R. Weismantel
Mixed integer cutting plane theory:
a geometric view
Otto-von-Guericke-Universität Magdeburg
Robert Weismantel
1 / 15
Mixed Integer Programming and cutting planes
A mixed integer linear program
max
st.
cT x
Ax = b,
xi ∈ Z+ , i ∈ I , xi ∈ R+ , i ∈
/ I.
A notation
Often the variables are (x, y ). Then
x ∈ Zn and y ∈ Rd .
For a closed convex set C , let
CI := {(x, y ) ∈ C ∩ (Zn × Rd )}.
The simplex tableau representation
What is a cutting plane?
max
st.
c̄NT xN
ĀN xN ≤ b̄,
xi ∈ R+ , xi ∈ Z+ , i ∈ I .
For S ⊆ Zn × Rd a linear inequality
aT (x, y ) ≤ b is valid, i.e.,
S ⊆ {(x, y ) | aT (x, y ) ≤ b}
and,
Ni+
Ni−
= {j ∈ N | aij ≥ 0},
= {j ∈ N | aij < 0},
f (g ) = (gj − bgj c) for g ∈ Rd .
Robert Weismantel
and cuts off the LP optimum
(x ∗ , y ∗ ), i.e, aT (x ∗ , y ∗ ) > b.
2 / 15
Mixed Integer Programming and cutting planes
A mixed integer linear program
max
st.
cT x
Ax = b,
xi ∈ Z+ , i ∈ I , xi ∈ R+ , i ∈
/ I.
A notation
Often the variables are (x, y ). Then
x ∈ Zn and y ∈ Rd .
For a closed convex set C , let
CI := {(x, y ) ∈ C ∩ (Zn × Rd )}.
The simplex tableau representation
What is a cutting plane?
max
st.
c̄NT xN
ĀN xN ≤ b̄,
xi ∈ R+ , xi ∈ Z+ , i ∈ I .
For S ⊆ Zn × Rd a linear inequality
aT (x, y ) ≤ b is valid, i.e.,
S ⊆ {(x, y ) | aT (x, y ) ≤ b}
and,
Ni+
Ni−
= {j ∈ N | aij ≥ 0},
= {j ∈ N | aij < 0},
f (g ) = (gj − bgj c) for g ∈ Rd .
Robert Weismantel
and cuts off the LP optimum
(x ∗ , y ∗ ), i.e, aT (x ∗ , y ∗ ) > b.
2 / 15
Mixed Integer Programming and cutting planes
A mixed integer linear program
max
st.
cT x
Ax = b,
xi ∈ Z+ , i ∈ I , xi ∈ R+ , i ∈
/ I.
A notation
Often the variables are (x, y ). Then
x ∈ Zn and y ∈ Rd .
For a closed convex set C , let
CI := {(x, y ) ∈ C ∩ (Zn × Rd )}.
The simplex tableau representation
What is a cutting plane?
max
st.
c̄NT xN
ĀN xN ≤ b̄,
xi ∈ R+ , xi ∈ Z+ , i ∈ I .
For S ⊆ Zn × Rd a linear inequality
aT (x, y ) ≤ b is valid, i.e.,
S ⊆ {(x, y ) | aT (x, y ) ≤ b}
and,
Ni+
Ni−
= {j ∈ N | aij ≥ 0},
= {j ∈ N | aij < 0},
f (g ) = (gj − bgj c) for g ∈ Rd .
Robert Weismantel
and cuts off the LP optimum
(x ∗ , y ∗ ), i.e, aT (x ∗ , y ∗ ) > b.
2 / 15
Mixed Integer Programming and cutting planes
A mixed integer linear program
max
st.
cT x
Ax = b,
xi ∈ Z+ , i ∈ I , xi ∈ R+ , i ∈
/ I.
A notation
Often the variables are (x, y ). Then
x ∈ Zn and y ∈ Rd .
For a closed convex set C , let
CI := {(x, y ) ∈ C ∩ (Zn × Rd )}.
The simplex tableau representation
What is a cutting plane?
max
st.
c̄NT xN
ĀN xN ≤ b̄,
xi ∈ R+ , xi ∈ Z+ , i ∈ I .
For S ⊆ Zn × Rd a linear inequality
aT (x, y ) ≤ b is valid, i.e.,
S ⊆ {(x, y ) | aT (x, y ) ≤ b}
and,
Ni+
Ni−
= {j ∈ N | aij ≥ 0},
= {j ∈ N | aij < 0},
f (g ) = (gj − bgj c) for g ∈ Rd .
Robert Weismantel
and cuts off the LP optimum
(x ∗ , y ∗ ), i.e, aT (x ∗ , y ∗ ) > b.
2 / 15
The fundamental questions in mixed integer cutting plane theory
The point of departure
`
´
For a polyhedron P ⊆ Rn+d , the set S = conv(P ∩ Z n × Rd is a polyhedron. Hence,
c ∗ = max c T x + g T y
“
”
(x, y ) ∈ P ∩ Z n × Rd
=
max c T x + g T y
(x, y ) ∈ S.
(1) Which geometric tools are needed in order to understand S?
(2) Which algebraic tools are needed to generate cutting planes?
(3) Can (1) and (2) be turnt into a finite algorithm that computes c ∗ ?
The pure integer setting
Rounding
of hyperplanes:
P
Pi ai xi ≤ a0 turns into
i bai cxi ≤ ba0 c.
Cutting plane proofs [Chv 73].
Finiteness [Gomory 58].
Robert Weismantel
The mixed integer setting
In the mixed 0-1-case, things are
nice.
In general, nothing extends easily.
Why?
3 / 15
The fundamental questions in mixed integer cutting plane theory
The point of departure
`
´
For a polyhedron P ⊆ Rn+d , the set S = conv(P ∩ Z n × Rd is a polyhedron. Hence,
c ∗ = max c T x + g T y
“
”
(x, y ) ∈ P ∩ Z n × Rd
=
max c T x + g T y
(x, y ) ∈ S.
