On the Unique-Lifting Property Amitabh Basu Joint Work with - PowerPoint PPT Presentation

On the Unique-Lifting Property Amitabh Basu Joint Work with Gennadiy Averkov 18th Aussois Combinatorial Optimization Workshop

Cut Generating Functions for Mixed-Integer Linear Programs ◮ Solve problem as Linear Program and obtain optimal simplex tableau. ◮ If some basic variables are non-integral, “apply” Cut Generating Functions to one or more rows of the tableau obtain Cutting Planes. ◮ Add these cutting planes and re-iterate (usually combined with some enumeration scheme).

General Framework for Cut Generating Functions + � k + � ℓ j =1 r j s j j =1 p j y j x = f x ∈ Z q s ∈ R k y ∈ Z ℓ + +

General Framework for Cut Generating Functions + � k + � ℓ j =1 r j s j j =1 p j y j = x f x ∈ Z q s ∈ R k y ∈ Z ℓ + + We seek pair of functions ψ f : R q → R π f : R q → R such that the inequality k ℓ � ψ f ( r j ) s j + � π f ( p j ) y j ≥ 1 j =1 j =1 is valid for any k , ℓ, r j , p j .

General Framework for Cut Generating Functions + � k + � ℓ j =1 r j s j j =1 p j y j x = f x ∈ Z q s ∈ R k y ∈ Z ℓ + + We seek pair of functions ψ f : R q → R π f : R q → R such that the inequality k ℓ � ψ f ( r j ) s j + � π f ( p j ) y j ≥ 1 j =1 j =1 is valid for any k , ℓ, r j , p j . Want minimal valid pairs to remove redundancies.

What are we after ? • Johnson’s theorem. Structural characterization of all minimal valid pairs.

What are we after ? • Johnson’s theorem. Structural characterization of all minimal valid pairs. • MAIN GOAL: Find closed form formulas for these functions.

What are we after ? • Johnson’s theorem. Structural characterization of all minimal valid pairs. • MAIN GOAL: Find closed form formulas for these functions. • Let B ∈ R q be a maximal lattice-free convex polytope with f ∈ int ( B ): B = { x ∈ R q : a i ( x − f ) ≤ 1 , i ∈ I } . Define the function ∀ r ∈ R q . φ f , B ( r ) = max i ∈ I a i r , Then, ψ f = π f = φ f , B is a valid pair, but not minimal.

What are we after ? • Johnson’s theorem. Structural characterization of all minimal valid pairs. • MAIN GOAL: Find closed form formulas for these functions. • Let B ∈ R q be a maximal lattice-free convex polytope with f ∈ int ( B ): B = { x ∈ R q : a i ( x − f ) ≤ 1 , i ∈ I } . Define the function ∀ r ∈ R q . φ f , B ( r ) = max i ∈ I a i r , Then, ψ f = π f = φ f , B is a valid pair, but not minimal. • The idea is to start with some maximal lattice-free B , define ψ f = φ f , B and find a minimal function π f such that it forms a valid pair with ψ f . These are called minimal liftings of ψ f .

What are we after ? • Johnson’s theorem. Structural characterization of all minimal valid pairs. • MAIN GOAL: Find closed form formulas for these functions. • Let B ∈ R q be a maximal lattice-free convex polytope with f ∈ int ( B ): B = { x ∈ R q : a i ( x − f ) ≤ 1 , i ∈ I } . Define the function ∀ r ∈ R q . φ f , B ( r ) = max i ∈ I a i r , Then, ψ f = π f = φ f , B is a valid pair, but not minimal. • The idea is to start with some maximal lattice-free B , define ψ f = φ f , B and find a minimal function π f such that it forms a valid pair with ψ f . These are called minimal liftings of ψ f . • For some special f , B ’s, minimal liftings are unique. Gives us formulas for minimal valid pairs.

Recognizing pairs f , B with unique minimal liftings 1. Given f , B f

Recognizing pairs f , B with unique minimal liftings 1. Given f , B . 2. For every facet F 3 F , construct P F := F 1 P F 1 P F 3 f conv ( F ∪ { f } ). P F 2 F 2

Recognizing pairs f , B with unique minimal liftings 1. Given f , B . 2. For every facet F , construct P F := conv ( F ∪ { f } ). z 1 z 3 3. For each S z 1 ,F 1 F 3 S z 3 ,F 3 F 1 z ∈ Z q ∩ relint ( F ), f construct S z 2 ,F 2 S F , z ( f ) := P F ( f ) ∩ z 2 F 2 ( z + f − P F ( f )).

Recognizing pairs f , B with unique minimal liftings 1. Given f , B . 2. For every facet F , construct P F := conv ( F ∪ { f } ). z 1 z 3 3. For each S z 1 ,F 1 F 3 S z 3 ,F 3 F 1 z ∈ Z q ∩ relint ( F ), f construct S z 2 ,F 2 S F , z ( f ) := P F ( f ) ∩ z 2 F 2 ( z + f − P F ( f )). � � R ( f , B ) = S z , F ( f ) z ∈ Z q ∩ relint ( F ) Facets F

Recognizing pairs f , B with unique minimal liftings 1. Given f , B . 2. For every facet F 3 F , construct P F := z 3 S z 3 ,F 3 f conv ( F ∪ { f } ). S z 2 ,F 2 xz 2 3. For each F 2 S z 1 ,F 1 z ∈ Z q ∩ relint ( F ), F 1 construct z 1 S F , z ( f ) := P F ( f ) ∩ ( z + f − P F ( f )). � � R ( f , B ) = S z , F ( f ) z ∈ Z q ∩ relint ( F ) Facets F

Recognizing pairs f , B with unique minimal liftings � � R ( f , B ) = S z , F ( f ) Facets F z ∈ Z q ∩ relint ( F ) THEOREM Basu, Campelo, Conforti, Cornu´ ejols, Zambelli 2011 f , B has the unique-lifting property if and only if R ( f , B ) + Z q = R q .

