Elementary Closures in Nonlinear Integer

Programming

Daniel Dadush – CWI

Santanu Dey – Georgia Tech

Juan Pablo Vielma – MIT

Cutting plane: valid inequality for integer hull.

Use to get tighter relaxations – important component in practical solvers.

Very well developed theory within ILP..

Cutting Planes for Integer Linear Programs

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Many other important Integer Programming models.

Substantial Progress in recent years:

Solvers, Complexity, Heuristics, etc.

Cutting planes poorly understood, even when objective and constraints are convex.

Mixed and Non Linear Models?

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Convex Integer Programming

Goal: understand basic behavior for fundamental classes of cutting planes.

Issue: most current theory depends heavily on rational linear structure.

Plan: Develop stronger / more robust Analytical Tools.

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Chvátal-Gomory (CG) Cuts

compact convex

Support function of :

For cut:

𝑣𝐾

𝑣𝑣

𝑥1+𝑥2≤0

𝑥1+𝑥2≤4.8

𝑥1+𝑥2≤4

Chvátal-Gomory Closure

compact convex

closure of :

𝑣 2

𝑣 4

𝑣 6

𝑣 1

𝐾

𝑣 3𝑣 5

Chvátal-Gomory Closure

compact convex

closure of :

𝐾 𝑣 2

𝑣 4

𝑣 6

𝑣 1

𝑣 3𝑣 5

Chvátal-Gomory Closure

compact convex

closure of :

Chvátal `73:Converge to integer hullafter finitely many iterations.

CG (𝐾 ) 𝑣 2

𝑣 4

𝑣 6

𝑣 1

𝑣 3𝑣 5

𝑥1+𝑥2≤3

Split Cuts

compact convex

Elementary disjunction:

𝐾

𝑥1+𝑥2≥4

compact convex

Elementary disjunction:

Split Cut:

𝑥1+𝑥2≤3

Split Cuts

𝐾

𝑥1+𝑥2≥4

Split Closure

compact convex

Split Closure:

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Split Closure

compact convex

Split Closure:

CG (𝐾 )𝑥1+𝑥2≤1

𝑥1+𝑥2≥2

Split Closure

compact convex

Split Closure:

SC(𝐾 )

Generating the CG Closure

Schrijver 1980Theorem: If is a rational polyhedron, then is finitely generated.

𝑣 2

𝑣 3

𝑣 4

𝑣 1

𝐾

Generating the CG Closure

Schrijver 1980Theorem: If is a rational polyhedron, then is finitely generated.

𝑣 2𝑣 3

𝑣 4

𝑣 1

CG (𝐾 )

Schrijver 1980Theorem: If is a rational polyhedron, then is finitely generated.

Question: Is this true in general?

𝑣 2𝑣 3

𝑣 4

𝑣 1

CG (𝐾 )

Generating the CG Closure

Generating the CG Closure

Schrijver 1980Theorem: If is a rational polyhedron, then is finitely generated.

Question: Is this true in general?

Not always.

Generating the Split Closure

Cook, Kannan, Schrijver 1990Theorem: If is a rational polyhedron, then is finitely generated.

𝑣 2

𝑣 3

𝑣 4

𝑣 1

𝐾

Generating the Split Closure

Cook, Kannan, Schrijver 1990Theorem: If is a rational polyhedron, then is finitely generated.

𝑣 2𝑣 3

𝑣 4

𝑣 1

SC(𝐾 )

Generating the Split Closure

Cook, Kannan, Schrijver 1990Theorem: If is a rational polyhedron, then is finitely generated.

Question:

Is the Split Closure always polyhedral?

No.

𝑣 2𝑣 3

𝑣 4

𝑣 1

Non Polyhedral Split Closure

D., Dey, Vielma 11Theorem:The split closureof an ellipsoid canbe non polyhedral.

Generating the CG Closure

D., Dey, Vielma 11Theorem: If is compact convex, then is proper and finitely generated.

𝐾𝑣 2

𝑣 3

𝑣 4

𝑣 1

Generating the Split Closure

D., Dey, Vielma 11Theorem: If is compact strictly convex, then is proper and finitely generated.

is strictly convex if does notcontain lines. 𝐾

𝑣 2𝑣 3

𝑣 4

𝑣 1

Generating the CG Closure

D., Dey, Vielma 13Theorem: If is a thin, totally closed convex set with rational polyhedral , then is proper and finitely generated.

If is closed convex with proper, non-empty & rational polyhedral then:1. is thin.2. is rational polyhedral.

𝐾

𝑣 3

𝑣 4

𝑣 1

Thin Convex Sets

Definition: Closed convex set is thin if for some finite radius .

Thin Not Thin

Thin Convex Sets

Definition: Closed convex set is thin if for some finite radius .

Lemma: If is polyhedral, then is thin for generators cone for some constant .

Totally Closed Sets

Definition: A convex set is totally closed if is closed for any subspace of .

Not Totally Closed

Totally Closed

Totally Closed Sets

Definition: A convex set is totally closed if is closed for any subspace of .

Thin &Totally Closed

Totally Closed Sets

Definition: A convex set is totally closed if is closed for any subspace of .

Lemma: If is polyhedral, then is totally closed for every face of , the projection of onto span(F) is closed.

denotes the partial closure.

1. Achieve Containment:

Building the CG Closure

𝐾

Most difficult step (trivial for rational polytopes). Need tools from Convex Analysis and Geometry of Numbers.

denotes the partial closure.

1. Achieve Containment:

2. Complete the Boundary:

Building the CG Closure

𝐾

Only a finite number of faces of need fixing. Use induction on each face.

𝐶𝐺 (𝐾 ,𝑆 )∩𝜕𝐾

denotes the partial closure.

1. Achieve Containment:

2. Complete the Boundary:

3. Complete the Closure: Add all cuts separatinga vertex of .

Building the CG Closure

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All remaining cuts have bounded norm.

𝛿𝛿

𝛿

𝛿𝛿

Lifting CG cuts

D., Dey, Vielma 11Theorem: Let be an exposed face of . Then

Goal: Lift to .

1. Find : on “rational” part of .2. Remove “irrational” parts of using additional cuts.

𝐹

𝐾

𝑣𝑣𝑣 ′𝑣 ′

𝑣𝑣

Removing Irrational Parts

0ℤ𝑛𝐹

( √32, √23

)

Since is an irrational line segment, a rational polytope contained in can only intersect at .

Need CG cuts that remove irrational parts of .

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Rational Polytope

Removing Irrational Parts

0ℤ𝑛𝐹

( √32, √23

)

Kronecker’s Approximation TheoremFractional part is dense.

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Removing Irrational Parts

0ℤ𝑛𝐹

( √32, √23

)

Integers close to line give“nearly” same inequality for K.

𝐾

Removing Irrational Parts

0ℤ𝑛𝐹

( √32, √23

)

Use density to find “close” cuts biased towards opposite sides of .Cuts remove irrational parts of .

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Extending to General Convex Sets

Main Technical Issues:

1. Can only build CG cuts in integer directions where the objective value is finite.

(need thinness and rational polyhedral recession cone)

2. To lift CG cuts from faces must show fine continuity properties of the objective function.

(need total closedness)

OpenQuestion

Is the split closure of a compact convex set finitely generated?

Thank You!