Linear Programming

Abstract

Although the origin of linear programming as a mathematical discipline is quite recent, linear programming is now well established as an important and very active branch of applied mathematics. The wide applicability of linear programming models and the rich mathematical theory underlying these models and the methods developed to solve them have been the driving forces behind the rapid and continuing evolution of the subject. Linear programming problems involve the optimization of a linear function, called the objective function, subject to linear constraints, which may be either equalities or inequalities, in the unknowns. The recognition of the importance of linear programming models, especially in the areas of economic analysis and planning, coincided with the development of both an effective method, the ‘simplex method’ of G.B. Dantzig, for solving linear programming problems, (Dantzig 1951) and a means, the digital computer, for doing so. A major part of the foundation of linear programming was laid in an amazingly short period of intense research and development between 1947 and 1949, as the above three key factors converged. Prior to 1947 mathematicians had studied systems of linear inequalities, starting with Fourier (1826), and optimality conditions for systems with inequality constraints within the classical theory of the calculus of variations (Bolza 1914; Valentine 1937). For the finite dimensional case, the first general result of the latter type appeared in a master’s thesis by Karush (1939). (See also (John 1948).) Also, as early as 1939, L.V. Kantorovich had proposed linear programming models for production planning and a rudimentary algorithm for their solution (Kantorovich