(1) Which geometric tools are needed in order to understand S?
(2) Which algebraic tools are needed to generate cutting planes?
(3) Can (1) and (2) be turnt into a finite algorithm that computes c ∗ ?
The pure integer setting
Rounding
of hyperplanes:
P
Pi ai xi ≤ a0 turns into
i bai cxi ≤ ba0 c.
Cutting plane proofs [Chv 73].
Finiteness [Gomory 58].
Robert Weismantel
The mixed integer setting
In the mixed 0-1-case, things are
nice.
In general, nothing extends easily.
Why?
3 / 15
The fundamental questions in mixed integer cutting plane theory
The point of departure
`
´
For a polyhedron P ⊆ Rn+d , the set S = conv(P ∩ Z n × Rd is a polyhedron. Hence,
c ∗ = max c T x + g T y
“
”
(x, y ) ∈ P ∩ Z n × Rd
=
max c T x + g T y
(x, y ) ∈ S.
(1) Which geometric tools are needed in order to understand S?
(2) Which algebraic tools are needed to generate cutting planes?
(3) Can (1) and (2) be turnt into a finite algorithm that computes c ∗ ?
The pure integer setting
Rounding
of hyperplanes:
P
Pi ai xi ≤ a0 turns into
i bai cxi ≤ ba0 c.
Cutting plane proofs [Chv 73].
Finiteness [Gomory 58].
Robert Weismantel
The mixed integer setting
In the mixed 0-1-case, things are
nice.
In general, nothing extends easily.
Why?
3 / 15
The dilemma for general mixed integer programs
An intriguing small example [Cook, Kannan, Schrijver 90]
max
111
000
000
111
+y
− x1
+y ≤0
−x2 + y ≤ 0
+ x1
x1 ∈ Z+ ,
111
000
111
000
111
000
111
000
111
000
000
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000
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111
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000
+x2 + y ≤ 2
x2 ∈ Z+ ,y ≥ 0.
111
000
000
111
111
000
000
111
111
000
111
000
with fractional optimum ( 23 , 23 , 23 ).
111
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000
In understanding this example . . ., we need
˘
¯
For (x, y ) ∈ S ⊂ Rn+d , its projection is
projx (S) = {x ∈ Rn | ∃y such that (x, y ) ∈ S} .
The projection-operation preserves polyhedrality and convexity.
Robert Weismantel
4 / 15
The dilemma for general mixed integer programs
An intriguing small example [Cook, Kannan, Schrijver 90]
max
111
000
000
111
+y
− x1
+y ≤0
−x2 + y ≤ 0
+ x1
x1 ∈ Z+ ,
111
000
111
000
111
000
111
000
111
000
000
111
111
000
000
111
111
000
111
000
111
000
000
111
111
000
000
111
111
000
000
111
111
000
111
000
111
000
111
000
111
000
000
111
111
000
000
111
111
000
111
000
111
000
111
000
111
000
000
111
111
000
000
111
111
000
111
000
111
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111
000
111
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000
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000
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111
000
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111
000
000
111
111
000
000
111
111
000
111
000
111
000
111
000
111
000
111
000
+x2 + y ≤ 2
x2 ∈ Z+ ,y ≥ 0.
111
000
000
111
111
000
000
111
111
000
111
000
with fractional optimum ( 23 , 23 , 23 ).
111
000
111
000
111
000
000
111
111
000
111
000
111
000
111
000
111
000
000
111
111
000
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111
111
000
111
000
111
000
000
111
111
000
000
111
111
000
111
000
111
000
111
000
In understanding this example . . ., we need
˘
¯
For (x, y ) ∈ S ⊂ Rn+d , its projection is
projx (S) = {x ∈ Rn | ∃y such that (x, y ) ∈ S} .
The projection-operation preserves polyhedrality and convexity.
Robert Weismantel
4 / 15
The mixed 0 − 1-case and disjunctive programming
The Lift-and-Project Algorithm
Ingredients
2
z ∈ {0, 1} iff z =
z.
Step 1: Select j ∈ N.
Step 2: Generate the nonlinear system Qj ,
For polyhedra
P1 , P2 ⊆ Rn ,
conv(P1 ∪ P2 ) can
be compactly
described. [Balas
79,85].
xj (Ax − b) ≤ 0, (1 − xj )(Ax − b) ≤ 0.
.
Step 3: Linearize Qj by yi := xi xj for i 6= j and xj2 = xj .
Step 4: Project the linearized system onto x-space.
Theorem [Balas, Ceria, Cornuejols 93]
The system
˘
S = x ∈ {0, 1}n × Rd+ :
¯
Ax ≤ b .
Robert Weismantel
projx (Qj ) = conv {{Ax ≤ b, xj = 0} ∪ {Ax ≤ b, xj = 1}} .
`
´
conv(S) = . . . (projx (Q1 ))2 . . . n .
Similar approaches: [Lovasz, Schrijver 91] [Sherali, Adams 90]
5 / 15
The mixed 0 − 1-case and disjunctive programming
The Lift-and-Project Algorithm
Ingredients
2
z ∈ {0, 1} iff z =
z.
Step 1: Select j ∈ N.
Step 2: Generate the nonlinear system Qj ,
For polyhedra
P1 , P2 ⊆ Rn ,
conv(P1 ∪ P2 ) can
be compactly
described. [Balas
79,85].
xj (Ax − b) ≤ 0, (1 − xj )(Ax − b) ≤ 0.
.
Step 3: Linearize Qj by yi := xi xj for i 6= j and xj2 = xj .
Step 4: Project the linearized system onto x-space.
Theorem [Balas, Ceria, Cornuejols 93]
The system
˘
S = x ∈ {0, 1}n × Rd+ :
¯
Ax ≤ b .
