.
...
Quasi-Lov´ asz extensions and their symmetric counterparts
Miguel Couceiro
Joint work with Jean-Luc Marichal
University of Luxembourg
Order simplexes
Letσbe a permutation on [n] ={1,. . .,n} (σ∈Sn) Rnσ =
{
x= (x1,. . .,xn)∈Rn : xσ(1)6· · ·6xσ(n)
}
[0, 1]nσ = Rnσ∩[0, 1]n
Example : n=2 (2!=2 permutations ⇒ 2 simplexes!)
- 6
x16x2
x1>x2
In general: The hypercube [0, 1]n has exactlyn!simplexes, and
Order simplexes
Letσbe a permutation on [n] ={1,. . .,n} (σ∈Sn) Rnσ =
{
x= (x1,. . .,xn)∈Rn : xσ(1)6· · ·6xσ(n)
}
[0, 1]nσ = Rnσ∩[0, 1]n
Example : n=2 (2!=2 permutations ⇒ 2 simplexes!)
- 6
x16x2
x1>x2
In general: The hypercube [0, 1]n has exactlyn!simplexes, and each simplex [0, 1]nσ has exactlyn+1 vertices.
Order simplexes
Letσbe a permutation on [n] ={1,. . .,n} (σ∈Sn) Rnσ =
{
x= (x1,. . .,xn)∈Rn : xσ(1)6· · ·6xσ(n)
}
[0, 1]nσ = Rnσ∩[0, 1]n
Example : n=2 (2!=2 permutations ⇒ 2 simplexes!)
- 6
x16x2
x1>x2
In general: The hypercube [0, 1]n has exactlyn!simplexes, and
Lov´ asz extension
Letφ:{0, 1}n→R be a (pseudo-Boolean) function s. t. φ(0) =0.
.Definition (Lov´asz, 1983) ..
...
TheLov´asz extension of φ:{0, 1}n → R is the function fφ: Rn → R whose restriction to eachRnσ is the unique linear function which coincides withφat then+1 vertices of the simplex[0, 1]nσ
In particular: fφ|{0,1}n =φ
Lov´ asz extension
Example :
- 6
r r
r r
φ(0, 0) =0 φ(1, 0) =1 φ(0, 1) =5 φ(1, 1) =3
x1>x2 ⇒ fφ(x1,x2) =x1+2x2 x16x2 ⇒ fφ(x1,x2) =−2x1+5x2
OnR2 :
fφ(x1,x2) = x1+5x2−3 min(x1,x2)
Representations of Lov´ asz extensions
In general: fφ can always be written in the form fφ(x) =
∑
S⊆[n]
aφ(S) min
i∈Sxi (x∈Rn)
where the coefficientsaφ(S) are given by theM¨obius transformof φ
Consequence: fφ is always piecewise linear and continuous!
Representations of Lov´ asz extensions
... and on each order simplexRnσ ?
In general :
fφ(x) =xσ(1)φ(1) +
∑
n i=2(xσ(i)−xσ(i−1))φ(1{σ(i),...,σ(n)}) (x∈Rnσ)
Choquet integral
f:Rn→Ris aLov´asz extensionif there is φ:{0, 1}n→Rs.t. f =fφ.
.Definition ..
...AChoquet integralis a nondecreasing Lov´asz extension (vanishing at 0).
Generalization: Quasi-Lov´ asz extensions
LetI be a real interval containing 0.
.Definition ..
...
Aquasi-Lov´asz extensionis a functionf:In→Rdefined by f = L◦ϕ = L◦(ϕ,. . .,ϕ),
where .
1.. L:Rn→Ris a Lov´asz extension .
2.. ϕ:I →Ris a nondecreasing function satisfying ϕ(0) =0.
In DMU: ϕis a utility function andf an overall preference functional.
Symmetric Lov´ asz extension
Forx∈Rn, set x+ =x∨0 and x−= (−x)+
.Definition ..
...
Thesymmetric Lov´asz extensionof φ:{0, 1}n→Ris defined by fˇφ(x) = fφ(x+) − fφ(x−)
For a (nonsymmetric) Lov´asz extension:
fφ(x) = fφ(x+) − fφd(x−)
where φd(1A) =φ(1)−φ(1−1A) =φ(1)−φ(1[n]\A).
Symmetric Lov´ asz extension
Forx∈Rn, set x+ =x∨0 and x−= (−x)+
.Definition ..
...
