Wardrop vs Nesterov
traffic equilibrium concept.
Georg Still
University of Twente
joint work with Walter Kern
(9th International Conference on Operations Research,
Havana, February, 2010)
p 1/24
Dutch Highway System
p 2/24
Traffic Network:
• (sw , tw ) :
• dw :
• xe , fp :
• ce (x) ∈ C :
V: nodes; E: edges (roads)
origin-destination nodes, w ∈ W
traffic demands (cars/hour)
edge-, path-flow (cars/hour)
travel time (“costs”) on edge e ∈ E
f1
s1
t1
f2
e x
e
p 3/24
• Pw : set
P of (sw , tw )-paths p , P = ∪w Pw
cp (x) = e∈p ce (x), p ∈ P: path costs
feasible flow: (x, f ) ∈ RE × RP satisfying
Λf
∆f − x
f
= d
= 0
≥ 0
| Λ path-demand incidence| ∆ path-edge incidence matrix
Notation: given demand d
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(x, f ) ∈ Fd : feasible set
I
x ∈ Xd : projection of Fd onto x-space.
p 4/24
Wardrop Equilibrium (52): (x, f ) ∈ Fd is WE if
∀w ∈ W , p, q ∈ Pw
fp > 0 ⇒
cp (x) = cq (x)
cp (x) ≤ cq (x)
if fq > 0
if fq = 0
Meaning: For each used path p ∈ Pw between
O-D pairs (sw , tw ) the path-costs must be the same.
“traffic user equilibrium”, (Nash-equilibrium)
p 5/24
Relation: Wardrop-equilibrium ↔ optimization
Assume ce (x) = ce (xe ), increasing. Consider the program
P:
min N(x) :=
x,f
XZ
e∈E
0
xe
ce (t)dt
s.t. (x, f ) ∈ Fd
KKT conditions: (x, f ) ∈ Fd is sol. of P iff
c(x) = λ
0 = ΛT γ − ∆T λ + µ
T
f µ = 0
f, µ ≥ 0
or equivalentely: for any path p ∈ Pw
½
= cp (x)
T
γw = [∆ c(x)]p − µp
≤ cp (x)
if fp > 0
if fp = 0
These are the W-equilibrium conditions.
p 6/24
Th.1 The following are equivalent for x ∈ Xd
(i)
(ii)
(iii)
(iv)
I
x is an W-equilibrium flow.
c(x)T (x − x) ≥ 0 ∀x ∈ Xd .
x solves min{c(x)T x | x ∈ Xd }.
[in case ce (x) = ce (xe )] x minimizes N(x) on Xd
More generally, (iv) holds if there exists N(x) such that
∇N(x) = c(x)
By Poincaré’s Lemma this holds (on convex sets) if:
c(x) ∈ C 1 and
∂ci
∂xj
=
∂cj
∂xi
Existence of a Wardrop equilibrium x ?
I
I
case c(x) = ∇N(x):
general case:
By the Weierstrass Theorem
By a Fixed Point Theorem
p 7/24
Existence Theorem: (Stampacchia 1966) Let c : X → Rm
be continuous on the convex, compact set X ⊂ Rm . Then
there exists a vector x ∈ X such that
c(x)T (x − x) ≥ 0 ∀x ∈ X
Stampacchia’s Lemma ←→ Brouwer’s Fixed Point Theorem
Brouwer’s Fixed Point Theorem:
Let f : X → X be continuous , X ⊂ Rm convex, compact.
Then f has a fixed point x ∈ X : f (x) = x
Pf. “→”: Choose
c(y) := −[f (y) − y].
Then, there exists x ∈ X such that
c(x)T (x − x) = −[f (x) − x]T (x − x) ≥ 0 ∀x ∈ X
Choose x = f (x) ∈ X
−→ −kf (x) − xk2 ≥ 0
−→
f (x) = x
p 8/24
Objectives: user (Nash-eq.)
