1
1
0
0
0
0
0

1
0
0
0
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 mid1990s, Gomory cuts were shown to be very effective in
combination with branchandbound.
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 cutSuppose an LP relaxation is solved, the optimal solutionx*doesnot satisfy the integrality constraint, and the optimal simplexdictionary is the following:ζ=c>BA1Bb+ (c>Nc>BA1BAN)xNxB=A1BbA1BANxN.Letx*ibe a variable that violates the integrality constraint, and letthe following equation denote the corresponding row in thedictionary:xi=ai,0Xj∈Nai,jxj.Lizhi Wang ([email protected])
IE 534 Linear Programming
October 31, 2012
15 / 21
You've reached the end of your free preview.
Want to read all 23 pages?
 Spring '12
 lizhiwang
 Linear Programming, Vector Space, Optimization, Linear Programming Relaxation, Constraint, Lizhi Wang