Cutting planes from lattice-point-free polyhedra
Quentin Louveaux
Universit´e catholique de Louvain - CORE - Belgium
January 2007
Outline
Split cuts
Lattice-point-free polyhedra
High-dimensional split cuts and the d -dimensional split closure
Some steps to prove that the d -dimensional split closure is a polyhedron Previous work on the 2-dimensional split cuts
Conclusion
Outline
Split cuts
Lattice-point-free polyhedra
High-dimensional split cuts and the d -dimensional split closure
Some steps to prove that the d -dimensional split closure is a polyhedron Previous work on the 2-dimensional split cuts
Outline
Split cuts
Lattice-point-free polyhedra
High-dimensional split cuts and the d -dimensional split closure
Some steps to prove that the d -dimensional split closure is a polyhedron Previous work on the 2-dimensional split cuts
Conclusion
Outline
Split cuts
Lattice-point-free polyhedra
High-dimensional split cuts and the d -dimensional split closure
Some steps to prove that the d -dimensional split closure is a polyhedron
Previous work on the 2-dimensional split cuts Conclusion
Outline
Split cuts
Lattice-point-free polyhedra
High-dimensional split cuts and the d -dimensional split closure
Some steps to prove that the d -dimensional split closure is a polyhedron Previous work on the 2-dimensional split cuts
Conclusion
Outline
Split cuts
Lattice-point-free polyhedra
High-dimensional split cuts and the d -dimensional split closure
Some steps to prove that the d -dimensional split closure is a polyhedron Previous work on the 2-dimensional split cuts
Split cuts
The algebra
Based on a disjunction
πTx ≤ π0 or πTx ≥ π0+ 1
is valid for x ∈ Znwhen π, π0are integer. The geometry
Split cuts
The algebra
Based on a disjunction
πTx ≤ π0 or πTx ≥ π0+ 1
is valid for x ∈ Znwhen π, π0are integer. The geometry
Split cuts
The algebra
Based on a disjunction
πTx ≤ π0 or πTx ≥ π0+ 1
is valid for x ∈ Znwhen π, π0are integer. The geometry 0 1 01 01 01 0 1 01 01 01 01 0 1 01 01 01 01 0 1 01 01 01 01 0 1 0 1 0 1 0 1 0 1 0 1
Split cuts
The algebra
Based on a disjunction
πTx ≤ π0 or πTx ≥ π0+ 1
is valid for x ∈ Znwhen π, π0are integer. The geometry 0 1 01 01 01 0 1 01 01 01 01 0 1 01 01 01 01 0 1 01 01 01 01 0 1 0 1 0 1 0 1 0 1 0 1 SPLIT
Split cuts
The algebra
Based on a disjunction
πTx ≤ π0 or πTx ≥ π0+ 1
is valid for x ∈ Znwhen π, π0are integer. The geometry 0000000 0000000 0000000 0000000 0000000 0000000 0000000 1111111 1111111 1111111 1111111 1111111 1111111 1111111 00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 11111111111 11111111111 11111111111 11111111111 11111111111 11111111111 11111111111 11111111111 11111111111 11111111111 11111111111 0 1 01 01 0 1 01 01 01 01 0 1 01 01 01 01 0 1 0 1 01 01 01 0 1 0 1 0 1 0 1 0 1 0 1 0 1
Split cuts
The algebra
Based on a disjunction
πTx ≤ π0 or πTx ≥ π0+ 1
is valid for x ∈ Znwhen π, π0are integer. The geometry Split cut 0 1 01 01 01 0 1 01 01 01 01 0 1 01 01 01 01 0 1 01 01 01 01 0 1 0 1 0 1 0 1 0 1 0 1
More about split cuts
The split closure
Consider a polyhedron P ⊆ Rn, the intersection of all split cuts of P is called the (first)
split closureof P, denoted by SC(P).
Some previous results
Cook, Kannan, Schrijver [1990]The split closure is apolyhedron
Lift-and-project, Chv´atal-Gomory cuts are split cuts
Nemhauser, Wolsey [1988] MIR inequalities are split cuts andMIR closure and split closureare equivalent
Cook, Kannan, Schrijver [1990]The number of rounds of split cuts to apply to obtain the integer hull of a polyhedron might beinfinite
Balas, Saxena [2006] Optimizing over the split closure
Andersen, Cornu´ejols, Li [2005]Every split cut of P is also a split cut of abasisof P (maybe infeasible).