Robert Weismantel
projx (Qj ) = conv {{Ax ≤ b, xj = 0} ∪ {Ax ≤ b, xj = 1}} .
`
´
conv(S) = . . . (projx (Q1 ))2 . . . n .
Similar approaches: [Lovasz, Schrijver 91] [Sherali, Adams 90]
5 / 15
The mixed 0 − 1-case and disjunctive programming
The Lift-and-Project Algorithm
Ingredients
2
z ∈ {0, 1} iff z =
z.
Step 1: Select j ∈ N.
Step 2: Generate the nonlinear system Qj ,
For polyhedra
P1 , P2 ⊆ Rn ,
conv(P1 ∪ P2 ) can
be compactly
described. [Balas
79,85].
xj (Ax − b) ≤ 0, (1 − xj )(Ax − b) ≤ 0.
.
Step 3: Linearize Qj by yi := xi xj for i 6= j and xj2 = xj .
Step 4: Project the linearized system onto x-space.
Theorem [Balas, Ceria, Cornuejols 93]
The system
˘
S = x ∈ {0, 1}n × Rd+ :
¯
Ax ≤ b .
Robert Weismantel
projx (Qj ) = conv {{Ax ≤ b, xj = 0} ∪ {Ax ≤ b, xj = 1}} .
`
´
conv(S) = . . . (projx (Q1 ))2 . . . n .
Similar approaches: [Lovasz, Schrijver 91] [Sherali, Adams 90]
5 / 15
The mixed 0 − 1-case and disjunctive programming
The Lift-and-Project Algorithm
Ingredients
2
z ∈ {0, 1} iff z =
z.
Step 1: Select j ∈ N.
Step 2: Generate the nonlinear system Qj ,
For polyhedra
P1 , P2 ⊆ Rn ,
conv(P1 ∪ P2 ) can
be compactly
described. [Balas
79,85].
xj (Ax − b) ≤ 0, (1 − xj )(Ax − b) ≤ 0.
.
Step 3: Linearize Qj by yi := xi xj for i 6= j and xj2 = xj .
Step 4: Project the linearized system onto x-space.
Theorem [Balas, Ceria, Cornuejols 93]
The system
˘
S = x ∈ {0, 1}n × Rd+ :
¯
Ax ≤ b .
Robert Weismantel
projx (Qj ) = conv {{Ax ≤ b, xj = 0} ∪ {Ax ≤ b, xj = 1}} .
`
´
conv(S) = . . . (projx (Q1 ))2 . . . n .
Similar approaches: [Lovasz, Schrijver 91] [Sherali, Adams 90]
5 / 15
Mixed integer rounding: a basic one-row-principle [Nemhauser, Wolsey 88]
The basic model
X = {(x, y ) ∈ Z × R+ |x + y ≥ b}
The only missing inequality is
x+
1
y ≥ dbe
1 − f (b)
Mixed integer rounding can be applied to general
models by aggregating variables,
X
z :=
gi xi .
i∈S
Lemma. The Gomory-fractional cut is obtained from mixed integer rounding.

ff
X
X f (b i )aij
f (b i )(1 − f (aij ))
min f (aij ),
xj +
aij xj −
xj ≥ f (b i ).
1
−
f
(b
)
1 − f (b i )
i
+
−
j∈N∩I
X
j∈Ni \I
Robert Weismantel
j∈Ni \I
6 / 15
Mixed integer rounding: a basic one-row-principle [Nemhauser, Wolsey 88]
The basic model
X = {(x, y ) ∈ Z × R+ |x + y ≥ b}
y
The only missing inequality is
x+
1
y ≥ dbe
1 − f (b)
Mixed integer rounding can be applied to general
models by aggregating variables,
X
z :=
gi xi .
x
i∈S
Lemma. The Gomory-fractional cut is obtained from mixed integer rounding.

ff
X
X f (b i )aij
f (b i )(1 − f (aij ))
min f (aij ),
xj +
aij xj −
xj ≥ f (b i ).
1
−
f
(b
)
1 − f (b i )
i
+
−
j∈N∩I
X
j∈Ni \I
Robert Weismantel
j∈Ni \I
6 / 15
Mixed integer rounding: a basic one-row-principle [Nemhauser, Wolsey 88]
The basic model
X = {(x, y ) ∈ Z × R+ |x + y ≥ b}
y
The only missing inequality is
x+
1
y ≥ dbe
1 − f (b)
Mixed integer rounding can be applied to general
models by aggregating variables,
X
z :=
gi xi .
x
i∈S
Lemma. The Gomory-fractional cut is obtained from mixed integer rounding.

ff
X
X f (b i )aij
f (b i )(1 − f (aij ))
min f (aij ),
xj +
aij xj −
xj ≥ f (b i ).
1
−
f
(b
)
1 − f (b i )
i
+
−
j∈N∩I
X
j∈Ni \I
Robert Weismantel
j∈Ni \I
6 / 15
Mixed integer rounding: a basic one-row-principle [Nemhauser, Wolsey 88]
The basic model
X = {(x, y ) ∈ Z × R+ |x + y ≥ b}
y
The only missing inequality is
x+
1
y ≥ dbe
1 − f (b)
Mixed integer rounding can be applied to general
models by aggregating variables,
X
z :=
gi xi .
x
i∈S
Lemma. The Gomory-fractional cut is obtained from mixed integer rounding.

ff
X
X f (b i )aij
f (b i )(1 − f (aij ))
min f (aij ),
xj +
aij xj −
xj ≥ f (b i ).
1
−
f
(b
)
1 − f (b i )
i
+
−
j∈N∩I
X
j∈Ni \I
Robert Weismantel
j∈Ni \I
6 / 15
Cuts from two or more rows of a simplex tableau
A basic model for two and more row -relaxations:
(
f + CI =
(x, s) | x = f +
n
X
)
rj sj , x ∈ Zd , s ∈ Qn+
.
j=1
The structure of valid inequalities of f + CI [Andersen, Louveaux, W, Wolsey 06]
A non trivial inequality is of the kind
P
j∈N αj sj ≥ 1 where αj ≥ 0.
