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Dantzig-Wolfe Decomposition MS&E348 Winter 2011/2012 Professor Gerd Infanger

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Dantzig-Wolfe Decomposition

MS&E348Winter 2011/2012

Professor Gerd Infanger

Page 2: Dantzig-Wolfe Decomposition - Stanford Universityweb.stanford.edu/class/msande348/handouts/Dantzig-Wolfe Decomposition.pdf · Dantzig-Wolfe Decomposition MS&E348 Winter 2011/2012

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Primal block angular structure

Page 3: Dantzig-Wolfe Decomposition - Stanford Universityweb.stanford.edu/class/msande348/handouts/Dantzig-Wolfe Decomposition.pdf · Dantzig-Wolfe Decomposition MS&E348 Winter 2011/2012

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The problem

Page 4: Dantzig-Wolfe Decomposition - Stanford Universityweb.stanford.edu/class/msande348/handouts/Dantzig-Wolfe Decomposition.pdf · Dantzig-Wolfe Decomposition MS&E348 Winter 2011/2012

The feasible region of the subproblems

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Page 5: Dantzig-Wolfe Decomposition - Stanford Universityweb.stanford.edu/class/msande348/handouts/Dantzig-Wolfe Decomposition.pdf · Dantzig-Wolfe Decomposition MS&E348 Winter 2011/2012

The full master problem

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Page 6: Dantzig-Wolfe Decomposition - Stanford Universityweb.stanford.edu/class/msande348/handouts/Dantzig-Wolfe Decomposition.pdf · Dantzig-Wolfe Decomposition MS&E348 Winter 2011/2012

Restricted master problem

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Page 7: Dantzig-Wolfe Decomposition - Stanford Universityweb.stanford.edu/class/msande348/handouts/Dantzig-Wolfe Decomposition.pdf · Dantzig-Wolfe Decomposition MS&E348 Winter 2011/2012

Pricing out

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Page 8: Dantzig-Wolfe Decomposition - Stanford Universityweb.stanford.edu/class/msande348/handouts/Dantzig-Wolfe Decomposition.pdf · Dantzig-Wolfe Decomposition MS&E348 Winter 2011/2012

Subproblems

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Page 9: Dantzig-Wolfe Decomposition - Stanford Universityweb.stanford.edu/class/msande348/handouts/Dantzig-Wolfe Decomposition.pdf · Dantzig-Wolfe Decomposition MS&E348 Winter 2011/2012

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Page 10: Dantzig-Wolfe Decomposition - Stanford Universityweb.stanford.edu/class/msande348/handouts/Dantzig-Wolfe Decomposition.pdf · Dantzig-Wolfe Decomposition MS&E348 Winter 2011/2012

Optimal solution

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Page 11: Dantzig-Wolfe Decomposition - Stanford Universityweb.stanford.edu/class/msande348/handouts/Dantzig-Wolfe Decomposition.pdf · Dantzig-Wolfe Decomposition MS&E348 Winter 2011/2012

Considering unboundedness of subs

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Page 12: Dantzig-Wolfe Decomposition - Stanford Universityweb.stanford.edu/class/msande348/handouts/Dantzig-Wolfe Decomposition.pdf · Dantzig-Wolfe Decomposition MS&E348 Winter 2011/2012

Initial master

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Page 13: Dantzig-Wolfe Decomposition - Stanford Universityweb.stanford.edu/class/msande348/handouts/Dantzig-Wolfe Decomposition.pdf · Dantzig-Wolfe Decomposition MS&E348 Winter 2011/2012

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D-W bounds

Page 14: Dantzig-Wolfe Decomposition - Stanford Universityweb.stanford.edu/class/msande348/handouts/Dantzig-Wolfe Decomposition.pdf · Dantzig-Wolfe Decomposition MS&E348 Winter 2011/2012

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D-W(2) versus Benders decomposition

B D

AT BT

Benders D-W

Page 15: Dantzig-Wolfe Decomposition - Stanford Universityweb.stanford.edu/class/msande348/handouts/Dantzig-Wolfe Decomposition.pdf · Dantzig-Wolfe Decomposition MS&E348 Winter 2011/2012

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Page 16: Dantzig-Wolfe Decomposition - Stanford Universityweb.stanford.edu/class/msande348/handouts/Dantzig-Wolfe Decomposition.pdf · Dantzig-Wolfe Decomposition MS&E348 Winter 2011/2012

D-W on the dual problem

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Page 17: Dantzig-Wolfe Decomposition - Stanford Universityweb.stanford.edu/class/msande348/handouts/Dantzig-Wolfe Decomposition.pdf · Dantzig-Wolfe Decomposition MS&E348 Winter 2011/2012

D-W full master problem

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Page 18: Dantzig-Wolfe Decomposition - Stanford Universityweb.stanford.edu/class/msande348/handouts/Dantzig-Wolfe Decomposition.pdf · Dantzig-Wolfe Decomposition MS&E348 Winter 2011/2012

Observe:

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Page 19: Dantzig-Wolfe Decomposition - Stanford Universityweb.stanford.edu/class/msande348/handouts/Dantzig-Wolfe Decomposition.pdf · Dantzig-Wolfe Decomposition MS&E348 Winter 2011/2012

Continuing with D-W:

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D-W bounds revisited

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