The structure of the infinite models in integer …Approach in Mixed Integer Programming: Theory, Computation and Perspectives, Fifty years of Integer Programming 1958-2008: From early

The structure of the infinite models in integerprogramming

Amitabh Basu, MC, Marco Di Summa, Joseph Paat.

January 10, 2017

Amitabh Basu, MC, Marco Di Summa, Joseph Paat. The structure of the infinite models in integer programming

Not Again!Well, Aussois is partly responsible..... Some years ago:



Tuesday, January 8, 2008Chair: Laurence Wolsey

17:15-18:00 Jean-Philippe Richard Group Relaxations for IntegerProgramming

18:00-18:30 Santanu Dey Facets of High-Dimensional InfiniteGroup Problem






Jean-Philippe P. Richard, Santanu S. Dey The Group-TheoreticApproach in Mixed Integer Programming: Theory, Computationand Perspectives, Fifty years of Integer Programming 1958-2008:From early years to the state-of-the-art (M. Juenger, T. Liebling,D. Naddef, G. Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi,and L. Wolsey (eds.)), December 2009 (Springer).






Jean-Philippe P. Richard, Santanu S. Dey The Group-TheoreticApproach in Mixed Integer Programming: Theory, Computationand Perspectives, Fifty years of Integer Programming 1958-2008:From early years to the state-of-the-art (M. Juenger, T. Liebling,D. Naddef, G. Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi,and L. Wolsey (eds.)), December 2009 (Springer).

a lot of research since then...Amitabh Basu, MC, Marco Di Summa, Joseph Paat. The structure of the infinite models in integer programming

MIPs in tableau form

b ∈ Rn \ Zn.

xB +∑

r∈R

rs(r) +∑

p∈P

py(p) = b, xB ∈ Zn, s(r) ∈ R+, y(p) ∈ Z+



b ∈ Rn \ Zn.

xB +∑

r∈R

rs(r) +∑

p∈P

py(p) = b, xB ∈ Zn, s(r) ∈ R+, y(p) ∈ Z+

∑

r∈R

rs(r) +∑

p∈P

py(p)∈ b + Zn, s(r) ∈ R+, y(p) ∈ Z+



b ∈ Rn \ Zn.

xB +∑

r∈R

rs(r) +∑

p∈P

py(p) = b, xB ∈ Zn, s(r) ∈ R+, y(p) ∈ Z+

∑

r∈R

rs(r) +∑

p∈P

py(p)∈ b + Zn, s(r) ∈ R+, y(p) ∈ Z+

BFS: s(r) = y(p) = 0, xB = b. Want xB ∈ Zn


The mixed-integer model

Mixed-integer infinite group relaxation b ∈ Rn \ Zn, s : Rn → R+

and y : Rn → Z+

Mb = {s, y ∈ R(Rn)+ × R

(Rn)+ :

∑

r∈Rn

rs(r) +∑

p∈Rn

py(p) ∈ b + Zn}.

R(Rn) is the set of finite support functions from Rn to R. R(Rn)+ .


The mixed-integer model

Mixed-integer infinite group relaxation b ∈ Rn \ Zn, s : Rn → R+

and y : Rn → Z+

Mb = {s, y ∈ R(Rn)+ × R

(Rn)+ :

∑

r∈Rn

rs(r) +∑

p∈Rn

py(p) ∈ b + Zn}.

R(Rn) is the set of finite support functions from Rn to R. R(Rn)+ .

(ψ, π, α), ψ, π : Rn → R, α ∈ R,

Hψ,π,α :=

{

(s, y) ∈ R(Rn) × R(Rn) :∑

r∈Rn

ψ(r)s(r) +∑

p∈Rn

π(p)y(p) ≥ α

}

(ψ, π, α) is a valid tuple (functions) for Mb if Mb ⊆ Hψ,π,α.equivalently: conv(Mb). α ∈ {−1, 0, 1}.Amitabh Basu, MC, Marco Di Summa, Joseph Paat. The structure of the infinite models in integer programming

A face: The pure integer model

The pure integer infinite group relaxation y : Rn → Z+.

