A Panorama of Pure Mathematics (Pure and Applied Mathematics by Jean A. Dieudonne

By Jean A. Dieudonne

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R 3. Falls ΔzN ≤ 0, dann ist die Aufgabe unzul¨ assig. Ende. zjr 4. Berechnung der Schrittweite sz = min{ Δzr : j ∈ N, Δzjr > 0}; Bestimmung des in die j Basis eintretenden Index, d. h. Wahl eines j0 , welches dieses Minimum realisiert. r r r := zN − sz ΔzN 5. zN zi0 := sz B := B \ {i0 } ∪ {j0 } xB := A−1 B b Fortsetzung bei 1. 4 Sensitivit¨ at und Postoptimalit¨ at In der primalen Simplexmethode sind alle verwendeten Basen primal zul¨assig. Sind in einer Iteration die reduzierten Kosten nichtnegativ – d.

B. B. 5 = 6 = −1 = 15 ≥ 0. Entartung Die vorangegangenen Kapitel geben Auskunft dar¨ uber, wie in der primalen Simplexmethode zur L¨osung einer Aufgabe in Normalform min{c x : Ax = b, x ≥ 0} die Unzul¨assigkeit oder Unbeschr¨anktheit dieser Aufgabe bzw. die Optimalit¨at einer Basis erkannt wird. Es bleibt die Frage zu beantworten, ob eine derartige zum Abbruch f¨ uhrende Entscheidung stets nach einer endlichen Anzahl von Basiswechseln getroffen werden kann. Es werde dazu angenommen, dass in Phase 2 – analoge Bemerkungen gelten f¨ ur Phase 1 – ur jeden Basiswechsel vom Algorithmus die Folge B (1) , B (2) , .

1. Falls xB ≥ 0, dann ist B optimale Basis. Stopp. 2. Bestimmung des austretenden Index, d. h. Wahl eines i0 ∈ B mit xi0 < 0; Berechnung r von ΔzN = −(A−1 B AN ) ei0 . r 3. Falls ΔzN ≤ 0, dann ist die Aufgabe unzul¨ assig. Ende. zjr 4. Berechnung der Schrittweite sz = min{ Δzr : j ∈ N, Δzjr > 0}; Bestimmung des in die j Basis eintretenden Index, d. h. Wahl eines j0 , welches dieses Minimum realisiert. r r r := zN − sz ΔzN 5. zN zi0 := sz B := B \ {i0 } ∪ {j0 } xB := A−1 B b Fortsetzung bei 1. 4 Sensitivit¨ at und Postoptimalit¨ at In der primalen Simplexmethode sind alle verwendeten Basen primal zul¨assig.

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