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 $ε$, where we think of $ε$ 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 $Δ$ 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 ≤ $ε$) at which Z is stationary.