A set intersection theorem and applications |
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Authors: | John Freidenfelds |
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Affiliation: | 1. Bell Telephone Laboratories, Whippany, N.J., USA
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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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