Simplicial Decomposition with Disaggregated Representation for the Traffic Assignment Problem

Abstract

The class of simplicial decomposition (SD) schemes have shown to provide efficient tools for nonlinear network flows. When applied to the traffic assignment problem, shortest route subproblems are solved in order to generate extreme points of the polyhedron of feasible flows, and, alternately, master problems are solved over the convex hull of the generated extreme points. We review the development of simplicial decomposition and the closely related column generation methods for the traffic assignment problem; we then present a modified, disaggregated, representation of feasible solutions in SD algorithms for convex problems over Cartesian product sets, with application to the symmetric traffic assignment problem. The new algorithm, which is referred to as disaggregate simplicial decomposition (DSD), is given along with a specialized solution method for the disaggregate master problem. Numerical results for several well known test problems and a new one are presented. These experimentations indicate that only few shortest route searches are needed; this property is important for large-scale applications. The modification proposed maintains the advantages of SD, and the results show that the performance of the new algorithm is at least comparable to that of state-of-the-art codes for traffic assignment. Moreover, the reoptimization capabilities of the new scheme are significantly better; this is a main motive for considering it. The reoptimization facilities, especially with respect to changes in origin-destination flows and network topology, make the new approach highly profitable for more complex models, where traffic assignment problems arise as subproblems.

A generic column generation principle: derivation and convergence analysis

“…Especially, for nonlinear network flow problems, the simplicial decomposition methods have shown to be efficient computational tools (e.g., [38,66,45]). …”

“…Larsson and Patriksson [45] extend the simplicial decomposition strategy to take full advantage of Cartesian product structures, resulting in the disaggregate simplicial decomposition (DSD) algorithm. Ventura and Hearn [89] extend the restricted simplicial decomposition method to convexly constrained problems, and Feng and Li [24] analyze the effect of approximating the restricted master problem by a quadratic one.…”

A survey on the continuous nonlinear resource allocation problem

“…(With b = 1, it is known as the unit, or canonical, simplex.) Solving a projection problem over such a set described by a linear equality constraint and bounded variables has been considered, for example, in matrix updates in quasi-Newton methods ( [CaM87]), in gradient projection methods for a class of mathematical programs with equilibrium constraints (MPECs) arising in material and shape optimization problems in structural mechanics ( [FJR05]), in subgradient algorithms within right-hand side allocation methods for linear multicommodity network flow problems ([HWC74, KeS77, AHKL80, HKL80, LPS96]), in equilibration procedures for traffic flows ( [DaS69,BeG82,DaN89,LaP92,Lot06]), primal feasibility procedures within Lagrangian dual algorithms for classes of integer programs ( [KLN00]) and in Lagrangian dual methods for quadratic transportation problems, also known as constrained matrix problems ([BaK78, BaK80, OhK80, OhK81, OhK84, CDZ86, Ven89, ShM90, Ven91, NiZ92]); see further below.…”

Using ACCPM in a simplicial decomposition algorithm for the traffic assignment problem

“…The SD algorithm, applied to the asymmetric traffic assignment problem in [5,21,22,38,39], is a column generation method where feasible flows are written as convex combinations of the extreme points of Y (see [35] for a detailed description of algorithmic alternatives). Let E ∈ IR m×t be a matrix with all the t extreme flows of Y .…”