↔
minimize:
↔
N(x)
t
Braes example:
edge costs:
flow :
s-t demand:
government
P
S(x) := e ce (xe )xe
x
1
1, 1, c,x
x
1
u
v
c
x
1
s
c ≥ 1: Nash flow x,
S(x) = 3/2
p1 = s−u −t ,
f1 = 1/2,
c1 = 3/2
p2 = s−v −t ,
f2 = 1/2,
c2 = 3/2
c = 0: Nash flow x̂,
p = s−u −v −t,
S(x̂) = 2
f = 1,
c1 = 2
p 9/24
Nesterov’s new model (2000):
Based on the “queering model”
½
te for 0 ≤ xe < ue
ce (xe ) =
M for xe = ue
I
ue : max capacity of e ∈ E
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te : costs (travel times) without
congestion (e ∈ E).
modified, generalized concept (with te (x) ∈ C)
½
te (x) for 0 ≤ xe ≤ ue
ce (x) =
M
for xe > ue
This function is lower semicontinuous (lsc).
p 10/24
Def. Nesterov E. (NE): Given costs t : RE+ → RE+ in C, a
, x ∈ Xd is a NE if
capacity vector u, then (x, t) ∈ RE+E
+
1. x ≤ u , t ≥ t(x) and
2. x is a WE relative to the costs t.
Related capacity constr. program: Find x solving
Pt (x) : min t(x)T x
x,f
s.t.
Λf
∆f
− x
x
f
=
=
≤
≥
d
0
u |ν
0
Changes in KKT-condition compared with WE:
t(x) = λ − ν and (u − x)T ν = 0
or eqivalentely for any path p ∈ Pw
γw = [∆T (t(x) + ν)]p − µp
p 11/24
So: consider costs t = t(x) + ν.
Th.2 (x, t) is a NE if and only if x (with L-multiplier ν) is a
solution of Pt (x) and t = t(x) + ν.
I
The existence of a NE follows also by Stampacchia’s
Lemma.
p 12/24
Wardrop’s model for non-continuous costs
Def. lower-, upper limit, ce− , ce+ :
ce− (x) := lim
inf ce (x n )
n
x →x
ce+ (x) := lim sup ce (x n )
Similarly:
x n →x
−
cp (x), cp+ (x)
for pathcosts.
Model conditions: If fp > 0, p ∈ Pw , then:
•
cp (x) ≤ cq+ (x) ∀q ∈ Pw
should be a necessary condition and
•
cp (x) ≤ lim inf cq (x + ε1q − ε1p )
ε↓0
∀q ∈ Pw a sufficient condition for “stability”
p 13/24
This leads to the assumpions:
ce (x) are lower semicontinuous (lsc), i.e.
ce (x) ≤ ce− (x), ∀x
and satisfy the regularity condition: ∀q, p ∈ Pw ,
e ∈ q, e ∈
/p
(?)
ce+ (x) ≤ lim inf ce (x + ε1q − ε1p )
ε↓0
Def. (Wardrop equilibrium:) Suppose the ce ’s are lsc
and satisfy
the link regularity (?). We then call
P
x = p fp 1p ∈ Xd a Wardrop equilibrium if:
fp > 0 ⇒ cp (x) ≤ cq+ (x) ∀q ∈ Pw
p 14/24
Th.3 Let the link costs ce be lsc and satisfy the link
regularity condition. Assume x ∈ Xd and c ∈ [c(x), c + (x)].
Then (iii) ⇔ (ii) ⇒ (i) holds for
(i) x is a Wardrop equilibrium.
(ii) c T (x − x) ≥ 0 ∀x ∈ Xd .
(iii) x solves min{c T x | x ∈ Xd }
Def. A WE satisfying the sufficient condition (ii) (or (iii)) of
the theorem is called a strong WE.
Th.4 For lsc regular link costs strong Wardrop equilibria
exist.
Pf. Use cek (x) ↑ ce (x) with continuous cek (x) and the
existence of WE wrt. cek (x).
p 15/24
NE as special case of WE: NE is based on costs,
½
(?)
ce (x) =
te (x) for 0 ≤ xe ≤ ue
M
for xe > ue
This function is lsc and regular. By comparing the
KKT-conditions for a strong WE wrt. the costs (?):
0 = ΛT γ − ∆T λ + µ
½
for x e < ue
= te (x)
+
c ∈ [c(x), c (x)] , c e
∈ [te (x), M] if x e = ue
c=λ
and
with the KKT-conditions for the NE-program Pt (x) we
directly find:
Cor.1 (x, t) is a Nesterov equilibrium (wrt. te (x) and u)
if and only if x is a strong WE (wrt. ce (x) in (?)).
p 16/24
Parametric Aspects:
How do the equilibrated travel times γw (·) depend on
changes in the demand d and/or costs ce (x)?
