1 1 1 lizhi wang lzwangiastateedu ie 534 linear

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Lizhi Wang ([email protected]) IE 534 Linear Programming October 31, 2012 11 / 21

Outline
1
Brief introduction
2
Gomory cut
Lizhi Wang ([email protected]) IE 534 Linear Programming October 31, 2012 12 / 21

Cutting plane algorithm
Step 1:
Solve LP relaxation and go to Step 2.
Step 2:
if LP relaxation infeasible then
2(a) return ILP is infeasible.
else if x
L
is integral then
2(b) return x
L
is optimal to ILP.
else
2(c) add a cut and go to Step 1.
end
Lizhi Wang ([email protected]) IE 534 Linear Programming October 31, 2012 13 / 21

Ralph Edward Gomory
Ralph Edward Gomory (1923 – ).
American applied mathematician and
executive.
President of the Sloan Foundation,
1989 – 2007.
He proposed the Gomory cut method for ILP in the 1950s.
I
Most experts, including Gomory himself, considered them to be
impractical.
I
In the mid-1990s, Gomory cuts were shown to be very effective in
combination with branch-and-bound.
I
Nowadays, all commercial MILP solvers use Gomory cuts in one
way or another.
Lizhi Wang ([email protected]) IE 534 Linear Programming October 31, 2012 14 / 21

Gomory cut
Suppose an LP relaxation is solved, the optimal solution x
*
does
not satisfy the integrality constraint, and the optimal simplex
dictionary is the following:
ζ = c
>
B
A
- 1
B
b + ( c
>
N
- c
>
B
A
- 1
B
A
N
) x
N
x
B
= A
- 1
B
b - A
- 1
B
A
N
x
N
.
Let x
*
i
be a variable that violates the integrality constraint, and let
the following equation denote the corresponding row in the
dictionary:
x
i
= a
i, 0
-
X
j ∈N
a
i,j
x
j
.
Lizhi Wang ([email protected]) IE 534 Linear Programming October 31, 2012 15 / 21

Lizhi Wang ([email protected]) IE 534 Linear Programming October 31, 2012 16 / 21

Lizhi Wang ([email protected]) IE 534 Linear Programming October 31, 2012 17 / 21