Ib = {y : (0, y) ∈ Mb} = {y ∈ R(Rn)+ :

∑

p∈Rn

py(p) ∈ b + Zn}.


A face: The pure integer model

The pure integer infinite group relaxation y : Rn → Z+.

Ib = {y : (0, y) ∈ Mb} = {y ∈ R(Rn)+ :

∑

p∈Rn

py(p) ∈ b + Zn}.

(π, α), π : Rn → R, α ∈ R,

Hπ,α :=

{

y ∈ R(Rn) :∑

p∈Rn

π(p)y(p) ≥ α

}

(π, α) is a valid tuple for Ib if Ib ⊆ Hπ,α.


Gomory functionsn = 1, α = 1, very simple to describe (0 < b < 1).

ψ(r) =

{

rb

if r ≥ 0−r1−b

if r < 0π(p) =

{

f (p)b

if f (p) ≤ b1−f (p)1−b

if f (p) > b



ψ(r) =

{

rb

if r ≥ 0−r1−b

if r < 0π(p) =

{

f (p)b

if f (p) ≤ b1−f (p)1−b

if f (p) > b

1

b

1

b 1



ψ(r) =

{

rb

if r ≥ 0−r1−b

if r < 0π(p) =

{

f (p)b

if f (p) ≤ b1−f (p)1−b

if f (p) > b

1

b

1

b 1

Subadditive if φ(r1) + φ(r2) ≥ φ(r1 + r2), r1, r2 ∈ Rn. (ψ, π)



ψ(r) =

{

rb

if r ≥ 0−r1−b

if r < 0π(p) =

{

f (p)b

if f (p) ≤ b1−f (p)1−b

if f (p) > b

1

b

1

b 1

Subadditive if φ(r1) + φ(r2) ≥ φ(r1 + r2), r1, r2 ∈ Rn. (ψ, π)Positively homogenous if φ(λr) = λφ(r) r ∈ Rn and λ ≥ 0. (ψ)



ψ(r) =

{

rb

if r ≥ 0−r1−b

if r < 0π(p) =

{

f (p)b

if f (p) ≤ b1−f (p)1−b

if f (p) > b

1

b

1

b 1

Subadditive if φ(r1) + φ(r2) ≥ φ(r1 + r2), r1, r2 ∈ Rn. (ψ, π)Positively homogenous if φ(λr) = λφ(r) r ∈ Rn and λ ≥ 0. (ψ)Sublinear subadditive + positive homogenous. (ψ)



ψ(r) =

{

rb

if r ≥ 0−r1−b

if r < 0π(p) =

{

f (p)b

if f (p) ≤ b1−f (p)1−b

if f (p) > b

1

b

1

b 1

Subadditive if φ(r1) + φ(r2) ≥ φ(r1 + r2), r1, r2 ∈ Rn. (ψ, π)Positively homogenous if φ(λr) = λφ(r) r ∈ Rn and λ ≥ 0. (ψ)Sublinear subadditive + positive homogenous. (ψ)Periodic φ(r) = φ(r + z), r ∈ Rn and z ∈ Zn.(π)



ψ(r) =

{

rb

if r ≥ 0−r1−b

if r < 0π(p) =

{

f (p)b

if f (p) ≤ b1−f (p)1−b

if f (p) > b

1

b

1

b 1

Subadditive if φ(r1) + φ(r2) ≥ φ(r1 + r2), r1, r2 ∈ Rn. (ψ, π)Positively homogenous if φ(λr) = λφ(r) r ∈ Rn and λ ≥ 0. (ψ)Sublinear subadditive + positive homogenous. (ψ)Periodic φ(r) = φ(r + z), r ∈ Rn and z ∈ Zn.(π)Symmetry condition φ satisfies φ(r) + φ(b − r) = 1. (π)Amitabh Basu, MC, Marco Di Summa, Joseph Paat. The structure of the infinite models in integer programming

Attractiveness of valid functions: Plug and play

−3.6s1 + 1.7s2 + .2y1 − .2y2 = .4 + Z

s1, s2 ∈ R+, y1, y2 ∈ Z+



−3.6s1 + 1.7s2 + .2y1 − .2y2 = .4 + Z

s1, s2 ∈ R+, y1, y2 ∈ Z+

....not so easy as it seems, but attractive nevertheless.



