By Ulrich Eckhardt (auth.), Prof. Dr. Günter Fandel, Prof.Dr. Hermann Gehring (eds.)
Das vorliegende Buch behandelt allgemeine Ansätze und spezielle Methoden des Operations examine für volks- und betriebswirtschaftliche Anwendungen. Beiträge von über 30 Forschern informieren den Leser über den Stand und neuere Entwicklungen auf dem Gebiet der quantitativen Wirtschaftsforschung. Es ergibt sich ein aktuelles Bild des Operations examine, wie es sich derzeit an in- und ausländischen Universitäten präsentiert. Neue Ergebnisse und Methoden betreffen u.a. neue Konzepte der Produktionsplanung, spezielle Ansätze der Wissensverarbeitung, die Problemlösung mit informierten Suchstrategien und die Modellierung von Ungewißheit.
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Corollary 1. Given a program Stat {f(x)lx > OJ, x* is a minimal point if 4>0= all =0, aXj x' x/=O= all aXj x' ~o j=1, ... ,n. If (x*, A*) is a minimal point of the Lagrangian, equation (3), then x* is a minimal point of Stat{L(x, A*)Ix E Q2} and A* is a maximal point of Stat{L(x*, A)IA ~ OJ. Remark 2. Few additional explanatory words are necessary to define the primal constraints. Given a Lagrangian function L(x, A) = I(xo) + I L Ajgj(Xj) x E Q A~ 0 Without loss of generality it is assumed that j, gj j= 1, ...
U. PASSY and S. YOTAV 38 (b) If (y*, 1'/*) is minimal then + b(K) = -Vw(y*)AT(K) + b(K) ~ -Vw(y*)AT(K) ° ° Since V Tw(y*) = x*, the point x* is primal feasible. Also (Y*,1'/k) solves D(K)-[3sk,uk,Ak>03(x*,sk,uk,Ak) solves PL(K)]-[x*,sk=O, uk=-bK , Ak>O solves PdK)]-[s*=O, u*=-b, Ak> 0, AE = 0, x* solves PL] - [x* solves P and is minimal]. (c) f(x*) + w(y*) = (x*)T(y*) or f(x*) + w(y*) + b T1'/* = (x*,b)T(y*,1'/*) = (x*,bK )T(Y*,1'/k) = (x*)T[I,A T(K)](Y*,1'/k) = 0. Example 3. 4-YV8+31'/11'/+y,>0 1'/+Y2>0 1'/~0} Each program has four minimal points: Primal xi xi xi xi ° ° = x!
For all Aj 3 Xjl(Aj) 3 Zjt/Aj = Vgj{Xjl(Aj)}. The dual program can be represented differently if the indices are rearranged. To do so the following notation is adopted: (a) q(j) = n(j) - 1 j = 0, ... , K (b) Yjl = [Yo,j. YI,J. , Ym(J),j] = [Zm(J) + I,j, ZI,j, ... , zm(j),j] E tfm(j) + I E tfn(J)-1 (c) Yj2 = [Ym(J) + I,j, ... , Ym(j)+q(J),j] = [Zm(J)+2,J. , zm(J)+n(J),j] (d) Cj = = tfq(j) [CO,j, ... , CI,J, ... , Cq(j),j]. Then it is possible to eliminate the dual variables Aj j=O, ... ,K from D2, thus obtaining D 1 Dl: Stat Lt, lj(Yj' YOj)IY E Q!