2013/04/30 by Ke Han, Terry L. Friesz, W.Y. Szeto +2 · 53 citations
Engineering · Mathematics · Social Sciences · #Algorithm #Applied mathematics #Computation #Convergence (economics) #Dykstra's projection algorithm #Economics #Fixed point #Fixed-point theorem #Hilbert space #Mathematical analysis #Mathematical optimization #Mathematics #Monotonic function #Projection (relational algebra) #Projection method #Traffic control and management #Transportation Planning and Optimization #Urban Transport and Accessibility #Variational inequality #math.OC #msc:90B06 #msc:90B10 #msc:90B20 #msc:90C90
paper · pdf · doi:10.1016/j.trb.2015.07.008
published in Transportation Research Part B Methodological 81, 183-209 (Elsevier BV) · 32 pages, 6 figures, 2 tables
openalex publication_date 2015/09/24 · arxiv created 2016/03/24 · arxiv updated 2016/03/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
This paper is concerned with dynamic user equilibrium with elastic travel demand (E-DUE) when the trip demand matrix is determined endogenously. We present an infinite-dimensional variational inequality (VI) formulation that is equivalent to the conditions defining a continuous-time E-DUE problem. An existence result for this VI is established by applying a fixed-point existence theorem (Browder, 1968) in an extended Hilbert space. We present three algorithms based on the aforementioned VI and its re-expression as a differential variational inequality (DVI): a projection method, a self-adaptive projection method, and a proximal point method. Rigorous convergence results are provided for these methods, which rely on increasingly relaxed notions of generalized monotonicity, namely mixed strongly-weakly monotonicity for the projection method; pseudomonotonicity for the self-adaptive projection method, and quasimonotonicity for the proximal point method. These three algorithms are tested and their solution quality, convergence, and computational efficiency compared. Our convergence results, which transcend the transportation applications studied here, apply to a broad family of infinite-dimensional VIs and DVIs, and are the weakest reported to date.