Split cuts areintersection cuts
More about split cuts
The split closure
Consider a polyhedron P ⊆ Rn, the intersection of all split cuts of P is called the (first)
split closureof P, denoted by SC(P).
Some previous results
Cook, Kannan, Schrijver [1990]The split closure is apolyhedron
Lift-and-project, Chv´atal-Gomory cuts are split cuts
Nemhauser, Wolsey [1988] MIR inequalities are split cuts andMIR closure and split closureare equivalent
Cook, Kannan, Schrijver [1990]The number of rounds of split cuts to apply to obtain the integer hull of a polyhedron might beinfinite
Balas, Saxena [2006] Optimizing over the split closure
Andersen, Cornu´ejols, Li [2005]Every split cut of P is also a split cut of abasisof P (maybe infeasible).
More about split cuts
The split closure
Consider a polyhedron P ⊆ Rn, the intersection of all split cuts of P is called the (first)
split closureof P, denoted by SC(P).
Some previous results
Cook, Kannan, Schrijver [1990]The split closure is apolyhedron Lift-and-project, Chv´atal-Gomory cuts are split cuts
Nemhauser, Wolsey [1988] MIR inequalities are split cuts andMIR closure and split closureare equivalent
Cook, Kannan, Schrijver [1990]The number of rounds of split cuts to apply to obtain the integer hull of a polyhedron might beinfinite
Balas, Saxena [2006] Optimizing over the split closure
Andersen, Cornu´ejols, Li [2005]Every split cut of P is also a split cut of abasisof P (maybe infeasible).
Split cuts areintersection cuts
More about split cuts
The split closure
Consider a polyhedron P ⊆ Rn, the intersection of all split cuts of P is called the (first)
split closureof P, denoted by SC(P).
Some previous results
Cook, Kannan, Schrijver [1990]The split closure is apolyhedron
Lift-and-project, Chv´atal-Gomory cuts are split cuts
Nemhauser, Wolsey [1988] MIR inequalities are split cuts andMIR closure and split closureare equivalent
Cook, Kannan, Schrijver [1990]The number of rounds of split cuts to apply to obtain the integer hull of a polyhedron might beinfinite
Balas, Saxena [2006] Optimizing over the split closure
Andersen, Cornu´ejols, Li [2005]Every split cut of P is also a split cut of abasisof P (maybe infeasible).
More about split cuts
The split closure
Consider a polyhedron P ⊆ Rn, the intersection of all split cuts of P is called the (first)
split closureof P, denoted by SC(P).
Some previous results
Cook, Kannan, Schrijver [1990]The split closure is apolyhedron
Lift-and-project, Chv´atal-Gomory cuts are split cuts
Nemhauser, Wolsey [1988] MIR inequalities are split cuts andMIR closure and split closureare equivalent
Cook, Kannan, Schrijver [1990]The number of rounds of split cuts to apply to obtain the integer hull of a polyhedron might beinfinite
Balas, Saxena [2006] Optimizing over the split closure
Andersen, Cornu´ejols, Li [2005]Every split cut of P is also a split cut of abasisof P (maybe infeasible).
Split cuts areintersection cuts
More about split cuts
The split closure
Consider a polyhedron P ⊆ Rn, the intersection of all split cuts of P is called the (first)
split closureof P, denoted by SC(P).
Some previous results
Cook, Kannan, Schrijver [1990]The split closure is apolyhedron
Lift-and-project, Chv´atal-Gomory cuts are split cuts
Nemhauser, Wolsey [1988] MIR inequalities are split cuts andMIR closure and split closureare equivalent
Cook, Kannan, Schrijver [1990]The number of rounds of split cuts to apply to
obtain the integer hull of a polyhedron might beinfinite
Balas, Saxena [2006] Optimizing over the split closure
Andersen, Cornu´ejols, Li [2005]Every split cut of P is also a split cut of abasisof P (maybe infeasible).
More about split cuts
The split closure
Consider a polyhedron P ⊆ Rn, the intersection of all split cuts of P is called the (first)
split closureof P, denoted by SC(P).
Some previous results
Cook, Kannan, Schrijver [1990]The split closure is apolyhedron
Lift-and-project, Chv´atal-Gomory cuts are split cuts
Nemhauser, Wolsey [1988] MIR inequalities are split cuts andMIR closure and split closureare equivalent
Cook, Kannan, Schrijver [1990]The number of rounds of split cuts to apply to
obtain the integer hull of a polyhedron might beinfinite
Balas, Saxena [2006] Optimizing over the split closure
Andersen, Cornu´ejols, Li [2005]Every split cut of P is also a split cut of abasisof P (maybe infeasible).
