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)

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Dutch Highway System

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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

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• 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 I

(x, f ) ∈ Fd : feasible set

I

x ∈ Xd : projection of Fd onto x-space.

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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)

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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) ´ Lemma this holds (on convex sets) if: By Poincare’s 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

ˆ, c = 0: Nash flow x p = s−u −v −t,

ˆ) = 2 S(x

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

I

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).

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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.

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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”

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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

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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).

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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 (?)).

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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)

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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

I

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

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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

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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.

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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.

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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 .

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Many other interesting aspects: I

Generalization to elastic demand is easy.

I

Tolling policy to ’improve’ the traffic flow.

I

Computation of traffic equilibria in large networks

I

Dynamic traffic equilibrium models (demand changes with time)

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