Thesymmetric Lov´asz extensionof φ:{0, 1}n→Ris defined by fˇφ(x) = fφ(x+) − fφ(x−)
For a (nonsymmetric) Lov´asz extension:
fφ(x) = fφ(x+) − fφd(x−)
where φd(1A) =φ(1)−φ(1−1A) =φ(1)−φ(1[n]\A).
Symmetric Lov´ asz extension
Forx∈Rn, set x+ =x∨0 and x−= (−x)+
.Definition ..
...
Thesymmetric Lov´asz extensionof φ:{0, 1}n→Ris defined by fˇφ(x) = fφ(x+) − fφ(x−)
Immediate consequences:
• fˇφ is piecewise linear and continuous
• fˇφ(cx) = cfˇφ(x)for everyc∈R
• Sipoˇˇ s (1979): Asymmetric Choquet integralis a nondecreasing symmetric Lov´asz extension
Symmetric Lov´ asz extension
Example :
- 6
r r
r r
φ(0, 0) =0 φ(1, 0) =1 φ(0, 1) =5 φ(1, 1) =3
fφ(x ,x ) =x +5x −3 min(x ,x ) fˇφ(x) =fφ(x+)−fφ(x−)
Symmetrizing quasi-Lov´ asz extensions
LetI be a real interval centered at 0: −x∈I wheneverx∈I.
.Definition ..
...
Asymmetric quasi-Lov´asz extensionis a functionf:In→Rdefined by f = Lˇ◦ϕ = Lˇ◦(ϕ,. . .,ϕ)
.
1.. Lˇ:Rn→Ris a symmetric Lov´asz extension .
2.. ϕ:I →Ris a nondecreasingoddfunction.
Comonotonic modularity
x,x0 ∈Inarecomonotonicif x,x0 ∈Iσn = In∩Rnσ for some σ∈Sn. .Definition
..
...
f:In→Riscomonotonically modular if for all comonotonicx,x0∈In f(x) +f(x0) = f(x∨x0) +f(x∧x0)
.
1.. If f(0) =0, then
f(x) = f(x+) + f(−x−) (takex0 =0)
.
2.. If f is odd, then
f(x) = f(x+) −f(x−)
Comonotonic modularity
x,x0 ∈Inarecomonotonicif x,x0 ∈Iσn = In∩Rnσ for some σ∈Sn. .Definition
..
...
f:In→Riscomonotonically modular if for all comonotonicx,x0∈In f(x) +f(x0) = f(x∨x0) +f(x∧x0)
.
1.. If f(0) =0, then
f(x) = f(x+) + f(−x−) (takex0 =0)
.
2.. If f is odd, then
f(x) = f(x+) −f(x−)
Comonotonic modularity
x,x0 ∈Inarecomonotonicif x,x0 ∈Iσn = In∩Rnσ for some σ∈Sn. .Definition
..
...
f:In→Riscomonotonically modular if for all comonotonicx,x0∈In f(x) +f(x0) = f(x∨x0) +f(x∧x0)
.
1.. If f(0) =0, then
f(x) = f(x+) + f(−x−) (takex0 =0)
.
2.. If f is odd, then
f(x) = f(x+) −f(x−)
Complete description of comonotonically modular functions
For 0∈I ⊆R, let I−=I∩R− and I+=I ∩R+
.Theorem: For anyf:In→R s.t. f(0) =0, T.F.A.E.:
..
. ...
.
1.. f is comonotonically modular.
.
2.. There are comonotonically modularg:I+n →Randh:I−n →R s.t.
f(x) = g(x+) +h(−x−).
Furthermore: For everyx∈Iσn s.t. xσ(p)<06xσ(p+1)
g(x+) =
∑
p+16i6n
(g(xσ(i)1{σ(i),...,σ(n)}) − g(xσ(i)1{σ(i+1),...,σ(n)}))
h(−x−) =
∑
16i6p
(h(xσ(i)1{σ(1),...,σ(i)}) − h(xσ(i)1{σ(1),...,σ(i−1)}))
In this case: We can choose g =f|I+n and h=f|I−n.
Complete description of comonotonically modular functions
For 0∈I ⊆R, let I−=I∩R− and I+=I ∩R+
.Theorem: For anyf:In→R s.t. f(0) =0, T.F.A.E.:
..
.
1.. f is comonotonically modular.
.
2.. There are comonotonically modularg:I+n →Randh:I−n →R s.t.
f(x) = g(x+) +h(−x−).
