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IE521 Advanced Optimization
Lecture 12
Dr. Zeliha Akc
¸a
January 2012
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View Full Document Large scale Linear Programming
I
Linear programs occurring in practice can be extremely large.
I
They may have a large number of variables or constraint.
I
For large LPs, the constraint matrix
A
can be too large to store
and process.
I
To solve a particular instance, we only need
I
Constraints that are binding at optimality.
I
Variables that are basic at optimality.
I
Therefore, only need constraints that have positive dual values
(binding constraints) and variables that have positive values at
optimality (basic variables).
I
With only these variables and constraints, we could solve the
problem very easily.
I
The problem is: which variables are basic at optimal and which
constraints are binding?
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Column Generation Procedure
I
For problems with
large numbers of variables
, two main
difﬁculties:
⇒
The time required to
generate the constraint matrix
A
.
⇒
The time required to calculate the reduced cost of each
variable (to prove the optimality) at each iteration.
I
These difﬁculties may be overcome with
column generation
.
I
Main steps of the procedure is:
⇒
Generate an initial
subset of columns.
⇒
Solve the LP with just these columns.
⇒
Search for the remaining columns and add those
with negative
reduced costs.
⇒
Iterate these steps until no column is found with negative
reduced cost.
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View Full Document Generic Column Generation Algorithm
I
We want to solve an LP with a large number of columns.
I
Consider the following LP:
(LP) min
X
i
∈
C
c
i
x
i
s.t.
X
i
∈
C
A
i
x
i
=
b
x
≥
0
where

C

is a
very large number.
I
Consider the
restricted problem
obtained by considering only the
subset of the columns
indexed by set
I
.
(RLP) min
X
i
∈
I
c
i
x
i
s.t.
X
i
∈
I
A
i
x
i
=
b
x
≥
0
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Generic Column Generation Algorithm
1.
Solve the restricted LP (RLP) and calculate the
optimal dual
solution (
p
>
=
c
B
B

1
.
)
2.
Generate a new column
A
j
with negative reduced cost for the
given dual solution (
p
>
):
⇒
satisfying
c
j

c
B
B

1
A
j
<
0
.
⇒
This column can be found by solving the
column generation
subproblem
which is also an
optimization problem:
w
*
=
min
a
∈
C
c
a

c
>
B
B

1
a
,
where
C
is the global set of columns.
3.
If the minimum value for the problem
w
*
is
<
0
, add the new
column to the set
I
and go to Step 1. If
w
*
>
=
0
, stop, we have
the optimal LP solution.
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View Full Document
There are some variants of the algorithm:
I
Only store the basic variable columns and delete the others from
set
I
.
I
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This note was uploaded on 01/09/2012 for the course IE 521 taught by Professor Zelihaakça during the Fall '11 term at Fatih Üniversitesi.
 Fall '11
 ZelihaAkça
 Optimization

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