Recognizing pairs f , B with unique minimal liftings � � R ( f , B ) = S z , F ( f ) Facets F z ∈ Z q ∩ relint ( F ) THEOREM Basu, Campelo, Conforti, Cornu´ ejols, Zambelli 2011 f , B has the unique-lifting property if and only if R ( f , B ) + Z q = R q . Main Credit for sparking this line of research: Santanu Dey and Laurence Wolsey 2009.

x 1 x 6 x 2 f x 3 x 4 x 5

x 1 x 6 x 2 R ( x 2 ) f x 3 x 4 x 5

x 1 x 6 x 2 R ( x 2 ) f x 3 x 4 x 5

x 1 x 6 x 2 R ( x 2 ) f x 3 x 4 x 5

R ψ + Z q = R q x 1 x 6 x 2 R ( x 2 ) f x 3 x 4 x 5

R ψ + Z q � = R q x 3 R ( x 3 ) R ( x 2 ) x 2 f B ψ R ( x 1 ) x 1

R ψ + Z q � = R q x 3 R ( x 3 ) R ( x 2 ) x 2 f B ψ R ( x 1 ) x 1

MORAL : 1. If the pair f , B has the unique-lifting property, then we get closed form formulas for a minimal valid pair. 2. The question of deciding if f , B has the unique-lifting property is equivalent to deciding if R ( f , B ) + Z q = R q . Potentially connects with a lot of research on coverings and tilings by star-shaped bodies, extensively studied in Geometry of Numbers.

Invariance of the Unique-lifting property For a fixed maximal lattice-free convex polytope B , R ( f , B ) (in fact ψ f itself) depends on the position of f in the interior. So, a priori , the same lattice-free set B might have the unique-lifting property when paired with one f 1 , and have the multiple-lifting property when paired with a different f 2 .

Invariance of the Unique-lifting property For a fixed maximal lattice-free convex polytope B , R ( f , B ) (in fact ψ f itself) depends on the position of f in the interior. So, a priori , the same lattice-free set B might have the unique-lifting property when paired with one f 1 , and have the multiple-lifting property when paired with a different f 2 . THEOREM Averkov, Basu 2013 Let B be a maximal lattice-free polytope in R q ( q ≥ 2). Then B has the unique-lifting property for all f ∈ int ( B ), or B has the multiple-lifting property for all f ∈ int ( B ).

Invariance of the Unique-lifting property THEOREM Averkov, Basu 2013 Let B be a maximal lattice-free polytope in R q ( q ≥ 2). Then B has the unique-lifting property for all f ∈ int ( B ), or B has the multiple-lifting property for all f ∈ int ( B ). OBSERVATION Deciding if R ψ + Z q = R q is the same as deciding if vol T q ( R ψ / Z q ) = 1.

Invariance of the Unique-lifting property THEOREM Averkov, Basu 2013 Let B be a maximal lattice-free polytope in R q ( q ≥ 2). Then B has the unique-lifting property for all f ∈ int ( B ), or B has the multiple-lifting property for all f ∈ int ( B ). OBSERVATION Deciding if R ψ + Z q = R q is the same as deciding if vol T q ( R ψ / Z q ) = 1. THEOREM Averkov, Basu 2013 Let B be a maximal lattice-free polytope and let f ∈ B . Then vol T q ( R ψ / Z q ) is an affine function of the coordinates of f .

Lattice Volume is an affine function THEOREM Averkov, Basu 2013 Let B be a maximal lattice-free polytope and let f ∈ B . Then vol T q ( R ψ / Z q ) is an affine function of the coordinates of f .

Lattice Volume is an affine function THEOREM Averkov, Basu 2013 Let B be a maximal lattice-free polytope and let f ∈ B . Then vol T q ( R ψ / Z q ) is an affine function of the coordinates of f . Proof Ingredient 1 : The volume of each S z , F ( f ) is an affine function of f . f 1

Lattice Volume is an affine function THEOREM Averkov, Basu 2013 Let B be a maximal lattice-free polytope and let f ∈ B . Then vol T q ( R ψ / Z q ) is an affine function of the coordinates of f . Proof Ingredient 1 : The volume of each S z , F ( f ) is an affine function of f . f 2 f 1

Lattice Volume is an affine function THEOREM Averkov, Basu 2013 Let B be a maximal lattice-free polytope and let f ∈ B . Then vol T q ( R ψ / Z q ) is an affine function of the coordinates of f . Proof Ingredient 1 : The volume of each S z , F ( f ) is an affine function of f . Proof Ingredient 2 : Take care of intersections due to lattice translations: We find a closed form integral expression for vol T q ( R ψ / Z q ).

Operations that preserve that Unique-lifting property Pyramid Construction. Let B ⊆ R q be a maximal lattice-free polytope. Consider B as embedded in R q +1 , i.e., B ⊆ R q × { 0 } ⊆ R q +1 . Let v ∈ R q +1 \ ( R q × { 0 } ). Let C ( B , v ) be the cone formed with B − v as base. We define Pyr( B , v ) = ( C ( B , v ) + v ) ∩ { x ∈ R q +1 : x q +1 ≥ − 1 } .