Dependence on c(x):
I
c(x) % ;
No monotonicity
γw % (see Braes)
p 17/24
Dependence on d: Let N(x) be convex with ∇N(x) = c(x),
i.e., c(x) satisfies the “monotonicity” condition
(?)
[c(x 0 ) − c(x)]T (x 0 − x) ≥ 0 ∀x 0 , x
Consider the W-equilibrium problem: d parameter
P(d) :
minx,f N(x) s.t. (x, f ) ∈ Fd
Λf = d
|γ
Parametric Opt.: For solutions x(d) with L-Mult. γ(d):
I the value function v(d) of P(d) is convex (in d).
I ∂v (d) = {γ(d)} (maximal) monotone:
[γ(d) − γ(d)]T (d − d) ≥ 0 ∀d, d
Even if the W-equilibrium cannot be modelled as an
optimization problem: Monotonicity of γ(d) still holds
under (?).
p 18/24
interpretation: of (Hall’s result)
[γ(d) − γ(d)]T (d − d) ≥ 0 ∀d, d
Let d d then
I
γw (d) ≥ γw (d) for at least one w ∈ W
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even if d > d: possibly
γw 0 (d) > γw 0 (d)
for one w 0 ∈ W
γw (d) < γw (d)
for the other w 6= w 0
p 19/24
Pf. of Hall’s result:
solutions x 0 , x corresp. to d 0 , d
[c(x 0 ) − c(x)]T (x 0 − x) ≥ 0 •
[c(x 0 ) − c(x)]T ∆(f 0 − f ) ≥ 0
[µ0 − µ]T (f 0 − f ) + [ΛT γ 0 − ΛT γ]T (f 0 − f ) ≥ 0
|
{z
}
≤0 by 3.
[γ 0 − γ]T Λ(f 0 − f ) ≥ 0
[γ 0 − γ]T (d 0 − d) ≥ 0 •
Use:
1. x = ∆f , Λf = d
2. ∆T c(x) = µ + ΛT γ
3. E.g.: fp0 > 0, fp = 0 ⇒ µ0p = 0, µp ≥ 0
⇒ ( µ0p −µp )(fp0 − fp ) ≤ 0
|{z}
|{z}
=0
=0
p 20/24
Monotonicity of γ(d) in the general W-concept?
For strong W-equilibria: Under the “monotonicity
condition”,
[c(x 0 ) − c(x)]T (x 0 − x) ≥ 0 ∀x 0 , x
monotonicity of the equilibrated travel-times γ(d) still
holds:
[γ(d) − γ(d)]T (d − d) ≥ 0 ∀d, d
However: For (weak) W-equilibria this monotonicity may
fail.
p 21/24
Example network with 4 O-D pairs; identify edge e:
e1
4
e2
e
e3
4
e
1
b1
1
2
b2 2
3
e
3
The link cost for e, ei bj are zero except for
½
0 0≤t ≤2
ce (t) :=
M else
cei (t) = t,
cbj (t) ≡ 1.
demands: d1 = d2 = d3 = 1, d4 = ε ≥ 0.
p 22/24
A weak W-E x and the unique strong equilibrium x:
x : 2
x : 2
2
3
1
+ 13 ε
2
3
1
+ 13 ε
1−ε
− 13 ε
1
3
1
3
ε
+ 23 ε
1
3
ε
+ 23 ε
Corresponding γ, γ
γ: 1 1
γ: 1 1
1−ε
− 13 ε
1
3
3 − ε ←− •
+ 13 ε
5
3
with objective
N(x) =
3
2
+ε+
1
2
ε2
,
N(x) =
7
6
+
5
3
ε+
1
6
ε2 .
p 23/24
Many other interesting aspects:
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Generalization to elastic demand is easy.
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Tolling policy to ’improve’ the traffic flow.
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Computation of traffic equilibria in large networks
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Dynamic traffic equilibrium models
(demand changes with time)
p 24/24