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On the complex complementarity problem

Published online by Cambridge University Press:  17 April 2009

Bertram Mond
Bertram Mond
Affiliation:
Department of Mathematics, La Trobe University, Bundoora, Victoria.
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Abstract

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The complex linear complementarity problem considered here is the following: Find z such that

where S is a polyhedral convex cone in Cp, S* the polar cone, M ∈ Cp×p and q ∈ Cp.

Generalizing earlier results in real and complex space, it is shown that if M satisfies RezHMz ≥ 0 for all z ∈ Cp and if the set satisfying Mz + q ∈ S*, z ∈ S is not empty, then a solution to the complex linear complementarity problem exists. If RezHMz > 0 unless z = 0, then a solution to this problem always exists.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 9 , Issue 2 , October 1973 , pp. 249 - 257
Copyright
Copyright © Australian Mathematical Society 1973

References

[1]Abrams, Robert A. and Ben-Israel, Adi, “A duality theorem for complex quadratic programming”, J. Optimization Theory Appl. 4 (1969), 244–252.CrossRefGoogle Scholar
[2]Ben-Israel, Adi, “Linear equations and inequalities on finite dimensional, real or complex, vector spaces: a unified theory”. J. Math. Anal. Appl. 27 (1969), 367–389.CrossRefGoogle Scholar
[3]Ben-Israel, A., “Theorems of the alternative for complex linear inequalities”, Israel J. Math. 7 (1969), 129–136.CrossRefGoogle Scholar
[4]Cottle, Richard W., “Note on a fundamental theorem in quadratic programming”, J. Soc. Indust. Appl. Math. 12 (1964), 663–665.CrossRefGoogle Scholar
[5]Dorn, W.S., “Self-dual quadratici programs”, J. Soc. Indust. Appl. Math. 9 (1961), 51–51.CrossRefGoogle Scholar
[6]Frank, Marguerite and Wolfe, Philip, “An algorithm for quadratic programming”, Naval Res. Logist. Quart. 3 (1956), 95–110.CrossRefGoogle Scholar
[7]Gale, David, The theory of linear economic models (McGraw-Hill, New York, Toronto, London, 1960).Google Scholar
[8]McCallum, Charles J. Jr, “Existence theory for the complex linear complementarity problem”, J. Math. Anal. Appl. 40 (1972), 738–762.CrossRefGoogle Scholar
[9]Mond, Bertram and Hanson, Morgan A., “Symmetric duality for quadratic programming in complex space”, J. Math. Anal. Appl. 23 (1968), 284–293.CrossRefGoogle Scholar
[10]Mond, Bertram and Hanson, Morgan A., “Some generalizations and applications of a complex transposition theorem”, Linear Algebra and Appl. 2 (1969), 401–411.CrossRefGoogle Scholar