Split cuts areintersection cuts
More about split cuts
The split closure
Consider a polyhedron P ⊆ Rn, the intersection of all split cuts of P is called the (first)
split closureof P, denoted by SC(P).
Some previous results
Cook, Kannan, Schrijver [1990]The split closure is apolyhedron
Lift-and-project, Chv´atal-Gomory cuts are split cuts
Nemhauser, Wolsey [1988] MIR inequalities are split cuts andMIR closure and split closureare equivalent
Cook, Kannan, Schrijver [1990]The number of rounds of split cuts to apply to
obtain the integer hull of a polyhedron might beinfinite
Balas, Saxena [2006] Optimizing over the split closure
Andersen, Cornu´ejols, Li [2005]Every split cut of P is also a split cut of abasisof P (maybe infeasible).
Split cuts are intersection cuts
Split cuts are intersection cuts
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 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 1 0 0 1 1 0 0 1 1Split cuts are intersection cuts
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 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 1 0 0 1 1 0 0 1 1Split cuts are intersection cuts
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 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 1 0 0 1 1 0 0 1 1Split cuts are intersection cuts
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 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 1 0 0 1 1 0 0 1 1Split cuts are intersection cuts
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 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 1 0 0 1 1 0 0 1 1Towards high-dimensional split cuts
Lattice-point-free polyhedra
A polyhedron P islattice-point-freewhen there is no integer pointin its interior.
Towards high-dimensional split cuts
Lattice-point-free polyhedra
A polyhedron P islattice-point-freewhen there is no integer pointin its interior.
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 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 1 0 0 1 1 0 0 1 1
Towards high-dimensional split cuts
Lattice-point-free polyhedra
A polyhedron P islattice-point-freewhen there is no integer pointin its interior.
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 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 1 0 0 1 1 0 0 0 1 1 1 0 0 0 1 1 1 000 000 000 000 000 000 000 000 000 000 111 111 111 111 111 111 111 111 111 111 v w 0 0 1 1 r conv{v , w } + span{r }
Towards high-dimensional split cuts
Lattice-point-free polyhedra
A polyhedron P islattice-point-freewhen there is no integer pointin its interior.
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 1
Towards high-dimensional split cuts
Lattice-point-free polyhedra
A polyhedron P islattice-point-freewhen there is no integer pointin its interior.
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 s 0 0 1 1 w 0 0 1 1 0000 0000 0000 0000 1111 1111 1111 1111 00 00 11 11 r v 0 0 0 1 1 1 0 0 0 1 1 1 conv{v , w } + span{r , s}
Towards high-dimensional split cuts
Lattice-point-free polyhedra
A polyhedron P islattice-point-freewhen there is no integer pointin its interior.
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 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 1 0 0 1 1 0 0 1 1
Towards high-dimensional split cuts
Lattice-point-free polyhedra
A polyhedron P islattice-point-freewhen there is no integer pointin its interior.
000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000000111111 000000111111 000 000 111 111 000 000 111 111
A triangle in R2can be lattice-point-free
It can be lifted to a lattice-point-free polyhedron in R3
Towards high-dimensional split cuts
Lattice-point-free polyhedra
A polyhedron P islattice-point-freewhen there is no integer pointin its interior.
000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000000111111 000000111111 w x v 000 000 111 111 r 000 000 111 111 conv{v , w , x } + span{r }
Towards high-dimensional split cuts
Lattice-point-free polyhedra
A polyhedron P islattice-point-freewhen there is no integer pointin its interior.
000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000 000 111 111 000000111111 000000111111 000000111111 000 000 111 111 000 000 111 111 000000111111 000 000 111 111 000 000 111 111 000 000 111 111 w x v 000 000 111 111 r 000 000 111 111 conv{v , w , x } + span{r }
Definition of thesplit dimension
A lattice-point-free polyhedron P ⊆ Rn can be written as
P = conv{v1, . . . , vp} + cone{w1
, . . . , wq} + span{r1
, . . . , rn−d}. Thesplit-dimensionof P isd.
