Generalized Lagrange Multiplier Method for Solving Problems of Optimum Allocation of Resources

TLDR

Lagrange multipliers are useful for constrained optimization beyond differentiable functions and are especially suited for allocating limited resources among independent activities. They can be applied to maximize any real‑valued objective over any set, subject to bounds on other finite real‑valued functions defined on the same set. Although not guaranteed to find a solution for all problems, the method is fail‑safe—any solution it finds is true—and, due to its simplicity, often succeeds in a surprising fraction of cases.

Abstract

The usefulness of Lagrange multipliers for optimization in the presence of constraints is not limited to differentiable functions. They can be applied to problems of maximizing an arbitrary real valued objective function over any set whatever, subject to bounds on the values of any other finite collection of real valued functions denned on the same set. While the use of the Lagrange multipliers does not guarantee that a solution will necessarily be found for all problems, it is “fail-safe” in the sense that any solution found by their use is a true solution. Since the method is so simple compared to other available methods it is often worth trying first, and succeeds in a surprising fraction of cases. They are particularly well suited to the solution of problems of allocating limited resources among a set of independent activities.