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

Abstract

We present versions of the Frank-Wolfe method for linearly constrained convex programs, in which consecutive search directions are made conjugate. Preliminary computational studies in a MATLAB environment applying pure Frank-Wolfe, Conjugate direction Frank-Wolfe (CFW), Bi-conjugate Frank-Wolfe (BFW) and "PARTANized" Frank-Wolfe methods to some classical Traffic Assignment Problems show that CFW and BFW compare favorably to the other methods. This spurred a more detailed study, comparing our methods to Bar-Gera's origin-based algorithm. This study indicates that our methods are competitive for accuracy requirements suggested by Boyce et al. We further show that CFW is globally convergent. We further point at independent studies by other researchers that show that our methods compare favourably with recent bush-based and gradient projection algorithms on computers with several cores.

A framework for and empirical study of algorithms for traffic assignment

“…Martimo [15], some of them try to improve the FW search direction (for example, see [18,19,20]) or step size (see [21,22]). Among these variations of FW, we are interested in conjugate (CFW) and bi-conjugate Frank-Wolfe (BFW) methods [20] as explained later in this section.…”

“…Among these variations of FW, we are interested in conjugate (CFW) and bi-conjugate Frank-Wolfe (BFW) methods [20] as explained later in this section. In Zhou and Martimo [15], the authors also categorise restricted simplicial decomposition (RSD) [23] and non-linear simplicial decomposition (NSD) [24] as link-based methods.…”

“…Both algorithms operate on auxiliary variables s i a , ∀a ∈ A called points of sight that store information about previously generated directions of descent (for details see Mitradjieva and Lindberg [20]). …”

“…This happens because the search direction is perpendicular to the gradient of the objective function when the algorithm approaches the equilibrium solution [13,20,29]. As mentioned in Bertsekas [41], when the constraint set is polyhedral, as is the case with TA, FW might converge sub-linearly.…”

Upright Stiff: subproblem updating in the FW method for traffic assignment

“…In a recent contribution, Mitradjieva and Lindberg (2013) show that the FW method is not altogether dead, but that the "stiff" is in fact moving.…”