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On duality in complex linear programming

Published online by Cambridge University Press:  09 April 2009

B. D. Craven  and
B. Mond
B. D. Craven
Affiliation:
University of MelbourneMelbourne, Australia.
B. Mond
Affiliation:
La Trobe University Bundoora, Vic., Australia.
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In [3], Levinson proved a duality theorem for linear programming in complex space. Ben-Israel [1] generalized this result to polyhedral convex cones in complex space. In this paper, we give a simple proof of Ben-Israel's result based directly on the duality theorem for linear programming in real space. The explicit relations shown between complex and real linear programs should be useful in actually computing a solution for the complex case. We also give a simple proof of Farkas' theorem, generalized to polyhedral cones in complex space ([1], Theorem 3.5); the proof depends only on the classical form of Farkas' theorem for real space.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 16 , Issue 2 , September 1973 , pp. 172 - 175
Copyright
Copyright © Australian Mathematical Society 1973

References

[1]Ben-Israel, A., ‘Linear equations and inequalities on finite dimensional, real or complex, spaces a unified theory’, J. Math. Anal. Appl. 27 (1969), 367–389.CrossRefGoogle Scholar
[2]Farkas, J., ‘Über die Theorie der einfachen Ungleichungen’, J. Reine angew. Math. 124 (1902), 1–14.Google Scholar
[3]Levinson, N., ‘Linear programming in complex space’, J. Math. Anal. Appl. 14 (1966), 44–62.CrossRefGoogle Scholar