Nonlinear Programming: A Historical Note

The paper [1] that first used the name ‘nonlinear programming’ was written 41 years ago. In the intervening period, I have learned a number of things about the influences, both mathematical and social, that have shaped the modern development of the subject. Some of these are quite old and long predate the differentiation of nonlinear programming as a separate area for research. Others are comparatively modern and culminate in the period 41 years ago when this differentiation took place. In order to discuss these influences in a precise context, a few key results will be stated and ’proved’. This will be done in an almost self-contained manner. These statements will allow the comparison of the results of various mathematicians who made early contributions to nonlinear programming. In reconstructing this story, I had the benefit of personal communications from A. W. Tucker, W. Karush, and F. John, who shared their memories of the relevant events. In Section 2, a definition of a nonlinear program is given. It will be seen to be a straightforward generalization of a linear program and those experienced in this field will recognize that the definition is far too broad to admit very much in the way of results. However, the immediate objective is the derivation of necessary conditions for a local optimum in the differentiable case. For this purpose, it will be seen that the definition includes situations in which these conditions are well known. On the other hand, it will be seen that the definition of a nonlinear program hides several implicit traps which have an important effect on the form of the correct necessary conditions. In Section 3, an account is given of the duality of linear programming as motivation for the generalization to follow. This duality, although it was discovered and explored with surprise and delight in the early days of linear programming, has ancient and honorable ancestors in pure and applied mathematics. Some of these are explored to round out this section. With the example of linear programming before us, the nonlinear program of Section 2 is subjected to a natural linearization which yields a set of likely necessary conditions for a local optimum in Section 4. Of course, these conditions do not hold in full generality without a regularity condition (conventionally called the constraint qualification). When it is invoked, the result is a theorem which has often been attributed to Kuhn and Tucker. This section is completed by a description of the background of the 1939 work of W. Karush [2] in which the theorem first appeared. As will be seen in Section 4, the motivation for Karush’s work was different from the spirit of mathematical programming that prevailed at the end of the 1940’s. In Section 5,