Generalized Chvátal-Gomory Closures for Integer Programs ... · Generalized Chv atal-Gomory...

Generalized Chvatal-Gomory Closures for Integer Programswith Bounds on Variables

Dabeen Lee

Carnegie Mellon University

January 7, 2018

Joint work with Sanjeeb Dash and Oktay Gunluk

Integer linear programming

Integer linear programming (ILP)

Given A ∈ Qm×n, b ∈ Qm, and w ∈ Qn, we want to solve

min wxs.t. Ax ≥ b

x ∈ Zn P = {x : Ax ≥ b}

• General-purpose cutting planes are developed for solving ILP.

• Split cuts, Gomory’s GMI cuts, Chvatal-Gomory cuts, etc.

• We will study a generalization of Chvatal-Gomory cuts.

min wxs.t. Ax ≥ b

x ∈ Zn P = {x : Ax ≥ b}

min wxs.t. Ax ≥ b

x ∈ Zn P = {x : Ax ≥ b}

min wxs.t. Ax ≥ b

x ∈ Zn P = {x : Ax ≥ b}

Chvatal-Gomory closure

• The Chvatal-Gomory closure of a rational polyhedron P ⊆ Rn is defined as

P ′ :=⋂c∈Zn

{x ∈ Rn : cx ≥ dmin{cy : y ∈ P}e}

Dabeen Lee Generalized Chvatal-Gomory Closures for Integer Programs with Bounds on Variables

P ′ :=⋂c∈Zn

Theorem [Chvatal, 1973, Schrijver, 1980]

Let P be a rational polyhedron. Then P ′ is also a rational polyhedron.

• We often face integer programs that involve constraints to bound thevalues of variables x (problems in practice and in combinatorialoptimization).

• Can we provide a generalization of Chvatal-Gomory closure closures forsuch integer programs?

P ′ :=⋂c∈Zn

Generalization of Chvatal-Gomory inequalities

min{cx : x ∈ P}

• S := {x ∈ Zn : ` ≤ x ≤ u}.• P ⊆ {x ∈ Rn : ` ≤ x ≤ u}.

min{cx : x ∈ P}

• S := {x ∈ Zn : ` ≤ x ≤ u}.• P ⊆ {x ∈ Rn : ` ≤ x ≤ u}.

dmin{cx : x ∈ P}e

• S := {x ∈ Zn : ` ≤ x ≤ u}.• P ⊆ {x ∈ Rn : ` ≤ x ≤ u}.

min{cx : x ∈ P}dmin{cx : x ∈ P}e

min{cz : cz ≥ min{cx : x ∈ P}, z ∈ S}

min{cx : x ∈ P}min{cz : cz ≥ min{cx : x ∈ P}, z ∈ Zn}

min{cz : cz ≥ min{cx : x ∈ P}, z ∈ S}

Generalization of Chvatal-Gomory closures

Let S ⊆ Zn be the set of integer points that satisfy all the bound constraints.

Given a valid inequality cx ≥ d for a polyhedron P,

• cx ≥ dde = min{cz : cz ≥ d , z ∈ Zn} is the Chvatal-Gomory inequalityobtained from it.

• However, cx ≥ min{cz : cz ≥ d , z ∈ S} is also valid for P ∩ Zn (⊆ S).

• We have min{cz : cz ≥ d , z ∈ S} ≥ dde = min{cz : cz ≥ d , z ∈ Zn}.

Given S such that P ⊆ conv(S) and S = conv(S) ∩ Zn, theS-Chvatal-Gomory-Gomory closure of a rational polyhedron P ⊆ Rn is definedas

P ′S :=⋂c∈Zn

x ∈ Rn : cx ≥ min{cz : cz ≥ min{cx : x ∈ P}, z ∈ S}︸︷︷︸S-Chvatal-Gomory inequality

• It was first studied by Dunkel and Schulz (2012) and Pokutta (2011).

P ′S :=⋂c∈Zn

• Remark that S-Chvatal-Gomory inequalities can be viewed as cuts from“wide split disjunctions” or “S-free split disjunctions”.

