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1 Dantzig-Wolfe Decomposition

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2 Outline block structure of a problem representing a point by extreme points Revised Simplex to the extreme point representation an example

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General Problem min s.t.s.t. B 1 x 1 = b 1, B 2 x 2 = b 2, … B K x K = b K, 0 x k, k = 1, 2, …, K. common in network- based problems distribution of K types of products B k x k = b k : constraints related to the flow of the kth type of products constraints of common resources for the K products: 3

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4 General Problem min s.t.s.t. B 1 x 1 = b 1, B 2 x 2 = b 2, …… B K x K = b K, 0 x k, k = 1, 2, …, K.

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5 General Problem min s.t. B 1 x 1 = b 1, B 2 x 2 = b 2, … B K x K = b K, 0 x k, k = 1, 2, …, K. Possible to avoid solving a large problem?

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6 A Numerical Example Problem P2: min–3x 1 – 2x 2 – 2x 3 – 4x 4, s.t. x 1 + x 2 + 2x 3 + x 4 10, x 1 + 2x 2 8, x 2 3, x 3 + 3x 4 6, x 3 4, x i 0.

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7 A Numerical Example feasible region: {(x 1, x 2 ): x 1 + 2x 2 8, x 2 3} extreme points: a feasible point: convex combination of the extreme points

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8 A Numerical Example feasible region: {(x 3, x 4 ): x 3 + 3x 4 6, x 3 4} extreme points: a feasible point: convex combination of the extreme points

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9 A Numerical Example Problem P3: problem in terms of extreme points min -3(8 13 ) – 2(3 14 ) – 2(4 23 ) – 4( 24 ), s.t. (8 13 ) + (3 14 ) + 2(4 23 ) + ( 24 ) 10, 11 + 12 + 13 + 14 = 1, 21 + 22 + 23 + 24 = 1, ij 0,i = 1, 2; j = 1, 2, 3, 4. a subspace B k x k = b k is represented by a single constraint n kn = 1

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10 A Numerical Example Problem P3: problem in terms of extreme points min –24 12 – 12 13 – 6 14 – 8 22 – (32/3) 23 – 8 24, s.t. 8 24 10, 11 + 12 + 13 + 14 = 1, 21 + 22 + 23 + 24 = 1, ij 0, i = 1, 2; j = 1, 2, 3, 4. The previous representation suits DW Decomposition more.

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11 Practical? impractical approach impossible to generate all the extreme points of a block B k x k = b k Dantzig-Wolfe Decomposition: check all extremely points without explicitly generating them

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12 General Problem K blocks, B k x k = b k, k = 1, …, K N k extreme points in the kth block dual variable 0 dual variable 1 dual variable 2 dual variable K …

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13 Reduced Cost for a Non-Basic Variable kn reduced cost of kn

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14 Most Negative Reduced Cost Check each extreme point for each block, which is equivalent to solving a linear program. Result: Solving K+1 small linear programs. where

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15 General Approach 1 Form the master program (MP) by the representation 2 Get a feasible solution of the MP; find the corresponding 3 Solve the subproblems to check the reduced costs of kn 3.1 stop if the MP is optimal; 3.2 else carry a standard revised simplex iteration (i.e., identifying the entering and leaving variables, stopping for an unbounded problem, and determining B -1 otherwise) 3.3 go back to 2 if the problem is not unbounded

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16 An Example of the Dantzig-Wolfe Approach An Example of the Dantzig-Wolfe Approach min –3y 1 – 2y 2 – 2y 3 – 4y 4, s.t. y 1 + y 2 + 2y 3 + y 4 10, y 1 + 2y 2 8, y 2 3, y 3 + 3y 4 6, y 3 4, y i 0.

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17 An Example of the Dantzig-Wolfe Approach An Example of the Dantzig-Wolfe Approach problem in general form with

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18 An Example of the Dantzig-Wolfe Approach An Example of the Dantzig-Wolfe Approach let be the extreme points of B 1 x 1 = b 1 any point in B 1 x 1 = b 1 : similarly, any point in B 2 x 2 = b 2 :

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19 An Example of the Dantzig-Wolfe Approach An Example of the Dantzig-Wolfe Approach in terms of the extreme points

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20 An Example of the Dantzig-Wolfe Approach An Example of the Dantzig-Wolfe Approach to solve the problem, introduce the slack variable x 5 and artificial variables x 6 and x 7

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21 An Example of the Dantzig-Wolfe Approach An Example of the Dantzig-Wolfe Approach initial basic variable x B = (x 5, x 6, x 7 ) T, B = I, (c B ) T = (0, M, M), b = (10, 1, 1) T = (c B ) T B = (0, M, M) for the kth subproblem, the reduced costs:

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