Cone fields and the cone projection method of designing signal settings and prices for transportation networks

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Published copy (DOI)



Publication details

Title of host publicationTRANSPORTATION PLANNING
DatePublished - 2002
Number of pages15
Place of PublicationDORDRECHT
EditorsM Patriksson, M Labbe
Original languageEnglish
ISBN (Electronic)978-0-306-48220-5
ISBN (Print)1-4020-0546-6

Publication series

NameApplied Optimization
ISSN (Print)1384-6485


This paper builds on ideas in Smale [13] and Smith et. al. [11, 12]. The paper utilises Smale’s cone fields rather than vector fields to impel disequilibrium steady state traffic-price-green-time distributions; and applies these ideas to the design of steady state signal controls and prices on transportation networks. The work is applied within a multi-modal equilibrium transportation model which contains elastic demands and deterministic choices. The model may readily be extended to include some stochastic route-choice or mode choice. Capacity constraints and queueing delays are permitted; and signal green-times and prices are explicitly included. The paper shows that, under natural linearity and monotonicity conditions, for fixed control parameters the set of equilibria is the intersection of convex sets. Using this result the paper outlines a cone field method of calculating equilibria and also an associated method of designing appropriate values for the control parameters; taking account of travellers’ choices by supposing that the network is in equilibrium. The method is shown to apply to certain non-linear monotone problems by linearising about a current point.
A rigorous proof of convergence to the set of equilibria is provided, for linear and some non-linear monotone problems. But only an outline of a potential proof of convergence to a (flow, control) pair which satisfies a Karush-Kuhn-Tucker necessary condition for local optimality is provided.

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