Using the lattice-point-free polyhedra to generate cuts
The algebra
Let P ⊆ Rn+m be a polyhedron and L ⊆ Rn be a lattice-point-free polyhedron. We define
a set of cuts, valid for {(x , y ) ∈ Rn+m
|x ∈ P ∩ Zn} as
cutsP(L) = conv{(x , y ) ∈ Rn+m|(x, y ) ∈ P and x 6∈ int(L)}. The geometry
Using the lattice-point-free polyhedra to generate cuts
The algebra
cutsP(L) = conv{(x , y ) ∈ Rn+m|(x, y ) ∈ P and x 6∈ int(L)}. The geometry
Using the lattice-point-free polyhedra to generate cuts
The algebra cutsP(L) = conv{(x , y ) ∈ R n+m |(x, y ) ∈ P and x 6∈ int(L)}. The geometryUsing the lattice-point-free polyhedra to generate cuts
The algebra
cutsP(L) = conv{(x , y ) ∈ Rn+m|(x, y ) ∈ P and x 6∈ int(L)}. The geometry
The high-dimensional split closure
Definition
Thed -dimensional split closureof P is the set of points in the intersection of all
high-dimensional split cuts obtained from P with asplit-dimension less or equal to d.
Steps to prove that the d -dimensional split closure is a polyhedron
Every d -dimensional split cut can be obtained from a simpler set (butnot a basis) Some d -dimensional splits generate cuts that are dominating compared to other d -dimensional splits
Mapping of integer points and edges to dominating splits
Finiteness on the number of coefficients of a given variable for fixed integer points and edges
The high-dimensional split closure
Definition
Thed -dimensional split closureof P is the set of points in the intersection of all
high-dimensional split cuts obtained from P with asplit-dimension less or equal to d.
Steps to prove that the d -dimensional split closure is a polyhedron
Every d -dimensional split cut can be obtained from a simpler set (butnot a basis) Some d -dimensional splits generate cuts that are dominating compared to other d -dimensional splits
Mapping of integer points and edges to dominating splits
Finiteness on the number of coefficients of a given variable for fixed integer points and edges
The high-dimensional split closure
Definition
Thed -dimensional split closureof P is the set of points in the intersection of all
high-dimensional split cuts obtained from P with asplit-dimension less or equal to d.
Steps to prove that the d -dimensional split closure is a polyhedron
Every d -dimensional split cut can be obtained from a simpler set (butnot a basis)
Some d -dimensional splits generate cuts that are dominating compared to other d -dimensional splits
Mapping of integer points and edges to dominating splits
Finiteness on the number of coefficients of a given variable for fixed integer points and edges
The high-dimensional split closure
Definition
Thed -dimensional split closureof P is the set of points in the intersection of all
high-dimensional split cuts obtained from P with asplit-dimension less or equal to d.
Steps to prove that the d -dimensional split closure is a polyhedron
Every d -dimensional split cut can be obtained from a simpler set (butnot a basis)
Some d -dimensional splits generate cuts that are dominating compared to other d -dimensional splits
Mapping of integer points and edges to dominating splits
Finiteness on the number of coefficients of a given variable for fixed integer points and edges
The high-dimensional split closure
Definition
Thed -dimensional split closureof P is the set of points in the intersection of all
high-dimensional split cuts obtained from P with asplit-dimension less or equal to d.
Steps to prove that the d -dimensional split closure is a polyhedron
Every d -dimensional split cut can be obtained from a simpler set (butnot a basis)
Some d -dimensional splits generate cuts that are dominating compared to other d -dimensional splits
Mapping of integer points and edges to dominating splits
Finiteness on the number of coefficients of a given variable for fixed integer points and edges
The high-dimensional split closure
Definition
Thed -dimensional split closureof P is the set of points in the intersection of all
high-dimensional split cuts obtained from P with asplit-dimension less or equal to d.
Steps to prove that the d -dimensional split closure is a polyhedron
Every d -dimensional split cut can be obtained from a simpler set (butnot a basis)
Some d -dimensional splits generate cuts that are dominating compared to other d -dimensional splits
Mapping of integer points and edges to dominating splits
Finiteness on the number of coefficients of a given variable for fixed integer points and edges
Some d -dimensional split cuts do not come from a basis
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 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 1 0 0 1 1 0 0 1 1Some d -dimensional split cuts do not come from a basis
Natural 2-dimensional split cut : x3≤ 0
Some d -dimensional split cuts do not come from a basis
Some d -dimensional split cuts do not come from a basis
The first basis is a feasible basis (the other feasible basis is symmetric).
Some d -dimensional split cuts do not come from a basis
The first basis is a feasible basis (the other feasible basis is symmetric).
Some d -dimensional split cuts do not come from a basis
The second basis is infeasible (and there is a symmetric one).
Some d -dimensional split cuts do not come from a basis
The second basis is infeasible (and there is a symmetric one). No cut generated !