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1
15.053
Tuesday, March 5
z
Duality
–
The art of obtaining bounds
–
weak and strong duality
z
Handouts:
Lecture Notes
2
Bounds
z
One of the great contributions of
optimization theory (and math
programming) is the providing of upper
bounds for maximization problems
z
We can prove that solutions are optimal
z
For other problems, we can bound the
distance from optimality
3
A 4variable linear program
maximize
z =
3x
1
+ 4x
2
+6x
3
+ 8x
4
subject to
x
1
+
x
2
+ x
3
+
x
4
=
1
2x
1
+ 3x
2
+4x
3
+ 5x
4
=
3
x
1
,
x
2
,
x
3
,
x
4
≥
0
David has minerals that he will mix together and sell
for profit.
The minerals all contain some gold
content, and he wants to ensure that the mixture has
3% gold, and each bag will weigh 1 kilogram.
Mineral 1:
2% gold, $3 profit/kilo
Mineral 2:
3% gold, $4 profit/kilo
Mineral 3:
4% gold, $6 profit/kilo
Mineral 4:
5% gold, $8 profit/kilo
4
A
4variable linear program
1
=
=
1
1
3
1
1
x
1
x
2
x
4
x
3
2
3
4
5
3
4
8
6
z
0
0
1
=
0
maximize
z =
3x
1
+ 4x
2
+6x
3
+ 8x
4
subject to
x
1
+
x
2
+ x
3
+
x
4
=
1
2x
1
+ 3x
2
+4x
3
+ 5x
4
=
3
x
1
,
x
2
,
x
3
,
x
4
≥
0
5
Obtaining a Bound
x
1
x
2
x
4
x
3
z
Subtract 8 times constraint 1 from the objective function.
z – 5 x
1
–4
±x
2
–2±x
3
= 8
Does this show that z
≤
8?
YES!
z + 5 x
1
+ 4x
2
+ 2 x
3
= 8
1=
=
1
1
3
1
1
2
3
4
5
0
0
5
4
0
2
8
6
Obtaining a Second Bound:
Treat the
operation as pricing out
x
1
x
2
x
4
x
3
z
Subtract 3 * constraint 1 and subtract constraint 2 from the
objective function.
z – 2 x
1
–2
2
–1±x
3
= 6
Thus z
≤
6!
Which bound is better: 6 or 8?
z + 2 x
1
+ 2x
2
+ 1 x
3
= 6
=
1
1
3
1
1
2
3
4
5
0
0
1
3
Prices
2
2
1
0
6
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Obtaining the Best Bound:
Formulate
the problem as an LP
x
1
x
2
x
4
x
3
z
1=
=
1
1
3
1
1
2
3
4
5
0
0
y
2
y
1
Prices
A
A:
3  y
1
2y
2
≤
0
Î
y
1
+ 2y
2
≥
3
C
C:
6 – y
1
–4y
2
≤
0
Î
y
1
+ 4y
2
≥
6
B
B:
4  y
1
3y
2
≤
0
Î
y
1
+ 3y
2
≥
4
y
1
3y
2
minimize
y
1
+ 3y
2
D
D:
8 – y
1
–5y
2
≤
0
Î
y
1
+ 5y
2
≥
8
8
The problem that we formed is called
the dual problem
Subject to
y
1
+ 2y
2
≥
3
y
1
+ 4y
2
≥
6
y
1
+ 3y
2
≥
4
y
1
+ 5y
2
≥
8
minimize
y
1
+ 3y
2
y
1
and y
2
are unconstrained in sign
9
Summary of previous slides
z
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 Spring '05
 Prof.JamesOrlin

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