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(1)

Cutting planes from lattice-point-free polyhedra

Quentin Louveaux

Universit´e catholique de Louvain - CORE - Belgium

January 2007

(2)

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

(3)

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

(4)

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

(5)

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

(6)

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

(7)

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

(8)

Split cuts

The algebra

Based on a disjunction

πTx ≤ π0 or πTx ≥ π0+ 1

is valid for x ∈ Znwhen π, π0are integer. The geometry

(9)

Split cuts

The algebra

Based on a disjunction

πTx ≤ π0 or πTx ≥ π0+ 1

is valid for x ∈ Znwhen π, π0are integer. The geometry

(10)

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

(11)

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

(12)

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

(13)

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

(14)

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

(15)

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).

(16)

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

(17)

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).

(18)

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

(19)

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).

(20)

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

(21)

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).

(22)

Split cuts are intersection cuts

(23)

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 1

(24)

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 1

(25)

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 1

(26)

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 1

(27)

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 1

(28)

Towards high-dimensional split cuts

Lattice-point-free polyhedra

A polyhedron P islattice-point-freewhen there is no integer pointin its interior.

(29)

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

(30)

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 }

(31)

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

(32)

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}

(33)

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

(34)

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

(35)

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 }

(36)

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.

(37)

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

(38)

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

(39)

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 geometry

(40)

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

(41)

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

(42)

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

(43)

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

(44)

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

(45)

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

(46)

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

(47)
(48)

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 1

(49)
(50)

Some d -dimensional split cuts do not come from a basis

Natural 2-dimensional split cut : x3≤ 0

(51)

Some d -dimensional split cuts do not come from a basis

(52)

Some d -dimensional split cuts do not come from a basis

The first basis is a feasible basis (the other feasible basis is symmetric).

(53)

Some d -dimensional split cuts do not come from a basis

The first basis is a feasible basis (the other feasible basis is symmetric).

(54)

Some d -dimensional split cuts do not come from a basis

The second basis is infeasible (and there is a symmetric one).

(55)

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 !

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