Summary
For a linear-programming-problem Max {p′x | Ax ≤ b, x ≥ 0} the so called dual problem is defined as Min {b′ y | y′ A ≥ p, y≥ 0}. The two problems are linked by the following: “Existence Theorem: If both problems have feasible solutions, then both have optimal solutions”; and the “Duality Theorem: If one of the problems has an optimal solution, then both have optimal solutions with equal optimal values”.
In this paper we show that all the different generalisations of these theorems to convex, nonlinear programming problems are special cases of two general theorems, which use the means of the theory of conjugate functions as set forth byFenchel.
For easy understanding we develop the necessary properties of conjugate functions and give proofs of these two general theorems, of which the duality theorem has not been proved before. In the second part we specialise these theorems to some non-linear programming problems and obtain all the different duality theorems ofDennis, Dorn, Hanson, Huard, Wolfe as special cases of these general theorems.
Résumé
On dit que les problèmes de programmation linéaire Max {p′x | Ax ≤ b, x ≧ 0} et Min {b′y | y′ A ≥ p, y ≥ 0} sont duals. Les deux problèmes sont liés par les théorèmes suivants: Théorème d'existence: «Si les deux problèmes ont des solutions possibles, alors ils ont tout deux des solutions optimales.» Théorème de dualité: «Si un problème a une solution optimale, alors les deux ont des solutions optimales de mêmes valeurs.» Dans ce travail nous montrons que toutes les differentes généralisations de ces théorèmes aux problèmes de programmations convexes non linéaires sont des cas particuliers de deux théorèmes généraux qui utilisent les méthodes de la theorie des fonctions conjuguées comme les poseFenchel.
Pour une compréhension plus facile nous développons les propriétés nécessaires des fonctions conjuguées et nous donnons des preuves de ces deux théorèmes généraux dont le théorème de dualité n'a pas été démontré précédemment. Dans la deuxième partie nous particularisons ces théorèmes à quelques problèmes de programmation non linéaire et nous obtenons tous les différents théorèmes de dualité deDennis, Dorn, Hanson, Huard, Wolfe, en tant que cas particulier de ces deux théorèmes généraux.
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Der Deutschen Forschungsgemeinschaft danke ich für finanzielle Unterstützung während der Abfassung dieser Arbeit.
Vorgel. v.:W. Krelle
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Dieter, U. Dualität bei konvexen Optimierungs-(Programmierungs-)Aufgaben. Unternehmensforschung Operations Research 9, 91–111 (1965). https://doi.org/10.1007/BF01919477
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DOI: https://doi.org/10.1007/BF01919477