The coefficients αj are the recipocal of
the distance from f along r j to the
boundary
body
`˘ of the projected facet ¯´
projx (x, s) ∈ f + C , αT s = 1 .
Robert Weismantel
7 / 15
Cuts from two or more rows of a simplex tableau
A basic model for two and more row -relaxations:
(
f + CI =
(x, s) | x = f +
n
X
)
rj sj , x ∈ Zd , s ∈ Qn+
.
j=1
The structure of valid inequalities of f + CI [Andersen, Louveaux, W, Wolsey 06]
A non trivial inequality is of the kind
P
j∈N αj sj ≥ 1 where αj ≥ 0.
The coefficients αj are the recipocal of
the distance from f along r j to the
boundary
body
`˘ of the projected facet ¯´
projx (x, s) ∈ f + C , αT s = 1 .
Robert Weismantel
7 / 15
Cuts from two or more rows of a simplex tableau
A basic model for two and more row -relaxations:
(
f + CI =
(x, s) | x = f +
n
X
)
j
d
r sj , x ∈ Z , s ∈
Qn+
.
j=1
The structure of valid inequalities of f + CI [Andersen, Louveaux, W, Wolsey 06]
r3
x
2
A non trivial inequality is of the kind
P
j∈N αj sj ≥ 1 where αj ≥ 0.
r 2
r
1
f
The coefficients αj are the recipocal of
the distance from f along r j to the
boundary
body
`˘ of the projected facet ¯´
projx (x, s) ∈ f + C , αT s = 1 .
Robert Weismantel
x
1
r 4
r5
7 / 15
The special case of two rows is geometrically tractable
Theorem [Andersen, Louveaux, W, Wolsey 06]
r
The projected
facet body
contains no
interior integer
points.Either 3
or 4 rays
(integer points)
determine its
vertices
(boundary).
3
x
2
r 2
r
1
f
x
1
r 4
r5
r3
x
2
r 2
r
1
f
x
1
r 4
r5
Classification of the facets by lattice point free polyhedra
11
00
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1
0
00
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000
11
11111111111
0000000
0000
1
00
11
0000
1111
0000
1111
0000
1111
0000
1111
1
0
0000
1111
0000
1111
0000
1111
0000
1111
Robert Weismantel
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
8 / 15
The special case of two rows is geometrically tractable
Theorem [Andersen, Louveaux, W, Wolsey 06]
r
The projected
facet body
contains no
interior integer
points.Either 3
or 4 rays
(integer points)
determine its
vertices
(boundary).
3
x
2
r 2
r
1
f
x
1
r 4
r5
r3
x
2
r 2
r
1
f
x
1
r 4
r5
Classification of the facets by lattice point free polyhedra
11
00
00
11
00
11
00
11
00
11
00
11
00
11
00
11
00
11
1
0
00
11
000
11
11111111111
0000000
0000
1
00
11
0000
1111
0000
1111
0000
1111
0000
1111
1
0
0000
1111
0000
1111
0000
1111
0000
1111
Robert Weismantel
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
8 / 15
Split cuts [Cook, Kannan, Schrijver 1990]
The algebra
Based on a disjunction
π T x ≤ π0
or
π T x ≥ π0 + 1
is valid for x ∈ Zn when π, π0 are integer.
The geometry
Robert Weismantel
9 / 15
Split cuts [Cook, Kannan, Schrijver 1990]
The algebra
Based on a disjunction
π T x ≤ π0
or
π T x ≥ π0 + 1
is valid for x ∈ Zn when π, π0 are integer.
The geometry
Robert Weismantel
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
9 / 15
Split cuts [Cook, Kannan, Schrijver 1990]
The algebra
Based on a disjunction
π T x ≤ π0
or
π T x ≥ π0 + 1
is valid for x ∈ Zn when π, π0 are integer.
The geometry
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
Robert Weismantel
SP
LI
T
1
0
9 / 15
Split cuts [Cook, Kannan, Schrijver 1990]
The algebra
Based on a disjunction
π T x ≤ π0
or
π T x ≥ π0 + 1
is valid for x ∈ Zn when π, π0 are integer.
The geometry
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
00000000000
11111111111
1111111
0000000
00000000000
11111111111
0000000
1111111
0000000
1111111
00000000000
11111111111
0000000
1111111
00000000000
11111111111
1 1111111
0
1
0
1
0
1
0
1
0
0000000
00000000000
11111111111
0000000
1111111
00000000000
11111111111
0000000
1111111
00000000000
11111111111
00000000000
11111111111
1
0
1
0
1
0
1
0
1
0
00000000000
11111111111
00000000000
11111111111
00000000000
11111111111
1
0
1
0
Robert Weismantel
1
0
1
0
1
0
1
0
9 / 15
Split cuts [Cook, Kannan, Schrijver 1990]
The algebra
Based on a disjunction
π T x ≤ π0
or
π T x ≥ π0 + 1
is valid for x ∈ Zn when π, π0 are integer.
The geometry
1
0
1
0
1
0
1
0
1
0
Split cut
Robert Weismantel
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
9 / 15
What is a split? two points of view
Lattice-point-free polyhedron
A polyhedron P is lattice-point-free when
there is no integer point in its interior.
Splits and lpf-polyhedra
A split cut is generated from a special
lattice point free polyhedron,
L = conv(v , w ) + span(z 1 , . . . , z n−1 ,
with z 1 , . . . , z n−1 ∈ Qn being linearly
independent.
Results on maximal lpf polyhedra
see survey of [Lovasz 87]
Splits and disjunctions
A split comes from a two-term
disjunction πx ≤ π0 , πx ≥ π0 + 1,
where π0 ∈ Z.
