Bilevel optimisation of prices and signals in transportation models

Mike Smith; Siriphong Lawphongpanich; Donald W. Hearn; Mike Smith

doi:10.1007/0-387-29645-X_8

Bilevel optimisation of prices and signals in transportation models

Mike Smith, Siriphong Lawphongpanich (Editor), Donald W. Hearn (Editor), Mike Smith (Editor)

Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Chapter

Abstract

We suppose given a variable demand model with some control parameters to represent prices, a smooth function V which measures departure from equilibrium and a smooth function Z which measures overall disbenefit. We suppose that we wish to minimise Z subject to the constraint that the disequilibrium function V is no more than e, where we think of e as a small positive number. The paper suggests a simultaneous descent direction to solve this bilevel optimisation problem; such a direction reduces Z and V simultaneously and may often be computed by simply bisecting the angle between -¿Z and -¿V. The paper shows that following a direction ¿ which employs the simultaneous descent direction as its central element leads, under natural conditions which preclude edge effects (where a flow may be zero or a price may be maximum), to the set of those approximate equilibria (where V = e) at which Z is stationary.
Then the method is extended on the one hand to deal with edge effects (allowing a route flow to be zero or a price to be the maximum permitted), by ensuring that the direction ¿ followed anticipates nearby edges of the feasible region, using reduced gradients instead of gradients, and on the other hand to deal with signal controls.
Within the optimisation procedure proposed here, optimisation and equilibration move in parallel and the need to compute a sequence of approximate equilibria is avoided.

Original language	English
Title of host publication	Mathematical and computational models for congestion charging
Publisher	Springer
Pages	159-200
Number of pages	42
Volume	101
ISBN (Electronic)	978-0-387-29645-6
ISBN (Print)	978-0-387-29644-9
DOIs	https://doi.org/10.1007/0-387-29645-X_8
Publication status	Published - 2006

Publication series

Name	Applied Optimization
Publisher	Springer

Keywords

Bilevel Optimisation
Transportation Networks
Pricing
Control
Equilibrium

Access to Document

10.1007/0-387-29645-X_8

Cite this

@inbook{bd42eb004fe54440be08279fba7f3c68,

title = "Bilevel optimisation of prices and signals in transportation models",

abstract = "We suppose given a variable demand model with some control parameters to represent prices, a smooth function V which measures departure from equilibrium and a smooth function Z which measures overall disbenefit. We suppose that we wish to minimise Z subject to the constraint that the disequilibrium function V is no more than e, where we think of e as a small positive number. The paper suggests a simultaneous descent direction to solve this bilevel optimisation problem; such a direction reduces Z and V simultaneously and may often be computed by simply bisecting the angle between -¿Z and -¿V. The paper shows that following a direction ¿ which employs the simultaneous descent direction as its central element leads, under natural conditions which preclude edge effects (where a flow may be zero or a price may be maximum), to the set of those approximate equilibria (where V = e) at which Z is stationary. Then the method is extended on the one hand to deal with edge effects (allowing a route flow to be zero or a price to be the maximum permitted), by ensuring that the direction ¿ followed anticipates nearby edges of the feasible region, using reduced gradients instead of gradients, and on the other hand to deal with signal controls. Within the optimisation procedure proposed here, optimisation and equilibration move in parallel and the need to compute a sequence of approximate equilibria is avoided. ",

keywords = "Bilevel Optimisation , Transportation Networks , Pricing, Control, Equilibrium",

author = "Mike Smith and Siriphong Lawphongpanich and Hearn, {Donald W.} and Mike Smith",

year = "2006",

doi = "10.1007/0-387-29645-X_8",

language = "English",

isbn = "978-0-387-29644-9",

volume = "101",

series = "Applied Optimization",

publisher = "Springer",

pages = "159--200",

booktitle = "Mathematical and computational models for congestion charging",

address = "Germany",

}

TY - CHAP

T1 - Bilevel optimisation of prices and signals in transportation models

AU - Smith, Mike

A2 - Lawphongpanich, Siriphong

A2 - Hearn, Donald W.

A2 - Smith, Mike

PY - 2006

Y1 - 2006

N2 - We suppose given a variable demand model with some control parameters to represent prices, a smooth function V which measures departure from equilibrium and a smooth function Z which measures overall disbenefit. We suppose that we wish to minimise Z subject to the constraint that the disequilibrium function V is no more than e, where we think of e as a small positive number. The paper suggests a simultaneous descent direction to solve this bilevel optimisation problem; such a direction reduces Z and V simultaneously and may often be computed by simply bisecting the angle between -¿Z and -¿V. The paper shows that following a direction ¿ which employs the simultaneous descent direction as its central element leads, under natural conditions which preclude edge effects (where a flow may be zero or a price may be maximum), to the set of those approximate equilibria (where V = e) at which Z is stationary. Then the method is extended on the one hand to deal with edge effects (allowing a route flow to be zero or a price to be the maximum permitted), by ensuring that the direction ¿ followed anticipates nearby edges of the feasible region, using reduced gradients instead of gradients, and on the other hand to deal with signal controls. Within the optimisation procedure proposed here, optimisation and equilibration move in parallel and the need to compute a sequence of approximate equilibria is avoided.

AB - We suppose given a variable demand model with some control parameters to represent prices, a smooth function V which measures departure from equilibrium and a smooth function Z which measures overall disbenefit. We suppose that we wish to minimise Z subject to the constraint that the disequilibrium function V is no more than e, where we think of e as a small positive number. The paper suggests a simultaneous descent direction to solve this bilevel optimisation problem; such a direction reduces Z and V simultaneously and may often be computed by simply bisecting the angle between -¿Z and -¿V. The paper shows that following a direction ¿ which employs the simultaneous descent direction as its central element leads, under natural conditions which preclude edge effects (where a flow may be zero or a price may be maximum), to the set of those approximate equilibria (where V = e) at which Z is stationary. Then the method is extended on the one hand to deal with edge effects (allowing a route flow to be zero or a price to be the maximum permitted), by ensuring that the direction ¿ followed anticipates nearby edges of the feasible region, using reduced gradients instead of gradients, and on the other hand to deal with signal controls. Within the optimisation procedure proposed here, optimisation and equilibration move in parallel and the need to compute a sequence of approximate equilibria is avoided.

KW - Bilevel Optimisation

KW - Transportation Networks

KW - Pricing

KW - Control

KW - Equilibrium

U2 - 10.1007/0-387-29645-X_8

DO - 10.1007/0-387-29645-X_8

M3 - Chapter

SN - 978-0-387-29644-9

VL - 101

T3 - Applied Optimization

SP - 159

EP - 200

BT - Mathematical and computational models for congestion charging

PB - Springer

ER -