Furthermore: For everyx∈Iσn s.t. xσ(p)<06xσ(p+1)
g(x+) =
∑
p+16i6n
(g(xσ(i)1{σ(i),...,σ(n)}) − g(xσ(i)1{σ(i+1),...,σ(n)}))
h(−x−) =
∑
16i6p
(h(xσ(i)1{σ(1),...,σ(i)}) − h(xσ(i)1{σ(1),...,σ(i−1)}))
In this case: We can choose g =f| n and h=f| n.
Complete description of comonotonically modular functions
For 0∈I ⊆R, let I−=I∩R− and I+=I ∩R+
.Theorem: For anyf:In→R s.t. f(0) =0, T.F.A.E.:
..
...
.
1.. f is comonotonically modular.
.
2.. There are comonotonically modularg:I+n →Randh:I−n →R s.t.
f(x) = g(x+) +h(−x−).
Furthermore: For everyx∈Iσn s.t. xσ(p)<06xσ(p+1)
g(x+) =
∑
p+16i6n
(g(xσ(i)1{σ(i),...,σ(n)}) − g(xσ(i)1{σ(i+1),...,σ(n)}))
h(−x−) =
∑
16i6p
(h(xσ(i)1{σ(1),...,σ(i)}) − h(xσ(i)1{σ(1),...,σ(i−1)}))
In this case: We can choose g =f|I+n and h=f|I−n.
Comonotonic modularity ⇔ comonotonic separability
.Corollary:
..
...
f is comonotonically modular iff it iscomonotonically separable:
for every σ∈Sn, there are fiσ:I →R, i∈[n], s.t.
f(x) =
∑
n i=1fiσ(xσ(i)) for x∈In∩Rnσ.
Axiomatization of symmetric quasi-Lov´ asz extensions
.Definition ..
...
f:In→Risoddly homogeneousif there is a nondecreasing odd function ϕ:I →R s.t. for every x∈I andA⊆[n]
f(x1A) = ϕ(x)f(1A)
.Theorem: Assume thatI is centered at 0 with[−1, 1]⊆I ⊆R...
..
...
Iff:In→Ris nonconstant, then T.F.A.E:
.
1.. f is symmetric quasi-Lov´asz with f(1A)6=0 for someA⊆[n]. .
2.. f is comonotonically modular and oddly homogeneous.
Moreover: f =Lˇf|
{0,1}n◦ϕf where ϕf(x) = ff(x1(1A)
A).
Axiomatization of symmetric quasi-Lov´ asz extensions
.Definition ..
...
f:In→Risoddly homogeneousif there is a nondecreasing odd function ϕ:I →R s.t. for every x∈I andA⊆[n]
f(x1A) = ϕ(x)f(1A)
.Theorem: Assume thatI is centered at 0 with[−1, 1]⊆I ⊆R...
..
...
Iff:In→Ris nonconstant, then T.F.A.E:
.
1.. f is symmetric quasi-Lov´asz with f(1A)6=0 for someA⊆[n]. .
2.. f is comonotonically modular and oddly homogeneous.
Moreover: f =Lˇf|
{0,1}n◦ϕf where ϕf(x) = ff(x1(1A)
A).
Axiomatization of symmetric quasi-Lov´ asz extensions
.Definition ..
...
f:In→Risoddly homogeneousif there is a nondecreasing odd function ϕ:I →R s.t. for every x∈I andA⊆[n]
f(x1A) = ϕ(x)f(1A)
.Theorem: Assume thatI is centered at 0 with[−1, 1]⊆I ⊆R...
..
...
Iff:In→Ris nonconstant, then T.F.A.E:
.
1.. f is symmetric quasi-Lov´asz with f(1A)6=0 for someA⊆[n]. .
2.. f is comonotonically modular and oddly homogeneous.
Moreover: f =Lˇf|
{0,1}n◦ϕf where ϕf(x) = ff(x1(1A)
A).
Final remarks
.
1.. Fornonsymmetricquasi-Lov´asz extensions “odd homogeneity” is replaced by: there is ϕ:I →Rnondecreasing s.t.
f(x1A) = sign(x)ϕ(x)f(sign(x)1A)
.
2.. ForsymmetricandnonsymmetricLov´asz extensions by:
f(x1A) =x f(1A) and f(x1A) =sign(x)x f(sign(x)1A)
.
3.. Conditionf(0) =0 can be dropped off: f0 = f −f(0)
