By Achim Bachem (auth.), Rudolf Henn, Bernhard Korte, Werner Oettli (eds.)
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Extra resources for Optimization and Operations Research: Proceedings of a Workshop Held at the University of Bonn, October 2–8, 1977
Example text
1- S i/j j which implies that I is a solution for the partition problem. 5. Results The results of section 3 and 4 are summarized in the tables polynomial solvable 2 I . 1 2 1m-dim. Euclid. I l: k 11-dim. Euclid. I l: k 11-dim. Euclid. I l: k 11-dim. Euclid. I max max dij N 1 lBvT l: dij l: d ij max dij The status of the 2 I· l: max d ij problem is not clear. All other noneuclidean problems belong to one of the classes PS or NPC. Another open question is which Euclidean problems are NP-complete.
According to [2], Satz 1, Gr f1 is to be described as the complement of a denumerable union of "rectangles· A n x B n with A n E Ct: and Bn E ~(E). In consequence of [4], Satz A, there exists a probability IJ with the properties IJCAxE) = pCA) for all A E Ot Hence it follows that and from this because of (**) that 1J*(Gr f) = IJ*CC) = 1. 54, remark to Theorem 1. 1 is fulfilled for (M,Ot to p ~n ,p); thus there exi~ts a probability P1 lif' which is equal &·~f. e. P1 1;'f is an element off(pIOl;it f ).
2 the result follows . . In general the mapping t ........... P t is not lower semicontinuous. We give now a sufficient condition for the lower semicontinuity of p t DEFINITION 3. A mapping t ~Zt is called a iff for every z,x Ux and Ut o in Zt and an a 40 > 0 (P)-mapping in to E T there exist °such that neighborhoods ! 5 Now we prove THEOREM 4. Let the (PLO)-Problem be given and let Zt be compact for each t E T. ,.. P t is lower semicontinuous. PROOF. If P is not lower semicontinuous in to there is a sequence (t n ) in T, an element v t such that t Choose v t n -> t 0 in P t o ' and an open set W containing v t 0 v & nE IN for each n E IN.