Robert Weismantel
10 / 15
What is a split? two points of view
Lattice-point-free polyhedron
A polyhedron P is lattice-point-free when
there is no integer point in its interior.
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
Splits and lpf-polyhedra
A split cut is generated from a special
lattice point free polyhedron,
L = conv(v , w ) + span(z 1 , . . . , z n−1 ,
with z 1 , . . . , z n−1 ∈ Qn being linearly
independent.
Results on maximal lpf polyhedra
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
see survey of [Lovasz 87]
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
Splits and disjunctions
A basic split set in R2
Robert Weismantel
A split comes from a two-term
disjunction πx ≤ π0 , πx ≥ π0 + 1,
where π0 ∈ Z.
10 / 15
What is a split? two points of view
Lattice-point-free polyhedron
A polyhedron P is lattice-point-free when
there is no integer point in its interior.
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
Splits and lpf-polyhedra
A split cut is generated from a special
lattice point free polyhedron,
L = conv(v , w ) + span(z 1 , . . . , z n−1 ,
with z 1 , . . . , z n−1 ∈ Qn being linearly
independent.
Results on maximal lpf polyhedra
r
1
0
1
0
000
111
000
111
000
v111
000
111
1
0
000
111
0
1
000
111
0
1
111
000
000
111
000
111
000
111
1
0
1
0
1
0
1
0
see survey of [Lovasz 87]
1
0
0
1
0w
1
1
0
0
1
1
0
0
1
1
0
0
1
Splits and disjunctions
conv{v , w } + span{r }
Robert Weismantel
A split comes from a two-term
disjunction πx ≤ π0 , πx ≥ π0 + 1,
where π0 ∈ Z.
10 / 15
What is a split? two points of view
Lattice-point-free polyhedron
A polyhedron P is lattice-point-free when
there is no integer point in its interior.
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
A basic split set in R3
Robert Weismantel
Splits and lpf-polyhedra
A split cut is generated from a special
lattice point free polyhedron,
L = conv(v , w ) + span(z 1 , . . . , z n−1 ,
with z 1 , . . . , z n−1 ∈ Qn being linearly
independent.
Results on maximal lpf polyhedra
see survey of [Lovasz 87]
Splits and disjunctions
A split comes from a two-term
disjunction πx ≤ π0 , πx ≥ π0 + 1,
where π0 ∈ Z.
10 / 15
What is a split? two points of view
Lattice-point-free polyhedron
A polyhedron P is lattice-point-free when
there is no integer point in its interior.
1
0
0
1
1
0
0
1
0
1
0
s 1
1
0
0
1
0
1
1
0
0
1
0
1
0
1
0
1
r
0
1
1
0000
1111
0
0w
1
0000
1111
0
0
0000 1
1111
01
1
0
1
0000
1111
0
1
11
00
0
1
11
00
v1
0
0
1
0
1
conv{v , w } + span{r , s}
Robert Weismantel
Splits and lpf-polyhedra
A split cut is generated from a special
lattice point free polyhedron,
L = conv(v , w ) + span(z 1 , . . . , z n−1 ,
with z 1 , . . . , z n−1 ∈ Qn being linearly
independent.
Results on maximal lpf polyhedra
see survey of [Lovasz 87]
Splits and disjunctions
A split comes from a two-term
disjunction πx ≤ π0 , πx ≥ π0 + 1,
where π0 ∈ Z.
10 / 15
What is a split? two points of view
Lattice-point-free polyhedron
A polyhedron P is lattice-point-free when
there is no integer point in its interior.
1
0
0
1
1
0
0
1
0
1
0
s 1
1
0
0
1
0
1
1
0
0
1
0
1
0
1
0
1
r
0
1
1
0000
1111
0
0w
1
0000
1111
0
0
0000 1
1111
01
1
0
1
0000
1111
0
1
11
00
0
1
11
00
v1
0
0
1
0
1
conv{v , w } + span{r , s}
Robert Weismantel
Splits and lpf-polyhedra
A split cut is generated from a special
lattice point free polyhedron,
L = conv(v , w ) + span(z 1 , . . . , z n−1 ,
with z 1 , . . . , z n−1 ∈ Qn being linearly
independent.
Results on maximal lpf polyhedra
see survey of [Lovasz 87]
Splits and disjunctions
A split comes from a two-term
disjunction πx ≤ π0 , πx ≥ π0 + 1,
where π0 ∈ Z.
10 / 15
What is a split? two points of view
Lattice-point-free polyhedron
A polyhedron P is lattice-point-free when
there is no integer point in its interior.
1
0
0
1
1
0
0
1
0
1
0
s 1
1
0
0
1
0
1
1
0
0
1
0
1
0
1
0
1
r
0
1
1
0000
1111
0
0w
1
0000
1111
0
0
0000 1
1111
01
1
0
1
0000
1111
0
1
11
00
0
1
11
00
v1
0
0
1
0
1
conv{v , w } + span{r , s}
Robert Weismantel
Splits and lpf-polyhedra
A split cut is generated from a special
lattice point free polyhedron,
L = conv(v , w ) + span(z 1 , . . . , z n−1 ,
with z 1 , . . . , z n−1 ∈ Qn being linearly
independent.
Results on maximal lpf polyhedra
see survey of [Lovasz 87]
Splits and disjunctions
A split comes from a two-term
disjunction πx ≤ π0 , πx ≥ π0 + 1,
where π0 ∈ Z.
10 / 15
A generalization of splits based on multi-term disjunctions
Definition
Let d 1 , . . . , d k ∈ Zn and
δ1 , . . . , δk ∈ Z. The family D(k, d, δ)
is a k-disjunction if for all x ∈ Zn
there exists i such that d i x ≤ δi .