−3.6s1 + 1.7s2 + .2y1 − .2y2 = .4 + Z

s1, s2 ∈ R+, y1, y2 ∈ Z+

....not so easy as it seems, but attractive nevertheless.

....Library of useful functions in IP solvers.


What are the ”important” functions?

What should we put in our library?




Nontrivial: Not valid for R(Rn)+ × R

(Rn)+ .

Minimal: Undominated in R(Rn)+ × R

(Rn)+ .

Extreme: Later.Facet: Later.





(Rn)+ .


(Rn)+ .


A (seemingly) technical detour: π ≥ 0 Why? There are minimalfunctions π 6≥ 0, but they are pathological: Every disc containssome (x , f (x)).





(Rn)+ .


(Rn)+ .


A (seemingly) technical detour: π ≥ 0 Why? There are minimalfunctions π 6≥ 0, but they are pathological: Every disc containssome (x , f (x)).

...but ignorance should not be an excuse.... We try to answer this.



Theorem (Yildiz and Cornuejols, related to Johnson.)

Let ψ : Rn → R, π : Rn → R be any functions, and α ∈ {−1, 0, 1}.Then (ψ, π, α) is a nontrivial minimal valid tuple for Mb if andonly if:

◮ π is subadditive;

◮ ψ(r) = supǫ>0π(ǫr)ǫ

= limǫ→0+π(ǫr)ǫ

= lim supǫ→0+π(ǫr)ǫ

forevery r ∈ Rn; sublinear

◮ π is Lipschitz continuous with Lipschitz constantL := max‖r‖=1 ψ(r);

◮ π ≥ 0, π(z) = 0 for every z ∈ Zn, and α = 1;

◮ π satisfies the symmetry condition (and is periodic).


(One of our) Goal(s)

Qb = R(Rn)+ × R

(Rn)+

⋂

(ψ,π,α) valid

Hψ,π,α

= R(Rn)+ × R

(Rn)+

⋂

(ψ,π,α) minimal, nontrivial

Hψ,π,α


(One of our) Goal(s)

Qb = R(Rn)+ × R

(Rn)+

⋂

(ψ,π,α) valid

Hψ,π,α

= R(Rn)+ × R

(Rn)+

⋂

(ψ,π,α) minimal, nontrivial

Hψ,π,α

Clearly conv(Mb) ⊆ Qb. But what is Qb?


Norms, closed sets...

While in finite dimensions all norms are equivalent to the Euclideannorm, In infinite dimensions this is not so.....Norm on R(Rn) × R(Rn) (BCCZ):

|(s, y)|∗ := |s(0)| +∑

r∈Rn

‖r‖|s(r)| + |y(0)|+∑

p∈Rn

‖p‖|y(p)|




|(s, y)|∗ := |s(0)| +∑

r∈Rn

‖r‖|s(r)| + |y(0)|+∑

p∈Rn

‖p‖|y(p)|

TheoremUnder the topology induced by |(·, ·)|∗,

Qb = cl(conv(Mb)) = conv(Mb) + R(Rn)+ × R

(Rn)+ .




|(s, y)|∗ := |s(0)| +∑

r∈Rn

‖r‖|s(r)| + |y(0)|+∑

p∈Rn

‖p‖|y(p)|

TheoremUnder the topology induced by |(·, ·)|∗,

Qb = cl(conv(Mb)) = conv(Mb) + R(Rn)+ × R

(Rn)+ .

conv(Mb) ( conv(Mb) +R(Rn)+ × R(Rn)


Liftable functions

Gb = {y ∈ R(Rn) : (0, y) ∈ Qb}(Minimal, nontrivial) (π, α) valid for Ib liftable if ∃ ψ s.t. (ψ, π, α)(Minimal, nontrivial) valid for Mb.

Gb = R(Rn)+ ∩

⋂

{Hπ,α : (π, α) minimal nontrivial liftable tuple}.


