  The paper presents various modifications of the Frank-Wolfe algorithm in the
equilibrium traffic assignment problem. The Beckman model is used as a model
for experiments. In this article, first of all, attention is paid to the choice
of the direction of the basic step of the Frank-Wolfe algorithm. Algorithms
will be presented: Conjugate Frank-Wolfe (CFW), Bi -conjugate Frank-Wolfe
(BFW), Fukushima Frank-Wolfe (FFW). Each modification corresponds to different
approaches to the choice of this direction. Some of these modifications are
described in previous works of the authors. In this article, following
algorithms will be proposed: N-conjugate Frank-Wolfe (NFW), Weighted Fukushima
Frank-Wolfe (WFFW). These algorithms are some ideological continuation of the
BFW and FFW algorithms. Thus, if the first algorithm used at each iteration the
last two directions of the previous iterations to select the next direction
conjugate to them, then the proposed algorithm NFW is using more than N
previous directions. In the case of Fukushima Frank-Wolfe, the average of
several previous directions is taken as the next direction. According to this
algorithm, a modification WFFW is proposed, which uses a exponential smoothing
from previous directions. Experiments with various modifications were carried
out on several datasets representing urban structures and taken from publicly
available sources. The relative gap value was taken as the quality metric. The
experimental results showed the advantage of algorithms using the previous
directions for step selection over the classic Frank-Wolfe algorithm. In
addition, an improvement in efficiency was revealed when using more than two
conjugate directions. For example, on various datasets, the modification 3FW
showed the best convergence. In addition, the proposed modification WFFW often
overtook FFW and CFW, although performed worse than NFW.