Let P ⊂ Rn be a polyhedron and
c T x ≤ γ be valid for PI . Then
c T x ≤ γ is a k-disjunctive cut for PI ,
if there exists a k-disjunction
D(k, d, δ) with
x ∈ P : c T x > γ =⇒ d i x > δi , ∀i.
Proposition
n+d
Let P ⊆ R
be a polyhedron. Every
valid inequality c T x ≤ γ for PI is a
2n -disjunctive cut for some k.
Robert Weismantel
1
0
1
0
1
0
1
0
1
0
Split cut
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
Theorem [Jörg 07]
There is a finite cutting plane algorithm
for any bounded mixed integer program
based on k-disjunctions.
11 / 15
A generalization of splits based on multi-term disjunctions
Definition
Let d 1 , . . . , d k ∈ Zn and
δ1 , . . . , δk ∈ Z. The family D(k, d, δ)
is a k-disjunction if for all x ∈ Zn
there exists i such that d i x ≤ δi .
Let P ⊂ Rn be a polyhedron and
c T x ≤ γ be valid for PI . Then
c T x ≤ γ is a k-disjunctive cut for PI ,
if there exists a k-disjunction
D(k, d, δ) with
x ∈ P : c T x > γ =⇒ d i x > δi , ∀i.
Proposition
n+d
Let P ⊆ R
be a polyhedron. Every
valid inequality c T x ≤ γ for PI is a
2n -disjunctive cut for some k.
Robert Weismantel
1
0
1
0
1
0
1
0
1
0
Split cut
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
Theorem [Jörg 07]
There is a finite cutting plane algorithm
for any bounded mixed integer program
based on k-disjunctions.
11 / 15
A generalization of splits based on multi-term disjunctions
Definition
Let d 1 , . . . , d k ∈ Zn and
δ1 , . . . , δk ∈ Z. The family D(k, d, δ)
is a k-disjunction if for all x ∈ Zn
there exists i such that d i x ≤ δi .
Let P ⊂ Rn be a polyhedron and
c T x ≤ γ be valid for PI . Then
c T x ≤ γ is a k-disjunctive cut for PI ,
if there exists a k-disjunction
D(k, d, δ) with
x ∈ P : c T x > γ =⇒ d i x > δi , ∀i.
Proposition
n+d
Let P ⊆ R
be a polyhedron. Every
valid inequality c T x ≤ γ for PI is a
2n -disjunctive cut for some k.
Robert Weismantel
1
0
1
0
1
0
1
0
1
0
Split cut
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
Theorem [Jörg 07]
There is a finite cutting plane algorithm
for any bounded mixed integer program
based on k-disjunctions.
11 / 15
A generalization of splits based on lpf-polyhedra
Observation
P = conv{v 1 , . . . , v p } + cone{w 1 , . . . , w s } + span{w s+1 , . . . , w q } ⊆ Rn is lpf if and
only if P 0 = conv{v 1 , . . . , v p } + span{w 1 , . . . , w q } ⊆ Rn is lpf.
A lpf polyhedron conv{v 1 , . . . , v p } + span{w 1 , . . . , w q } is called split body, the
number n − q the split-dimension.
Examples
A lpf triangle in R2 has split dimension
two.
Robert Weismantel
A lpf triangle lifted to R3 ,
conv{v , w , x} + span{r } has split
dimension two.
12 / 15
A generalization of splits based on lpf-polyhedra
Observation
P = conv{v 1 , . . . , v p } + cone{w 1 , . . . , w s } + span{w s+1 , . . . , w q } ⊆ Rn is lpf if and
only if P 0 = conv{v 1 , . . . , v p } + span{w 1 , . . . , w q } ⊆ Rn is lpf.
A lpf polyhedron conv{v 1 , . . . , v p } + span{w 1 , . . . , w q } is called split body, the
number n − q the split-dimension.
Examples
A lpf triangle in R2 has split dimension
two.
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
Robert Weismantel
A lpf triangle lifted to R3 ,
conv{v , w , x} + span{r } has split
dimension two.
12 / 15
A generalization of splits based on lpf-polyhedra
Observation
P = conv{v 1 , . . . , v p } + cone{w 1 , . . . , w s } + span{w s+1 , . . . , w q } ⊆ Rn is lpf if and
only if P 0 = conv{v 1 , . . . , v p } + span{w 1 , . . . , w q } ⊆ Rn is lpf.
A lpf polyhedron conv{v 1 , . . . , v p } + span{w 1 , . . . , w q } is called split body, the
number n − q the split-dimension.
Examples
A lpf triangle in R2 has split dimension
two.
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
Robert Weismantel
A lpf triangle lifted to R3 ,
conv{v , w , x} + span{r } has split
dimension two.
12 / 15
A generalization of splits based on lpf-polyhedra
Observation
P = conv{v 1 , . . . , v p } + cone{w 1 , . . . , w s } + span{w s+1 , . . . , w q } ⊆ Rn is lpf if and
only if P 0 = conv{v 1 , . . . , v p } + span{w 1 , . . . , w q } ⊆ Rn is lpf.
A lpf polyhedron conv{v 1 , . . . , v p } + span{w 1 , . . . , w q } is called split body, the
number n − q the split-dimension.
Examples
A lpf triangle in R2 has split dimension
two.
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
A lpf triangle lifted to R3 ,
conv{v , w , x} + span{r } has split
dimension two.
111
000
000
111
111
000
000
111
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
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0
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0
1
0
1
0
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0
1
0
111
000
111
000
111
000
111
000
111
000
111
000
v
111
000
111
000
111
000
000
111
000
w111
000
111
111
000
111
000
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000
111
000
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000
111
111
000
000
111
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000
000
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x
111
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111
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000
111
000
111
000
r
111
000
111
000
111
000
111
000
1
0
0
1
1
0
0
1
Robert Weismantel
1
0
0
1
1
0
0
1
111
000
000
111
111
000
000
111
111
000
000
111
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111
12 / 15
Split bodies give rise to cuts for convex mixed integer programs.
