wardrop vs nesterov traffic equilibrium concept.stillgj/lectures/havana.pdfdef. nesterov e. (ne):...
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Wardrop vs Nesterovtraffic equilibrium concept.
Georg StillUniversity 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: V: nodes; E: edges (roads)• (sw , tw) : origin-destination nodes, w ∈ W• dw : traffic demands (cars/hour)• xe, fp : edge-, path-flow (cars/hour)• ce(x) ∈ C : travel time (“costs”) on edge e ∈ E
e xe1
1f1
f2s
t
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• Pw : set of (sw , tw)-paths p , P = ∪w Pwcp(x) =
∑e∈p ce(x), p ∈ P: path costs
feasible flow: (x, f ) ∈ RE × RP satisfying
Λf = d | Λ path-demand incidence-∆f − x = 0 | ∆ path-edge incidence matrixf ≥ 0
Notation: given demand dI (x, f ) ∈ Fd : feasible setI 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) if fq > 0cp(x) ≤ cq(x) if fq = 0
Meaning: For each used path p ∈ Pw betweenO-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 : minx,f
N(x) :=∑
e∈E
∫ xe
0ce(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
f T µ = 0 f , µ ≥ 0
or equivalentely: for any path p ∈ Pw
γw = [∆T c(x)]p − µp
{= cp(x) if fp > 0≤ cp(x) if fp = 0
These are the W-equilibrium conditions.
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Th.1 The following are equivalent for x ∈ Xd
(i) x is an W-equilibrium flow.(ii) c(x)T (x − x) ≥ 0 ∀x ∈ Xd .(iii) x solves min{c(x)T x | x ∈ Xd}.(iv) [in case ce(x) = ce(xe)] x minimizes N(x) on Xd
I More generally, (iv) holds if there exists N(x) such that
∇N(x) = c(x)
By Poincare’s Lemma this holds (on convex sets) if:
c(x) ∈ C1 and∂ci
∂xj=
∂cj
∂xi
Existence of a Wardrop equilibrium x ?
I case c(x) = ∇N(x): By the Weierstrass TheoremI general case: By a Fixed Point Theorem
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Existence Theorem: (Stampacchia 1966) Let c : X → Rm
be continuous on the convex, compact set X ⊂ Rm. Thenthere 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 −→ −‖f (x)− x‖2 ≥ 0
−→ f (x) = x
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Objectives: user (Nash-eq.) ↔ government
minimize: N(x) ↔ S(x) :=∑
e ce(xe)xe
Braes example:
edge costs: 1, 1, c,xflow : x
s-t demand: 1
1
x 1
x
c
s
t
u v
c ≥ 1: Nash flow x , S(x) = 3/2
p1 = s−u−t , f1 = 1/2, c1 = 3/2p2 = s−v−t , f2 = 1/2, c2 = 3/2
c = 0: Nash flow x , S(x) = 2
p = s−u−v−t , f = 1, c1 = 2
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Nesterov’s new model (2000):
Based on the “queering model”
ce(xe) =
{te for 0 ≤ xe < ueM for xe = ue
I ue: max capacity of e ∈ EI te: costs (travel times) without
congestion (e ∈ E).modified, generalized concept (with te(x) ∈ C)
ce(x) =
{te(x) for 0 ≤ xe ≤ ueM for xe > ue
This function is lower semicontinuous (lsc).
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Def. Nesterov E. (NE): Given costs t : RE+ → RE
+ in C, acapacity vector u, then (x, t) ∈ RE+E
+ , x ∈ Xd is a NE if1. x ≤ u , t ≥ t(x) and2. x is a WE relative to the costs t .
Related capacity constr. program: Find x solving
Pt(x) : minx,f
t(x)T x s.t.
Λf = d∆f − x = 0
x ≤ u |νf ≥ 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
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So: consider costs t = t(x) + ν.
Th.2 (x, t) is a NE if and only if x (with L-multiplier ν) is asolution of Pt(x) and t = t(x) + ν.
I The existence of a NE follows also by Stampacchia’sLemma.
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Wardrop’s model for non-continuous costs
Def. lower-, upper limit, c−e , c+
e :c−
e (x) := lim infxn→x
ce(xn)
c+e (x) := lim sup
xn→xce(xn)
Similarly: c−p (x), c+
p (x) for pathcosts.
