Unternehmensforschung

, Volume 13, Issue 1, pp 29–36

# A simple proof of a general duality theorem of convex programming

• B. v. Hohenbalken
Abhandlungen

## Summary

The first part of this note presents concisely and partially proves in logical terms the relations betweenUzawa's andKuhn andTucker's equivalence theorems of nonlinear programming. In the second part we give simple mathematical proofs of two lemmata linking theKuhn-Tucker conditions and dual solutions and use them to establish a nonlinear duality theorem of considerable generality.

### Keywords

Nonlinear Programming Simple Proof Convex Programming Duality Theorem Mathematical Proof

## Zusammenfassung

Im ersten Abschnitt dieses Beitrags demonstrieren wir in kurzer, symbolischlogischer Form die Relationen zwischenUzawas undKuhn undTuckers Äquivalenzsätzen des nichtlinearen Programmierens. Im zweiten Teil geben wir mathematische Beweise zweier Lemmata, die dieKuhn-Tucker-Bedingungen und duale Lösungen in Zusammenhang bringen, und benützen sie und die logische Struktur des ersten Teils, um ein nichtlineares Dualitätstheorem von besonderer Allgemeinheit zu beweisen.

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### References

1. Dorn, W. S.: A Duality Theorem for Convex Programs; IBM Journal, October 1960, pp. 407–413.Google Scholar
2. Hanson, M. A.: A Duality Theorem in Non-Linear Programming with Non-Linear Constraints; Australian Journal of Statistics, 1961, Vol. III, pp. 64–72.Google Scholar
3. Huard, P.: Dual Programs; IBM Journal, January 1962, pp. 137–139.Google Scholar
4. Kuhn, H. W., andA. W. Tucker: Nonlinear Programming; Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, ed: J. Neyman; University of California Press, Berkeley and Los Angeles, 1951, pp. 481–492.Google Scholar
5. Uzawa, H.: The Kuhn-Tucker Theorem in Concave Programming; Studies in Linear and Nonlinear Programming. eds: K. J. Arrow, L. Hurwicz, H. Uzawa; Stanford University Press, Stanford 1958, Part I, Chapter 3, pp. 32–37.Google Scholar
6. Van Moeseke, P.: A General Duality Theorem of Convex Programming; Metroeconomica Sept.–Dec. 1965, Vol. XVII, No. 3, pp. 161–170.Google Scholar
7. Wolfe, P.: A Duality Theorem for Non-Linear Programming; Quarterly of Applied Mathematics, October 1961. Vol. XIX, No. 3, pp. 239–244.Google Scholar