A Full Analytical Implementation of the PARTAN/Frank–Wolfe Algorithm for Equilibrium Assignment

Abstract

We show that an essential step in the PARTAN variant of the Frank–Wolfe algorithm for equilibrium assignment, the calculation of a minimal step length for maintaining feasibility, can be accomplished using either analytical formulas or simple rules.

The Stiff Is Moving—Conjugate Direction Frank-Wolfe Methods with Applications to Traffic Assignment*

“…In the early papers these bounds were determined individually for the first few steps. Later, (Arezki & van Vliet, 1990) and (Florian, Guelat, & Spiess, 1987) determined analytical recursion formulas for these bounds. In spite of improved convergence characteristics, the PARTAN approach has not superseded the Frank-Wolfe method in the telecom and transportation areas.…”

Parallel implementation of a transportation network model

“…Chabini et al [14] also contains parallel computing implementations of a shortest paths algorithm due to Dijkstra [22]. The linear approximation method used to solve the fixed demand network equilibrium problem by Chabini et al [14], is an adaptation of the Frank-Wolfe algorithm [27,4]. In the sequential implementation of this algorithm a variant of Dijkstra's [22] algorithm is used to calculate shortest routes for all origin-destination pairs.…”

System-Optimal Routing of Traffic Flows with User Constraints in Networks with Congestion

“…As it is well-known that the standard Frank-Wolfe algorithm sometimes shows poor convergence (see, e.g., Sheffi 1985;Patriksson 1994;Florian and Hearn 1995), we consider an improved version called Partan that was proposed by LeBlanc, Helgason, and Boyce (1985) and further studied by Florian, Guélat, and Spiess (1987) and Arezki and Van Vliet (1990), among others. As we cannot explicitly work with all variables x P associated with paths P ∈ P ϕ , because there may be exponentially many, we only generate them when needed.…”