Model conditions: If fp > 0, p ∈ Pw , then:
• cp(x) ≤ c+q (x) ∀q ∈ Pw
should be a necessary condition and
• cp(x) ≤ lim infε↓0
cq(x + ε1q − ε1p)
∀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) ≤ c−e (x), ∀x
and satisfy the regularity condition: ∀q, p ∈ Pw ,e ∈ q, e /∈ p
(?) c+e (x) ≤ lim inf
ε↓0ce(x + ε1q − ε1p)
Def. (Wardrop equilibrium:) Suppose the ce’s are lscand satisfy the link regularity (?). We then callx =
∑p fp1p ∈ Xd a Wardrop equilibrium if:
fp > 0⇒ cp(x) ≤ c+q (x) ∀q ∈ Pw
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Th.3 Let the link costs ce be lsc and satisfy the linkregularity condition. Assume x ∈ Xd and c ∈ [c(x), c+(x)].Then (iii)⇔ (ii)⇒ (i) holds for
(i) x is a Wardrop equilibrium.(ii) cT (x − x) ≥ 0 ∀x ∈ Xd .(iii) x solves min{cT x | x ∈ Xd}
Def. A WE satisfying the sufficient condition (ii) (or (iii)) ofthe theorem is called a strong WE.
Th.4 For lsc regular link costs strong Wardrop equilibriaexist.
Pf. Use cke (x) ↑ ce(x) with continuous ck
e (x) and theexistence of WE wrt. ck
e (x).
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NE as special case of WE: NE is based on costs,
(?) ce(x) =
{te(x) for 0 ≤ xe ≤ ueM for xe > ue
This function is lsc and regular. By comparing theKKT-conditions for a strong WE wrt. the costs (?):
c = λ and 0 = ΛT γ −∆T λ + µ
c ∈ [c(x), c+(x)] , ce
{= te(x) for xe < ue∈ [te(x), M] if xe = ue
with the KKT-conditions for the NE-program Pt(x) wedirectly 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 onchanges in the demand d and/or costs ce(x)?
Dependence on c(x): No monotonicity
I c(x)↗ ; 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 ′)− c(x)]T (x ′ − x) ≥ 0 ∀x ′, 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 anoptimization problem: Monotonicity of γ(d) still holdsunder (?).
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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 ′(d) > γw ′(d) for one w ′ ∈ W
γw(d) < γw(d) for the other w 6= w ′
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Pf. of Hall’s result: solutions x ′, x corresp. to d ′, d
[c(x ′)− c(x)]T (x ′ − x) ≥ 0 •[c(x ′)− c(x)]T ∆(f ′ − f ) ≥ 0
[µ′ − µ]T (f ′ − f )︸ ︷︷ ︸≤0 by 3.
+ [ΛT γ′ − ΛT γ]T (f ′ − f ) ≥ 0
[γ′ − γ]T Λ(f ′ − f ) ≥ 0[γ′ − γ]T (d ′ − d) ≥ 0 •
Use:
1. x = ∆f , Λf = d 2. ∆T c(x) = µ + ΛT γ
3. E.g.: f ′p > 0, fp = 0⇒ µ′
p = 0, µp ≥ 0⇒ ( µ′
p︸︷︷︸=0
−µp)(f ′p − fp︸︷︷︸
=0
) ≤ 0
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Monotonicity of γ(d) in the general W-concept?
For strong W-equilibria: Under the “monotonicitycondition”,
[c(x ′)− c(x)]T (x ′ − x) ≥ 0 ∀x ′, x
monotonicity of the equilibrated travel-times γ(d) stillholds:
[γ(d)− γ(d)]T (d − d) ≥ 0 ∀d, d
However: For (weak) W-equilibria this monotonicity mayfail.
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Example network with 4 O-D pairs; identify edge e:
44
b1 b21 1 2 2 3 e 3
e2e1 e3
e e
The link cost for e, ei bj are zero except for
ce(t) :=
{0 0 ≤ t ≤ 2M 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 1 1 1− ε ε ε
x : 2 23 + 1
3ε 23 + 1
3ε 13 − 1
3ε 13 + 2
3ε 13 + 2
3ε
Corresponding γ, γ
γ : 1 1 1− ε 3− ε ←− •γ : 1 1 1
3 − 13ε 5
3 + 13ε
with objective
N(x) =32
+ ε +12ε2 , N(x) =
76
+53ε +
16ε2 .
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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 networksI Dynamic traffic equilibrium models
(demand changes with time)
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