, Volume 72, Issue 3, pp 459-486

Multilinear programming: Duality theories

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A type of nonlinear programming problem, called multilinear, whose objective function and constraints involve the variables through sums of products is treated. It is a rather straightforward generalization of the linear programming problem. This, and the fact that such problems have recently been encountered in several fields of application, suggested their study, with particular emphasis on the analogies between them and linear problems. This paper develops one such analogy, namely a duality concept which includes its linear counterpart as a special case and also retains essentially all of the desirable characteristics of linear duality theory. It is, however, found that a primal then has several duals. The duals are arrived at by way of a game which is closely associated with a multilinear programming problem, but which differs in some respects from those generally treated in game theory. Its generalizations may in fact be of interest in their own right.

Communicated by D. F. Shanno
Professor J. Stoer and an anonymous reviewer made helpful comments on an earlier version of this paper. Those comments are greatly appreciated.