Liftable functions


Gb = R(Rn)+ ∩

⋂


TheoremGb = cl(conv(Ib)) = conv(Ib) + R

(Rn)+ .


Liftable functions


Gb = R(Rn)+ ∩

⋂


TheoremGb = cl(conv(Ib)) = conv(Ib) + R

(Rn)+ .

conv(Ib) ( conv(Ib) + R(Rn)+


Canonical faces, Finite faces

Ultimately we want valid inequalities for Integer Programs (withrational data).Given R ,P ⊆ Rn, let

VR,P ={

(s, y) ∈ R(Rn) × R(Rn) : s(r) = 0 ∀r 6∈ R , y(p) = 0 ∀p 6∈ P}

.

A canonical face of conv(Mb) is F = conv(Mb) ∩ VR,P . When R ,P finite, F is a finite canonical face.




VR,P ={


.


Notice: conv(Mb) =⋃

F finite canonical face




VR,P ={


.


Notice: conv(Mb) =⋃

F finite canonical face

Same for conv(Ib). Finite canonical faces of conv(Ib) are thecorner polyhedra.


Rational finite faces

What happens when R , P ⊂ Qn?

TheoremLet R ,P ⊂ Qn. Then conv(Mb) ∩ VR,P = cl(conv(Mb)) ∩ VR,P .Let P ⊂ Qn. Then conv(Ib) ∩ VP = cl(conv(Ib)) ∩ VP .

Corollary

The restrictions of the minimal, nontrivial valid tuples give all the(nontrivial) facets of rational mixed-integer polyhedra.The restrictions of the minimal, nontrivial liftable functions give allthe (nontrivial) facets of rational corner polyhedra,


Extreme functions and facets

π ≥ 0 extreme if (π, 1) valid and π1 = π2 = π for every π1 ≥ 0,π2 ≥ 0 such that (π1, 1) (π2, 1) valid and π = 1

2π1 +12π2.

π ≥ 0 facet if (π, 1) valid and π1 = π for every π1 ≥ 0, such thatHπ1,1 ∩ Ib ⊂ Hπ1,1 ∩ Ib.

π facet → π extreme. The converse is not known.

Koppe and Zhou: Coincide for the case of continuous piecewiselinear functions On the notions of facets, weak facets, and extremefunctions of the Gomory-Johnson infinite group problem n = 1,

software to test extremality of piecewise linear functions.


Discontinuous extreme functions (n = 1)

Dey, Richard, Li and Miller: The following function is extreme:n = 1, 0 < b < 1

2 ,

π : π(r) =

{

rb

0 ≤ r ≤ br

1+bb < r < 1

1

b 1


More discontinuous functions (n = 1)

Letchford, Lodi ”Strong fractional functions” Minimal, dominatefractional functions.Dash, Gunluk ”Extended two-step MIR” (mixed integer rounding)functions. limit of sequences of two-step MIR functions, dominateLetcLo.Hildebrand , ”two-sided discontinuous at the origin with 1 or 2slopes”, extremeKoppe, Zhou: Extreme functions that are continuous but notLipschitz continuous.

see Koppe, Zhou Equivariant perturbation in Gomory andJohnson’s infinite group problem. vi. the curious case of two-sideddiscontinuous functions.


Not all extreme functions are needed

Summarizing our results:

cl(conv(Ib)) = conv(Ib) + R(Rn)+ =

= R(Rn)+ ∩⋂{Hπ,α : (π, α) minimal nontrivial liftable tuple}.

When P ⊆ Qn, we have that cl(conv(Ib)) ∩ VP = conv(Ib) ∩ VP

(π, α) minimal nontrivial liftable tuple:→ π ≥ 0, α = 1, πLipschitz continuous.

ONLY THESE ARE NEEDED


Cauchy equation and Hamel bases

The Cauchy functional equation in Rn:

θ(u) + θ(v) = θ(u + v) for all u, v ∈ Rn.

(subadditivity) θ(x) = cT x is obviously a solution to the equation.

A Hamel basis B in a basis of Rn over the field Q. i.e. a subset ofRn s.t. ∀x ∈ Rn, there exists a unique finite subset{β1, . . . , βt} ⊆ B and λ1, . . . , λt ∈ Q such that x =

∑ti=1 λiβi .

