Reconnect ‘04
A Couple of General Classes of Cutting Planes
Cynthia Phillips
Sandia National Laboratories
Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company,
for the United States Department of Energy under contract DE-AC04-94AL85000.
Knapsack Cover (KC) Inequalities
A
u(A) 
C
u
A
 D(C)
e A
residual D(A)  D  u(A)
uA (e)  min( ue ,D(A))
KC :
u
e C A
Slide 2
A
(e)x e  D(A)
Moving Away from Graphs
aT x  b
The cuts apply to more general
For this discussion, assume a  0 and x  0,1n


Let I be a set of variable indices such that
iI

residual r  b   ai
 iI
a
i  min( ai ,r)
cover cut :
 ax
j
j N I
Slide 3
j
r
ai  b
Cover Cuts
We can remove the assumption that a  0
Consider a general inequality a x   a x   b, a ,a  0
Set y   1 x   0

a  x   a 1 y   b



 
a x  a y   b  a



Apply a regular cover cut to a , y  and substitute 1 x  for y 


Slide 4

Review: Linear Programming Basis
What does a corner look like algebraically?
Ax=b
Partition A matrix into three parts
xB
B
L
U
xL
where B is nonsingular (invertible, square).
Reorder x: (xB, xL, xU)
We have BxB + LxL + UxU = b
Slide 5
xU
A Basic Solution
We have BxB + LxL + UxU = b
Set all members of xL to their lower bound.
Set all members of xU to their upper bound.
Let b b  LxL UxU (this is a constant because bounds and u are)
Thus we have Bx  b
Set x B  B b
1
B



So we
can express each basic variable in the current optimal LP solution
x* as a function of the nonbasic variables.
Slide 6
Gomory Cuts
Assume we have a pure integer program (not necessarily binary)
Express each basic variable in the current optimal LP solution x* as a
function of the nonbasic variables (tableau):
xi 
 g x
j
j 
j
x j x L
  g u
j
x j xU
*

x

x

j
j
i
fr(gj) is the fractional part of gj
Split
 gj into integral and fractional pieces:
xi 
g x
j
j
x j x L
 g u
j
x j xU
Slide 7

j
  fr g x
j
j
x j x L
j  x j 
 fr g u
j
x j xU

j

*

x

x

j
j
i
Gomory Cuts
g x
xi 
j
j

j
x j x L
 g u
j  x j 
g x

x j xU
j
j
j
x j x L

j
j
 fr g u
j
x j xU
 g u
j
j 
x j x L
j
j
x j xU
Slide 8

g x
j
x j x L
j 
j 
j

*

x

x

j
j
i
 x j  uj
g u
j
x j xU
j
*

x

x

j
j
i
  fr g u
 0 because
xi 

 x j 
j
x j xU
 fr g x

j
x j x L
j
xi 
  fr g x
*

x

x
j
j
i

Gomory Cuts
xi 
g x
j
j 
x j x L

j 
g u
j
x j xU
*

x

x
j
j
i
In a feasible solution xi is integral (pure integer program), so the whole
left side is integral. Thus the right side must be as well:
xi 
g x
j
x j x L
j 
j 
g u
j
x j xU
This is (one type of) Gomory Cut.
Slide 9
*

x

x

j
j
i
Global Validity
Cuts like the TSP subtour elimination cuts are globally valid (apply to all
subproblems).
• Can be shared
Recall the key step for Gomory cuts:
xi 
g x
j
j

j
x j x L
 fr g x
j
x j x L
 g u
j
x j xU
j

j
Slide 10

 x j 
  fr g u
 0 because

j
j
j
x j xU
j
 x j  uj
 x j  x *i
Global Validity
 x j  u j for the j and uj in effect at the subproblem
where the Gomory cut was generated.
• Gomory cuts are globally valid for binary variables
–Need fixed at 1 to be
fixed at upper and fixed at 0 to be at lower
• Gomory cuts are not generally valid for general integer variables
We require
Slide 11
j