An operation
For a split body L ⊆ Rn and a closed convex set C , let
R(L) := cl conv({(x, y ) ∈ C : x ∈
/ rint(L)}).
Then, conv(CI ) ⊆ R(L) ⊆ C .
Lemma [Andersen, Louveaux, W 07]
Let L ⊆ Rn be a split body.
For a closed convex set C ,
R(L) 6= C iff there exists an
extreme point (x, y ) of C such that
x is in the interior of L.
If C is a polyhedron, then R(L) is a
polyhedron.
Robert Weismantel
13 / 15
Split bodies give rise to cuts for convex mixed integer programs.
An operation
For a split body L ⊆ Rn and a closed convex set C , let
R(L) := cl conv({(x, y ) ∈ C : x ∈
/ rint(L)}).
Then, conv(CI ) ⊆ R(L) ⊆ C .
Lemma [Andersen, Louveaux, W 07]
Let L ⊆ Rn be a split body.
For a closed convex set C ,
R(L) 6= C iff there exists an
extreme point (x, y ) of C such that
x is in the interior of L.
If C is a polyhedron, then R(L) is a
polyhedron.
Robert Weismantel
13 / 15
Split bodies give rise to cuts for convex mixed integer programs.
An operation
For a split body L ⊆ Rn and a closed convex set C , let
R(L) := cl conv({(x, y ) ∈ C : x ∈
/ rint(L)}).
Then, conv(CI ) ⊆ R(L) ⊆ C .
Lemma [Andersen, Louveaux, W 07]
Let L ⊆ Rn be a split body.
For a closed convex set C ,
R(L) 6= C iff there exists an
extreme point (x, y ) of C such that
x is in the interior of L.
If C is a polyhedron, then R(L) is a
polyhedron.
Robert Weismantel
13 / 15
Split bodies give rise to cuts for convex mixed integer programs.
An operation
For a split body L ⊆ Rn and a closed convex set C , let
R(L) := cl conv({(x, y ) ∈ C : x ∈
/ rint(L)}).
Then, conv(CI ) ⊆ R(L) ⊆ C .
Lemma [Andersen, Louveaux, W 07]
Let L ⊆ Rn be a split body.
For a closed convex set C ,
R(L) 6= C iff there exists an
extreme point (x, y ) of C such that
x is in the interior of L.
If C is a polyhedron, then R(L) is a
polyhedron.
Robert Weismantel
13 / 15
Split bodies give rise to cuts for convex mixed integer programs.
An operation
For a split body L ⊆ Rn and a closed convex set C , let
R(L) := cl conv({(x, y ) ∈ C : x ∈
/ rint(L)}).
Then, conv(CI ) ⊆ R(L) ⊆ C .
Lemma [Andersen, Louveaux, W 07]
Let L ⊆ Rn be a split body.
For a closed convex set C ,
R(L) 6= C iff there exists an
extreme point (x, y ) of C such that
x is in the interior of L.
If C is a polyhedron, then R(L) is a
polyhedron.
Robert Weismantel
13 / 15
From split bodies to cutting plane proofs
The closure of split bodies
For a family F of split bodies, let
\
Cl(F , C ) :=
R(L).
L∈F
0
Letting C (F , C ) = C , define for i ≥ 1,
If Ax ≤ b is full dimensional and lpf,
then y ≤ 0 has split size n w.r.t.
Ax + 1y ≤ b, x ∈ Zn , y ≥ 0.
C i (F , C ) = Cl(F , C i−1 (F , C )).
Theorem [Cook, Kannan, Schrijver 90]
For a polyhedron P and the set F of split bodies
of split dimension one, Cl(F , P) is a polyhedron.
Definition
For an inequality c T x ≤ γ, valid for conv(CI ), a split body proof is a finite family F of
split bodies such that c T x ≤ γ is valid for C k (F , C ) for some k. The split size of the
proof is the largest split dimension of a split body in F .
Robert Weismantel
14 / 15
From split bodies to cutting plane proofs
The closure of split bodies
For a family F of split bodies, let
\
Cl(F , C ) :=
R(L).
L∈F
0
Letting C (F , C ) = C , define for i ≥ 1,
If Ax ≤ b is full dimensional and lpf,
then y ≤ 0 has split size n w.r.t.
Ax + 1y ≤ b, x ∈ Zn , y ≥ 0.
C i (F , C ) = Cl(F , C i−1 (F , C )).
Theorem [Cook, Kannan, Schrijver 90]
For a polyhedron P and the set F of split bodies
of split dimension one, Cl(F , P) is a polyhedron.
Definition
For an inequality c T x ≤ γ, valid for conv(CI ), a split body proof is a finite family F of
split bodies such that c T x ≤ γ is valid for C k (F , C ) for some k. The split size of the
proof is the largest split dimension of a split body in F .
Robert Weismantel
14 / 15
From split bodies to cutting plane proofs
The closure of split bodies
For a family F of split bodies, let
\
Cl(F , C ) :=
R(L).
L∈F
0
Letting C (F , C ) = C , define for i ≥ 1,
If Ax ≤ b is full dimensional and lpf,
then y ≤ 0 has split size n w.r.t.
Ax + 1y ≤ b, x ∈ Zn , y ≥ 0.
C i (F , C ) = Cl(F , C i−1 (F , C )).
Theorem [Cook, Kannan, Schrijver 90]
For a polyhedron P and the set F of split bodies
of split dimension one, Cl(F , P) is a polyhedron.
Definition
For an inequality c T x ≤ γ, valid for conv(CI ), a split body proof is a finite family F of
split bodies such that c T x ≤ γ is valid for C k (F , C ) for some k. The split size of the
proof is the largest split dimension of a split body in F .
Robert Weismantel
14 / 15
From split bodies to cutting plane proofs
The closure of split bodies
For a family F of split bodies, let
\
Cl(F , C ) :=
R(L).