(axiom of choice).


Cauchy equation and Hamel bases II

For β ∈ B , let c(β) ∈ R be a real number. Define θ as:

θ(x) =∑t

i=1 λic(βi ).

...θ solves the Cauchy equation.

TheoremLet B a Hamel basis of Rn. Then every solution to the Cauchyequation is of this form.


The affine hull of conv(Ib)

Theorem (Basu, Hildebrand Koppe)

The affine hull of conv(Ib) is described by the equations

∑

p∈Rn θ(p)y(p) = θ(b)

for all solutions θ : Rn → R of the Cauchy equation such thatθ(p) = 0 for every p ∈ Qn.

Extreme functions (without π ≥ 0) do not exist.....

aff(cl(conv(Ib))) = R(Rn)

aff(conv(Mb)) = aff(cl(conv(Mb))) = R(Rn) × R(Rn).


Every valid function is nonnegative.

TheoremFor every valid tuple (π, α) for Ib, there exists a unique solution ofthe Cauchy equation θ : Rn → R such that θ(p) = 0 for everyp ∈ Qn and the valid tuple (π′, α′) = (π + θ, α+ θ(b)) satisfiesπ′ ≥ 0.

→ Nonnegative valid functions form a compact, convex set. Itsextreme points are the extreme functions and suffice to describethis set. (.....but not all of them are necessary)


Finite faces and recession cones

√2y1 − .2y2 + (1−

√2)y3 ∈ .4 + Z

y1, y2, y3 ∈ Z+

y1 = y3, (1, 0, 1)



√2y1 − .2y2 + (1−

√2)y3 ∈ .4 + Z

y1, y2, y3 ∈ Z+

y1 = y3, (1, 0, 1)

L the linear space parallel to aff(conv(Ib))

TheoremFor every P ⊆ Rn finite:

◮ the face CP = conv(Ib) ∩ VP is a rational polyhedron in RP ;

◮ every extreme ray of CP is spanned by some r ∈ ZP+ such that

∑

p∈P pr(p) ∈ Zn;

◮ rec(CP) = (L ∩ VP) ∩ RP+.



TheoremThere are finite canonical faces of conv(Mb) that are not closed.All the finite canonical faces of conv(Ib) are rational polyhedra.


More work?n = 1: EVERY EXTREME FUNCTION π ≥ 0 IS ”NICE”.

Gomory Johnson: Every extreme function is piecewise linear. NOBasu, Conforti, Cornuejols, Zambelli.




Dey and Richard Aussois 2008 Construct extreme functions thatare piecewise linear and have > 4 slopes. YES

Hildebrand (2013) 6Koppe and Zhou (2015) 28. Computer search.BCDP (2015) For every k there exists an extreme function that ispiecewise linear with k slopes. The pointwise limit of this sequenceis extreme with ∞ slopes.




Dey and Richard Aussois 2008 Construct extreme functions thatare piecewise linear and have > 4 slopes. YES

Hildebrand (2013) 6Koppe and Zhou (2015) 28. Computer search.BCDP (2015) For every k there exists an extreme function that ispiecewise linear with k slopes. The pointwise limit of this sequenceis extreme with ∞ slopes.

Is every ”bad” function (discontinuous, non piecewise linear, ∞slopes) the pointwise limit of a sequence of ”good” functions?Amitabh Basu, MC, Marco Di Summa, Joseph Paat. The structure of the infinite models in integer programming

Maybe ”nice” functions suffice....

Is every facet of conv(Ib) ∩ VP , P finite (rational) the restriction ofa piecewise linear function?


Maybe ”nice” functions suffice....

Is every facet of conv(Ib) ∩ VP , P finite (rational) the restriction ofa piecewise linear function?

conv(Ib) = cl(conv(Ib)) ∩ aff(conv(Ib)) ?


THANK YOU FOR YOUR ATTENTION



Documents

The structure of the infinite models in integer …Approach in Mixed Integer Programming: Theory, Computation and Perspectives, Fifty years of Integer Programming 1958-2008: From early