Zusammenfassung
Betrachtet werden die Primär- und Dualbasen einer optimalen Basislösung eines linearen Programms mit einer gegebenen Parametermenge (d. h. Koeffizienten der Zielfunktion, Koeffizienten der Matrix und des Beschränkungsvektors). Der Kürze halber seien diese Basen selbst optimal genannt. Die Anfangsbasen bleiben optimal innerhalb gewisser Teilbereiche des Parameterraumes, bezeichnet als Optimalitätsbereiche der jeweiligen Basen, wenn die Parameter gewissen Variationen unterliegen (vorgegeben oder nicht, je nachdem, ob es sich um parametrisches oder stochastisches Programmieren handelt).
Es wird gezeigt, daß die Optimalitätsbereiche der Primär- und Dualbasen übereinstimmen.
Summary
Consider the primal and dual bases of a basic optimal solution to a linear-programming problem with a given set of parameters (coefficients of objective function, technology matrix, and restriction vector). For brevity, call those bases themselves optimal. If the parameters are subject to variation (controlled or uncontrolled according as one deals with parametric or stochastic programming, respectively) the initial bases are optimal throughout certain subregions of parameter space, termed optimality regions of the respective bases.
It is shown that the optimality regions of primal and dual bases are identical.
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This research has been carried out in association with, and with partial support from the National Science Foundation, Project Nr. 401-04-07 at Iowa State University.
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van Moeseke, P., Tintner, G. Base duality theorem for stochastic and parametric linear programming. Unternehmensforschung Operations Research 8, 75–79 (1964). https://doi.org/10.1007/BF01920923
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DOI: https://doi.org/10.1007/BF01920923