## 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.

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 |
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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 | |

Publication status | Published - 2006 |

### Publication series

Name | Applied Optimization |
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Publisher | Springer |

## Keywords

- Bilevel Optimisation
- Transportation Networks
- Pricing
- Control
- Equilibrium