Chapter 5: Algorithms for the Dynamic Route Choice Model

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Chapter 5Algorithms for the Dynamic Route
Choice Model

• Feike Brandt

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Contents

• The DUO route choice conditions
• The Variational Inequality Problem
• Convex optimization problem
• Relations between problems
• Link relaxations
• Nested diagonalization method
• Example

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The DUO route choice conditions
Find the flows such that where
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The Variational Inequality Problem
Find a solution u?? such that Or, in
expanded form Where ? is a subset of ? with

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Convex optimization problem
Find Where ? is nonemtpy, closed and convex
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Relations between problems

• The DUO route choice problem is equivalent to
  finding a solution u?? such that the VIP holds.
• The VIP can be written as a convex optimization
  problem if
• and the solution u of this convex optimization
  problem is also the solution of the VIP.

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Link relaxations (1)

• Problem 1
• The flow propagation constraints are essentially
  nonlinear and nonconvex.
• Solution
• Fix the actual link traveltimes (i.e. fixing the
  feasible time-space
• network) to obtain a convex feasible set with
  linear constraints.

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Link relaxations (2)

• Problem 2
• The travel time function is asymmetric in nature.
• Solution
• Temporarily fix the previously entered flows in
  the same physical link a
• (Jac(c) is symmetric).
• Cf. the Diagonalization Method (par. 2.3.2.)
• no topological interaction between different
  physical links
• when considering inflow on link a at interval t,
  fix all inflows on the other links at interval t
• This VIP can be solved using the translation to
  an optimization problem

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Nested Diagonalization Method (1)

• Initialization
• Let m0, n1
• Set initial vector of actual travel times
• find a initial feasible solution
• compute the associated link travel time

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Nested Diagonalization Method (2)

• First loop (iteration m)
• mm1
• Update estimated actual link travel times by
• Construct the corresponding feasible time-space
  network based on these link travel times (the
  network is now fixed in time)
• Let n1 and compute a new feasible solution
  based on this link travel times

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Nested Diagonalization Method (3)

• Second loop (iteration n1)
• Define travel time of each space-time link as a
  one-dimensional function of the inflow at this
  link at time interval t only, i.e.
• The Jacobian of the link travel times in a link a
  is symmetric and positive definite, so the VIP
  can be solved as a convex programming problem

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Nested Diagonalization Method (4)

• Third loop
• Solve the linearly contrained convex optimization
  problem, using e.g. the Frank-Wolfe method,
  yields .
• Compute the resulting travel times ,
  using .
• If un1 is not close to un then nn1, go to
  second loop.
• Elseif ?m is not close to cn1 then set nn1, go
  to first loop.
• Otherwise the solution is optimal.

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Nested Diagonalization Method (5)
Optimal solution
m0, n1, find t0, u1, c1
tm ? cn1
mm1, update tm find feasible time-space network
no
set n1, compute
un1 ? un
nn1
no
Solve the linear constrained convex optimization
problem using FW-method Compute
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Example (1)
2
b
c
1
3
a
Time depended O-D matrix (1-gt3)
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Example (2)
Initial traveltimes
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Example (3)
Using All-or- Nothing assignment method leads to
linkflows from node 1 to node 3 directly for the
4 timeperiods Cumputing the associated link
travel times gives
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Example (4)
Traveltimes after assignment 1
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Example (5)
T1
T2
T3
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Example (6)
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Example (7)
Traveltimes after assignment 2
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Example (8)
T1
T2
T3
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Example (9)
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Example (10)
Traveltimes after assignment 3
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Example (11)
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Example (12)
Traveltimes after assignment 4
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Example (13)
Check DUO conditions
FINISHED
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Questions?