Abstract
Using Scarf's algorithm for “computing” a fixed point of a continuous mapping, the following is proved: LetM 1 ⋯ M n be closed sets inR n which cover the standard simplexS, so thatM i coversS i , the face ofS opposite vertexi. We say a point inS iscompletely labeled if it belongs to everyM i andk-almost-completely labeled if it belongs to all butM k . Then there exists a closed setT ofk-almost-completely labeled points which connects vertexk with some completely labeled point.
This result is used to prove Browder's theorem (a parametric fixed-point theorem) inR n. It is also used to generate “algorithms” for the nonlinear complementarity problem which are analogous to the Lemke—Howson algorithm and the Cottle—Dantzig algorithm, respectively, for the linear complementarity problem.
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Freidenfelds, J. A set intersection theorem and applications. Mathematical Programming 7, 199–211 (1974). https://doi.org/10.1007/BF01585516
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DOI: https://doi.org/10.1007/BF01585516