If Ax ≤ b is full dimensional and lpf,
then y ≤ 0 has split size n w.r.t.
Ax + 1y ≤ b, x ∈ Zn , y ≥ 0.
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L∈F
Letting C 0 (F , C ) = C , define for i ≥ 1,
C i (F , C ) = Cl(F , C i−1 (F , C )).
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For a polyhedron P and the set F of split bodies
of split dimension one, Cl(F , P) is a polyhedron.
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Theorem [Cook, Kannan, Schrijver 90]
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Definition
For an inequality c T x ≤ γ, valid for conv(CI ), a split body proof is a finite family F of
split bodies such that c T x ≤ γ is valid for C k (F , C ) for some k. The split size of the
proof is the largest split dimension of a split body in F .
Robert Weismantel
14 / 15
Split bodies: a perspective
Theorem [Andersen, Louveaux, W
07]
T
Let c x ≤ γ be a nt valid inequality
for conv(CI ).
(i) It has a split body proof of split
size split-dim(c, γ).
(ii) There is no split body proof of
split size smaller than
split-dim(c, γ).
Corollary
Any cutting plane algorithm based on
split bodies of split-dimension less
than split-dim(c, γ) cannot solve the
optimization
problem ¯
˘
γ = max c T x, x ∈ CI in finitely
many rounds.
Robert Weismantel
Theorem [Andersen, Louveaux, W 07]
Let F be the optimal face of
max c T x | x ∈ P. F contains no mixed
integer points iff there exists a split body of
split size at most max {1, dim F } containing
F in its interior.
How to find split bodies?
Let (x ∗ , y ∗ ) be an optimal vertex for
max c T x + d T y : Ax + By ≤ b. Let I be the
tight rows at (x ∗ , y ∗ ) with [A, C ]I of rank
n + d. Then,
L∗ = { z ∈ QI |z T AI ∈ Zn ,
z T CI = 0}
is a lattice. Any basis {z1 , . . . , zk } of L∗
satisfies b T zi ∈ Z for all i iff x ∗ ∈ Zn .
15 / 15
Split bodies: a perspective
Theorem [Andersen, Louveaux, W
07]
T
Let c x ≤ γ be a nt valid inequality
for conv(CI ).
(i) It has a split body proof of split
size split-dim(c, γ).
(ii) There is no split body proof of
split size smaller than
split-dim(c, γ).
Corollary
Any cutting plane algorithm based on
split bodies of split-dimension less
than split-dim(c, γ) cannot solve the
optimization
problem ¯
˘
γ = max c T x, x ∈ CI in finitely
many rounds.
Robert Weismantel
Theorem [Andersen, Louveaux, W 07]
Let F be the optimal face of
max c T x | x ∈ P. F contains no mixed
integer points iff there exists a split body of
split size at most max {1, dim F } containing
F in its interior.
How to find split bodies?
Let (x ∗ , y ∗ ) be an optimal vertex for
max c T x + d T y : Ax + By ≤ b. Let I be the
tight rows at (x ∗ , y ∗ ) with [A, C ]I of rank
n + d. Then,
L∗ = { z ∈ QI |z T AI ∈ Zn ,
z T CI = 0}
is a lattice. Any basis {z1 , . . . , zk } of L∗
satisfies b T zi ∈ Z for all i iff x ∗ ∈ Zn .
15 / 15
Split bodies: a perspective
Theorem [Andersen, Louveaux, W
07]
T
Let c x ≤ γ be a nt valid inequality
for conv(CI ).
(i) It has a split body proof of split
size split-dim(c, γ).
(ii) There is no split body proof of
split size smaller than
split-dim(c, γ).
Corollary
Any cutting plane algorithm based on
split bodies of split-dimension less
than split-dim(c, γ) cannot solve the
optimization
problem ¯
˘
γ = max c T x, x ∈ CI in finitely
many rounds.
Robert Weismantel
Theorem [Andersen, Louveaux, W 07]
Let F be the optimal face of
max c T x | x ∈ P. F contains no mixed
integer points iff there exists a split body of
split size at most max {1, dim F } containing
F in its interior.
How to find split bodies?
Let (x ∗ , y ∗ ) be an optimal vertex for
max c T x + d T y : Ax + By ≤ b. Let I be the
tight rows at (x ∗ , y ∗ ) with [A, C ]I of rank
n + d. Then,
L∗ = { z ∈ QI |z T AI ∈ Zn ,
z T CI = 0}
is a lattice. Any basis {z1 , . . . , zk } of L∗
satisfies b T zi ∈ Z for all i iff x ∗ ∈ Zn .
15 / 15
Split bodies: a perspective
Theorem [Andersen, Louveaux, W
07]
T
Let c x ≤ γ be a nt valid inequality
for conv(CI ).
(i) It has a split body proof of split
size split-dim(c, γ).
(ii) There is no split body proof of
split size smaller than
split-dim(c, γ).
Corollary
Any cutting plane algorithm based on
split bodies of split-dimension less
than split-dim(c, γ) cannot solve the
optimization
problem ¯
˘
γ = max c T x, x ∈ CI in finitely
many rounds.
Robert Weismantel
Theorem [Andersen, Louveaux, W 07]
Let F be the optimal face of
max c T x | x ∈ P. F contains no mixed
integer points iff there exists a split body of
split size at most max {1, dim F } containing
F in its interior.
How to find split bodies?
Let (x ∗ , y ∗ ) be an optimal vertex for
max c T x + d T y : Ax + By ≤ b. Let I be the
tight rows at (x ∗ , y ∗ ) with [A, C ]I of rank
n + d. Then,
L∗ = { z ∈ QI |z T AI ∈ Zn ,
z T CI = 0}
is a lattice. Any basis {z1 , . . . , zk } of L∗
satisfies b T zi ∈ Z for all i iff x ∗ ∈ Zn .
15 / 15