Analysis and Computation of Fixed Points. Proceedings of a by Stephen M. Robinson

By Stephen M. Robinson

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Extra info for Analysis and Computation of Fixed Points. Proceedings of a Symposium Conducted by the Mathematics Research Center, the University of Wisconsin–Madison, May 7–8, 1979

Sample text

Both form locally, around the approximate fixed point, a well behaved homotopy, since the vertices on the lower level connected to those around the fixed point provide the same labeling match as in any stan­ dard homotopy algorithm. 4, when we drift away from the approximate fixed point, we get a pure restart algorithm, since the vertices on the lower level have artificial labels, and, in between, we have a continuous deformation from a homo­ topy algorithm (near the approximate fixed point) to a restart-type behaviour (far away from that point).

1 through no. 3 (Scarf, [8]). u at x € 8S, where hi TRIANGULATIONS FOR HOMOTOPY FIXED POINT ALGORITHMS i = miníx. = 0, x. n-simplex σ d , +1 \ + 1 > 0}). t. ν1, fixed point tices of [ where v 1 ( i = 0,1,... n) are the ver­ σ. The initial grid size was always chosen to be ~-(n+l) (k each 1-face of subdivisions, largest integer S n ) . The reason is that for sions there is only one interior point , k <_ -^-(n+l) , (n+1) on subdivi­ while for less there are none. Only when we use grid sizes of ~-(n+l) to 1 - 1 -~-(n+l) , does the simplex found on the first level yield any information about the function.

This vector is calculated dynamically, and cannot be specified in advance. Therefore, the triangulation of the space cannot be formed a priori, but has to be defined layer by layer. 40 SHLOMO SHAMIR The algorithm starts at level having n + 1 0, at a simplex of T vertices on level 0, and one vertex on level 1. Let us refer to this vertex as Ρ π ^ · +■ v» the algorithm climbs to the i level of approximate fixed point level i, and one AFP(i), (Pn(i+1)) we have on level Later on, when T and finds an n + 1 i + 1 vertices on (of_ T) .

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