Mthodes de dcomposition

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• Méthodes de décomposition

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• Overview of the main decomposition methods
  used
• To solve large scale problem
• To take advantage of structures
• embedded in the problem

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Outline

• Problem structures
• Column generation
• (Dantzig Wolfe decomposition)
• Constraint generation
• (Benders decomposition)
• Lagrangean relaxation
• Lagrangean decomposition

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Problem types

• Min f (x, y)
• Subject to
• F (x, y) 0
• x e X, y e Y
• x nice variables
• y annoying variables
• Constraint generation
• (Benders Decomposition)

• Min f (x)
• Subject to
• x e X1
• x e X2
• X1 nice constraints
• X2 annoying constraints
• Column generation
• (Dantzig Wolfe Decomposition)
• Lagrangean relaxation

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Problèmes linéaires

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Problème

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Problème équivalent

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Approche de résolution

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Illustration graphique

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• X

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Initialisation de la procédure

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Existence de bornes inférieure et supérieure sur
la valeur optimal

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Convergence de la procédure de Dantzig-Wolfe

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Exemple numérique

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Cas où ? nest pas borné

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Extension au cas non-linéaire

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Bornes sur la valeur optimale de (P)

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Convergence

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Advantages

• Solve a sequence of restricted (smaller) problems
• Mechanism to verify if the optimal solution of
  restricted problem is optimal for the original
  problem solving a problem specified with the
  nice constraints
• Same mechanism generates additional variables to
  modify the restricted problem and to get closer
  to the original one

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Advantage of the approach

• PRIMAL METHOD
• for any restricted problem
• the optimal solution ?i, ieG
• x ? ieG ?i xi feasible for the original
  problem

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References

• M. S. Bazaraa, J.J. Jarvis, H.D. Sherali, Linear
  Programming and Network Flows, Third Edition,
  Wiley (2005).
• G.B. Dantzig, P. Wolfe, Decomposition Principle
  for Linear Programs, Operations Research 8
  (1960), 101-111.
• A.M. Geoffrion, Elements of Large Scale
  Mathematical Programming, Part I Concepts Part
  II Synthesis of Algorithms and
  Bibliography,Management Science 16 (1970),
  652-691.