Dynamic traffic demand estimation posed as an infinite dimension bilevel programming problem

JARES, NICOLÁS; FERNÁNDEZ, DAMIÁN; PARENTE, LISANDRO A.

Córdoba

Workshop; VII Latin American Workshop on Optimization and Control; 2024

Given a traffic network represented as a directed graph G = (N, A), and a planning horizon T = [t0 , tf ], we assume to know traffic counts that are in the form of a family of functions x0: T → R≥0 , for each a ∈ A. We call x0 = x0 a∈A the vector containing all such functions.With x0 as the input data, we can write the dynamic traffic demand estimation problem as a bilevel programming problem of the form:minimizeh∈H,d∈D∥x0 − X(h)∥22 +∥d∥1subject to (A(h), h − v)H ≤ 0, ∀v ∈ ΛdHere d = (dw )w∈W is the vector of traffic demands, where W is the set of all OD pairs, and each dw is the departure profile from the origin of OD pair w. h = (hr )r∈R is the vector of route flow, where R is the set of all routes between all OD pairs, and H and D are appropriate Hilbert spaces. The operator X : H → L2 (T )|A| returns arc flows from route flows, and the operator A : H → H is the delay operator (the required time or cost to traverse each route on each possible departure time). The constraint of the problem 1 is a variational inequality with (·, ·)H being the usual inner product of H, and Λd ⊂ H being the set of all feasible route flows (those which satisfies the demand and are non-negative). This variational inequality solves the route choice dynamic user equilibrium problem (RC DUE) [1].Under reasonable hypothesis, we can write 1 as an infinite dimensional simple bilevel problem (SBP) with an convex upper level objective function and a convex feasible set. Most current state-of-the-art methods to solve BLP require a strongly convex function as the upper level objective (e.g. [2]). We were able to rely on some of the underlying ideas behind those methods to devise an iterative algorithm that converges to a solution of our problem.