Duality in Semi-Infinite Programs and some Works of Haar and Caratheodory
NORTHWESTERN UNIV EVANSTON IL TECHNOLOGICAL INST
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The following paper was stimulated by a paper of the Hungarian mathematician, A. Haar possibly one of the all-time great mathematicians. Because it was published in a relatively obscure journal, it has only recently been made generally available through the posthumous publication of his collected works. Since the theorems to be established rest heavily on this work, and Haars paper is published in German, we present it for ease of access in free translation in the appendix. Conjectured by von Neumann and proved by Gale, Kuhn and Tucker, the dual theorem of linear programming has been unique among dual extremal or variational principles see, for example, K. Friedrichs for classical mathematical physics principles, and J. Dennis and S. Dorn for more recent use of Legendre transformations to establish dual quadratic programming principles applying to general systems of constraints involving a finite number of variables in that neither principle contains the variables associated with the other. The theorem has also been shown to be as fundamental for the theory of linear inequalities see particularly Charnes and Cooper for this approach as the classic Farkas-Minkowski lemma.
- Theoretical Mathematics