method uses, which should not be confused with the J.A. Nelder and R. Mead's function-minimization method, 2 also associated with the word simplex. When Dantzig introduced his method in 1947, it was somewhat easier to sort out the details of the simplex method than to deal with the " where are we in the domain space " questions that are, in my opinion, the core of interior-point approaches. Easier does not mean simpler, however. Chapter and verse The LP problem is, in one of the simplex method's many forms, Minimize (with respect to x) c′ x (1a) subject to A x = b (1b) and x ≥ 0 (1c) This is not always the problem we want to solve, though—we might want to have the matrix A and vector b partitioned row-wise so that (A1) (b1) A = (A2) b = (b2) (A3) (b3) so that we can write an LP problem as George Dantzig created a simplex algorithm to solve linear programs for planning and decision-making in large-scale enterprises. The algorithm's success led to a vast array of specializations and generalizations that have dominated practical operations research for half a century.