We know that the Chvatal-Gomory closure of a rational polyhedron is arational polyhedron. What about the S-Chvatal-Gomory closure?

Theorem [Dash, Gunluk, L]

Let S =

z = (z1, z2, z3, z4) ∈ Zn :`1 ≤ z1 ≤ u1,`2 ≤ z2 ,

z3 ≤ u3

If P is a rational polyhedron contained in conv(S), then the S-Chvatal-Gomoryclosure of P is also a rational polyhedron.

We know that the Chvatal-Gomory closure of a rational polyhedron is arational polyhedron. What about the S-Chvatal-Gomory closure?

Let S =

z = (z1, z2, z3, z4) ∈ Zn :`1 ≤ z1 ≤ u1,`2 ≤ z2 ,

z3 ≤ u3

A difference

P ′S

• Some facets are not defined by S-Chvatal-Gomory inequalities.

• In this example, the inequality defining a facet is the “limit” of a sequenceof S-Chvatal-Gomory inequalities.

A difference

P ′S

A difference

P ′S

A difference

There is a more concrete example.

• Let S = {0, 1}4 and P be the convex hull of six points in [0, 1]4:

P = conv

2, 0, 0, 0

), (1, 0, 0, 0), (0, 1, 1, 0), (0, 1, 0, 1), (0, 0, 1, 1), (1, 1, 1, 1)

• 2x1 + x2 + x3 + x4 ≥ 2 is a facet-defining inequality for the integer hull ofP.

• However, 2x1 + x2 + x3 + x4 ≥ 1 is supporting P at ( 12, 0, 0, 0) and

(0, 1, 0, 0) is on its corresponding hyperplane.

• 2x1 + (1− δ)x2 + (1− δ)x3 + (1− δ)x4 ≥ 2− 2δ is the S-Chvatal-Gomoryinequality obtained from 2x1 + (1− δ)x2 + (1− δ)x3 + (1− δ)x4 ≥ 1 forδ < 1

A difference

P = conv

2, 0, 0, 0

), (1, 0, 0, 0), (0, 1, 1, 0), (0, 1, 0, 1), (0, 0, 1, 1), (1, 1, 1, 1)

A difference

P = conv

2, 0, 0, 0

), (1, 0, 0, 0), (0, 1, 1, 0), (0, 1, 0, 1), (0, 0, 1, 1), (1, 1, 1, 1)

A difference

P = conv

2, 0, 0, 0

), (1, 0, 0, 0), (0, 1, 1, 0), (0, 1, 0, 1), (0, 0, 1, 1), (1, 1, 1, 1)

Proof idea

Can we use techniques developed for proving polyhedrality of other closures?

• Chvatal-Gomory closure ([Chvatal, 1973], [Schrijver, 1980], [Dadush, Dey,Vielma, 2011], etc).

• Split closure ([Cook, Kannan, Schrijver, 1990], [Andersen, Cornuejols, Li,2005], [Dash, Gunluk, Lodi, 2007], etc).

• Closure from lattice-free sets with bounded max-facet-width ([Andersen,Louveaux, Weismantel, 2010], [Averkov, 2012], etc)

In fact, it seemed difficult to use the techniques from these papers.

• min{cz : cz ≥ miny∈P cy , z ∈ S} −miny∈P cy cannot be bounded by afixed constant.

• An S-free split strip (from disjunction) does not necessarily have boundedfacet-width.

Proof idea

Our proof is based on the following lemma, due to Dunkel and Schulz (2012):

Lemma [Dunkel, Schulz, 2012]

Let P ⊆ Rn be a rational polyhedron. Then⋂(α,β)∈H

{x ∈ P : αx ≥ β}

is a rational polyhedron if H ⊆ Rn+1 is a rational polyhedron.

We are going to write linear constraint on (α, β) ∈ Zn+1 for αx ≥ β to be anS-Chvatal-Gomory inequality.

Proof idea

For example, consider the case when S is finite.

