Komplementaritäts- und Fixpunktalgorithmen in der by Hans-Jakob Lüthi (auth.)

By Hans-Jakob Lüthi (auth.)

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Additional info for Komplementaritäts- und Fixpunktalgorithmen in der mathematischen Programmierung, Spieltheorie und Ökonomie

Example text

33) Y> ~ a(x, y> b(x. y) :s; b(x. y) Lemma 1. 2 ex. y) ist ein Gleichgewichtspunkt von rCA. B) genau. wenn ae (34) be (*) e n m n . ~ Ay :s; TB x. := (1. •• 1) E R n -T a := x Ay b -T := x By - 31 - Beweis Aus (33) folgt (34). wenn man fUr x, resp. y, aIle Verteilungen der Form x = (0,0, .. ,1, .. h. aIle reinen Strategien). Aus (34) folgt (33), indem man mit einer beliebigen gemischten Strategie x, resp. y, das Skalarprodukt bildet: x'Cae m ):

H. S = {X(t) It E R~}, X: R~ ~ R: stuckweise affin. Der Lemke-Algorithmus ist daher ein systematisches Verfahren, die Menge S explizite darzustellen (und ebenfalls ein konstruktiver Beweis des Satzes 1. 9. 3 fUr eine affine Funktion f). Als ein Beispiel nehme man f(x) = Mx+q, wobei M (siehe Figur 6) f l (x)=x 1+2x 2 - 2 f 2 (x) = x 1- 4x 2 +2) - 4 ' q - 2 ( 0 ), d - 47 - Das Problem (q 1M) besitzt keine L6sung, also ist die Menge S der stationaren Punkte unbeschrankt. (Die Umkehrung gilt bekanntlich nur fUr gewisse Klassen von Matrizen!

Beweis Die Notwendigkeit der Beziehung folgt unmittelbar aus der Anwendung der KuhnTucker-Bedingungen auf das lineare Optimierungsproblem (bei festem xl min y·f(x) unter den Nebenbedingungen y'd y ~ k ~ 0 und daraus, dass y = x eine Optimallosung sein solI. FUr ein lineares Optimierungsproblem sind die Kuhn-Tucker-Bedingungen aber ebenfalls hinreichend. Daraus folgt die Umkehrung des Satzes. Il. 1st in (54) x' d < k, so muss x ein stationarer Punkt von (f, R~ l sein. Umgekehrt gilt: Falls x ein stationarer Punkt von Cf, R~) ist und x' d ~ k, so ist x auch ein stationarer Punkt von (f, D~).

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