An Elementary Treatment of Lagrange Multipliers

Abstract

The problem of finding the infimum of a convex function f(x) subject to the constraint that one or more convex functions g(x) be non-positive can be treated by the Lagrange multiplier method. Such a treatment was revived by Kuhn and Tucker and further studied by many other scientists. These studies led to the following asso-ciated maximizing problem on the Lagrange function, L = f(x) + λg(x). First find the infimum of L with respect to λ and then take the supremum with respect to T, subject to λ≥ O. The minimizing problem and the associated maximizing problem are termed dual programs.

This paper is partly of an expository nature. The goal is to give a short and elementary proof that, under suitable qualifications, the infimum of the first program is equal to the supremum of the second program. The proof begins by using the Courant penalty function. No knowledge of linear programming is assumed. However, the duality theorem for linear programs is a special case of the duality theorem for convex programs developed.

This paper was prepared under Grant DAAG29-77-0024, Army Research Office, Research Triangle Park, North Carolina.