αx ≥ α0αx ≥ β

• If αx ≥ β is an S-Chvatal-Gomory inequality for some (α, β) ∈ Zn+1, thenit is obtained from a valid inequality αx ≥ α0 for P.

• S can be partitioned into G and L, where

G = {z ∈ S : αz ≥ β} and L = {z ∈ S : αz ≤ α0}

Proof idea

αx ≥ α0

αx ≥ β

G = {z ∈ S : αz ≥ β} and L = {z ∈ S : αz ≤ α0}

Proof idea

αx ≥ α0

αx ≥ β

(α, β) satisfies the following.

(α, α0) = (λA, λb),αz ≥ β, ∀z ∈ Gαz ≤ α0 − 1

∆, ∀z ∈ L

β ≥ α0

G = {z ∈ S : αz ≥ β} and L = {z ∈ S : αz ≤ α0}.

Proof idea

In fact,

Let P = {x ∈ Rn : Ax ≥ b} be rational polyhedron. The S-Chvatal-Gomoryclosure of P is obtained after applying αx ≥ β for all (α, β) ∈ Zn+1 satisfying

(α, α0) = (λA, λb),αz ≥ β, ∀z ∈ Gαz ≤ α0 − 1

∆, ∀z ∈ L

β ≥ α0

for some λ ∈ Rm+, α0 ∈ R, and some partition (G , L) of S . ∆ is the max size of

a sub-determinant of A.

The number of partitions (G , L) of S is finite. so the S-Chvatal-Gomory closureof P is a rational polyhedron when S is finite.

• However, if S is not finite, then the number of partitions of S is infinite.

• Besides, if S is not finite, either G or L is an infinite set.

Proof idea

In fact,

(α, α0) = (λA, λb),αz ≥ β, ∀z ∈ Gαz ≤ α0 − 1

∆, ∀z ∈ L

β ≥ α0

Proof idea

In fact,

(α, α0) = (λA, λb),αz ≥ β, ∀z ∈ Gαz ≤ α0 − 1

∆, ∀z ∈ L

β ≥ α0

Proof idea

In fact,

(α, α0) = (λA, λb),αz ≥ β, ∀z ∈ Gαz ≤ α0 − 1

∆, ∀z ∈ L

β ≥ α0

Proof idea

For the general case, let us first consider S = Zn+, which is infinite.

• First, take the Chvatal-Gomory closure of P: P ′.

• Apply S-Chvatal-Gomory inequalities cutting off some part of P ′.

Lemma [Dash, Gunluk, L]

If αx ≥ β cuts off some part of P ′, then either (α, β) ≥ 0 or (α, β) ≤ 0.

αx ≥ α0

αx ≥ β = dα0e

Proof idea

αx ≥ α0

αx ≥ β = dα0e

Proof idea

αx ≥ α0

αx ≥ β = dα0e

Proof idea

αx ≥ α0

αx ≥ β = dα0e

Proof idea

There exists a large constant M such that if αx ≥ β cuts off some part of P ′

and is non-redundant, then 0 ≤ 1βα ≤ M1.

• The intercepts of {x ∈ Rn : αx = β} with coordinate axes are allcontained in [0,M]n.

• Then, we can focus on the set of integer points contained in [0,M]n,which is finite.

Proof idea

Let S =

z = (z1, z2, z3, z4) ∈ Zn :`1 ≤ z1 ≤ u1,`2 ≤ z2 ,

z3 ≤ u3

A draft will be posted soon!

Thank you!

Chvatal, V. (1973).Edmonds polytopes and a hierarchy of combinatorial problems.Discrete Mathematics, 4(4):305 – 337.

Schrijver, A. (1980).On cutting planes.Annals of Discrete Mathematics, 9:291 – 296.

Generalized Chvátal-Gomory Closures for Integer Programs ... · Generalized Chv atal-Gomory...

Documents

Transcript of Generalized Chvátal-Gomory Closures for Integer Programs ... · Generalized Chv atal-Gomory...