It can be reformulated and solved as a MILP using bigM
max
x,y,λ,w
ζ
=
c
>
x
+
d
>
1
y
s
.
t
.
A
1
x
+
B
1
y
≤
b
1
B
>
2
λ
=
d
2
.
b
2

A
2
x

B
2
y
≥
0
b
2

A
2
x

B
2
y
≤
Mw
λ
≤
M
(1
m
2
×
1

w
)
λ
≥
0
w
∈ {
0
,
1
}
m
2
.
Lizhi Wang ([email protected])
IE 534 Linear Programming
November 9, 2012
8 / 28
Difficulties with the bigM method
We may not know how big is big enough.
I
If bigM is set too small, then it may reduce the feasible region.
I
If bigM is set too big, then it may cause computational problems
due to rounding errors.
A finite bigM may not exist. For example, when BLP is
unbounded.
Lizhi Wang ([email protected])
IE 534 Linear Programming
November 9, 2012
9 / 28
Solution technique: branch and bound
Define the LP relaxation as
max
x,y,λ
ζ
=
c
>
x
+
d
>
1
y
(11)
s
.
t
.
A
1
x
+
B
1
y
≤
b
1
(12)
A
2
x
+
B
2
y
≤
b
2
(13)
B
>
2
λ
=
d
2
(14)
λ
≥
0
.
(15)
If LP relaxation solution satisfies the complementarity constraints,
then it’s optimal to BLP.
Otherwise create two new nodes (assuming
i
th complementarity
constraint violated)
LP relaxation (11)(15)
(
A
2
x
+
B
2
y
)
i
= (
b
2
)
i
LP relaxation (11)(15)
(
λ
)
i
= 0
Lizhi Wang ([email protected])
IE 534 Linear Programming
November 9, 2012
10 / 28
Definitions:
B
(
l, u
)
and
L
(
l, u
)
B
(
l, u
)
max
x,y,λ
ζ
=
c
>
x
+
d
>
1
y
s
.
t
.
A
1
x
+
B
1
y
≤
b
1
l
≤
A
2
x
+
B
2
y
≤
b
2
0
≤
λ
≤
u
B
>
2
λ
=
d
2
0
≤
b
2

A
2
x

B
2
y
⊥
λ
≥
0
L
(
l, u
)
max
x,y,λ
ζ
=
c
>
x
+
d
>
1
y
s
.
t
.
A
1
x
+
B
1
y
≤
b
1
l
≤
A
2
x
+
B
2
y
≤
b
2
0
≤
λ
≤
u
B
>
2
λ
=
d
2
Lizhi Wang ([email protected])
IE 534 Linear Programming
November 9, 2012
11 / 28
Step 0: Initialization
Create the root node
B
(
l
1
=
∞
m
2
×
1
, u
1
=
∞
m
2
×
1
)
, which is
characterized by
(
l
1
, u
1
, z
1
=
∞
)
.
Initialize
x
*
=
∅
, y
*
=
∅
, λ
*
=
∅
, ζ
*
=
∞
, N
= 1
.
Go to Step 1.
Lizhi Wang ([email protected])
IE 534 Linear Programming
November 9, 2012
12 / 28
Step 1: Node management
For all
k
∈ {
1
, ..., N
}
such that
z
k
≤
ζ
*
, remove node
k
.
Update
N
as the number of remaining nodes.
if
N
= 0
then
if
x
*
6
=
∅
then
1(a)
return
(
x
*
, y
*
, λ
*
, ζ
*
)
is an optimal solution to the BLP
(4)(6).
else
1(b)
return
the BLP (4)(6) is infeasible.
end
else
1(c)
select a node
k
from
{
1
, ..., N
}
, set
(
ˆ
l
=
l
k
,
ˆ
u
=
u
k
)
, remove
node
k
, reorder the remaining nodes from 1 to
N

1
, reduce
N
by
1, and go to Step 2.
end
Lizhi Wang ([email protected])
IE 534 Linear Programming
November 9, 2012
13 / 28
Step 2: LP relaxation
Solve
L
(
ˆ
l,
ˆ
u
)
.
if
L
(
ˆ
l,
ˆ
u
)
is infeasible
then
2(a)
go to Step 1.
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 Spring '12
 lizhiwang
 Operations Research, Linear Programming, Optimization, Constraint, Lizhi Wang