dantzig -wolfe decomposition
DESCRIPTION
Dantzig -Wolfe Decomposition. Outline. block structure of a problem representing a point by extreme points Revised Simplex to the extreme point representation an example. General Problem. min s . t . B 1 x 1 = b 1 , B 2 x 2 = b 2 , … - PowerPoint PPT PresentationTRANSCRIPT
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Dantzig-Wolfe DecompositionDantzig-Wolfe Decomposition
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OutlineOutline
block structure of a problem
representing a point by extreme points
Revised Simplex to the extreme point representation
an example
General ProblemGeneral Problem
min
s.t.
B1x1 = b1,
B2x2 = b2,
…
BKxK = bK,
0 xk, k = 1, 2, …, K.
common in network-based problems distribution of K types of
products
Bkxk = bk: constraints related to the flow of the kth type of products
constraints of common resources for the K products:
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T T T1 1 2 2 ... ,K K c x c x c x
1 1 2 2 0... ,K K A x A x A x b
1 1 2 2 0... K K A x A x A x b
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General ProblemGeneral Problem
min s.t.
B1x1 = b1,
B2x2 = b2,
…
BKxK = bK,
0 xk, k = 1, 2, …, K.
T T T1 1 2 2 ... ,K K c x c x c x
1 1 2 2 0... ,K K A x A x A x b
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General ProblemGeneral Problem
min s.t.
B1x1 = b1,
B2x2 = b2,
…
BKxK = bK,
0 xk, k = 1, 2, …, K.
T T T1 1 2 2 ... ,K K c x c x c x
1 1 2 2 0... ,K K A x A x A x b
Possible to avoid solving a large problem?
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A Numerical Example A Numerical Example
Problem P2:
min –3x1 – 2x2 – 2x3 – 4x4,
s.t. x1 + x2 + 2x3 + x4 10,
x1 + 2x2 8,
x2 3,
x3 + 3x4 6,
x3 4,
xi 0.
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A Numerical Example A Numerical Example
feasible region: {(x1, x2): x1 + 2x2 8, x2 3} extreme points:
a feasible point: convex combination of the extreme points
0 8 2 0, , , .
0 0 3 3
111 12 13 14
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0 8 2 0;
0 0 3 3
x
x
11 12 13 14 1;
11 12 13 140 , , , .
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A Numerical Example A Numerical Example
feasible region: {(x3, x4): x3 + 3x4 6, x3 4} extreme points:
a feasible point: convex combination of the extreme points
0 4 4 0, , , .
0 0 2 / 3 2
321 22 23 242
4 3
40 8 0,
0 0 2
x
x
21 22 23 24 1,
21 22 23 240 , , , .
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A Numerical Example A Numerical Example
Problem P3: problem in terms of extreme points
min -3(812+213) – 2(313+314) – 2(422+423) – 4( 23+224),
s.t. (812+213) + (313+314) + 2(422+423) + ( 23+224) 10,
11 + 12 + 13 + 14 = 1,
21 + 22 + 23 + 24 = 1,
ij 0, i = 1, 2; j = 1, 2, 3, 4.
a subspace Bkxk = bk is represented by a single constraint n kn = 1
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A Numerical Example A Numerical Example
Problem P3: problem in terms of extreme points
min –2412 – 1213 – 614 – 822 – (32/3)23 – 824,
s.t. 812 + 513 + 314 + 822 +8 23 +224 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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Practical? Practical?
impractical approach
impossible to generate all the extreme points of a block Bkxk = bk
Dantzig-Wolfe Decomposition: check all extremely points without explicitly generating them
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General ProblemGeneral Problem
K blocks, Bkxk = bk, k = 1, …, K
Nk extreme points in the kth block
dual variable 0
dual variable 1
dual variable 2
dual variable K
…
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Reduced Cost Reduced Cost for a Non-Basic Variable for a Non-Basic Variable knkn
reduced cost of kn
T 1kn kn knc c Bc B P T T 1
ˆ
0
ˆ1
0
k kn
k kn
B
A X
= c X c B
T
0ˆ ˆ .k kn k kn k = c X X
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Most Negative Reduced CostMost Negative Reduced Cost
T0
,ˆ ˆmin k kn k kn k
k n c X A X
T0
ˆ ˆmin min k kn k kn kk n
c X A X
T0
ˆ ˆmin k kn k kn kn
c X A X
T0min
k k kk k k k k
B X bc X A X
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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General ApproachGeneral 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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An Example of An Example of the Dantzig-Wolfe Approachthe Dantzig-Wolfe Approach
min –3y1 – 2y2 – 2y3 – 4y4,
s.t. y1 + y2 + 2y3 + y4 10,
y1 + 2y2 8,
y2 3,
y3 + 3y4 6,
y3 4,
yi 0.
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An Example of An Example of the Dantzig-Wolfe Approachthe Dantzig-Wolfe Approach
problem in general form with1 3
1 22 4
, ,y y
y y
x x
T T1 2( 3, 2), ( 2, 4), c c
1 2 0(1,1), (1,1), 10, A A b
1 11 2 8
, ,0 1 3
B b 2 2
1 3 6, ,
1 0 4
B b
1 20 0
, .0 0
x x
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An Example of An Example of the Dantzig-Wolfe Approach the Dantzig-Wolfe Approach
let be the extreme points of B1x1 = b1
any point in B1x1 = b1:
similarly, any point in B2x2 = b2:
11 12 13 14ˆ ˆ ˆ ˆ, , ,X X X X
1
2
y
y
4 T1 1 1
1
ˆn n
n c X
3
4
y
y
4 T2 2 2
1
ˆn n
n c X
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An Example of An Example of the Dantzig-Wolfe Approach the Dantzig-Wolfe Approach
in terms of the extreme points
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An Example of An Example of the Dantzig-Wolfe Approach the Dantzig-Wolfe Approach
to solve the problem, introduce the slack variable x5 and artificial variables x6 and x7
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An Example of An Example of the Dantzig-Wolfe Approach the Dantzig-Wolfe Approach
initial basic variable xB = (x5, x6, x7)T, B = I,
(cB)T = (0, M, M), b = (10, 1, 1)T
= (cB)TB = (0, M, M)
for the kth subproblem, the reduced costs: T
0ˆ ˆmin k kn k kn k
n c X A X
T0min
k k kk k k k k
B X bc X A X