Abstract
This study proposes an improved truncated Newton (ITN) method for the logit-based stochastic user equilibrium problem. The ITN method incorporates a preprocessing procedure to the traditional truncated Newton method so that a good initial point is generated, on the basis of which a useful principle is developed for the choice of the basic variables. We discuss the rationale of both improvements from a theoretical point of view and demonstrate that they can enhance the computational efficiency in the early and late iteration stages, respectively, when solving the logit-based stochastic user equilibrium problem. The ITN method is compared with other related methods in the literature. Numerical results show that the ITN method performs favorably over these methods.
Highlights
Ese models are widely used in the field of transportation planning and network design
It only assumes an implicit path choice set, such as the use of all efficient paths (Dial [9]; Maher [10]), or all cyclic and acyclic paths (Bell [11]; Akamatsu [12]). e most well-known link-based algorithm is the method of successive averages (MSAs) proposed in Sheffi and Powell’s study [4]. is algorithm uses a stochastic loading procedure to produce an auxiliary link flow pattern, and the search direction equals the difference between the auxiliary link flow and the current link flow. e step size for MSA is a predetermined sequence that is decreasing towards zero, such as 1/k where k is the iteration index
In order to numerically justify the theoretical analysis conducted in this research, this section presents some performance comparisons between the improved truncated Newton (ITN) method and other typical methods in the literature. ese methods include the MSA method (Sheffi and Powell [4]), the modified truncated Newton (MTN) method (Zhou et al [16]), and the gradient projection (GP) method (Bekhor and Toledo [15])
Summary
By substituting equation (11) into equation (8), we obtain a minimization problem in terms of route flow variable xwr only. E preprocessing procedure is proposed to find a good initial point to start with It can largely replace the early iteration stage of the truncated Newton method. 5. A Maximal Flow Principle for the Choice of the Basic/Nonbasic Variables e second feature of the improved truncated Newton method is the development of a practical principle for choosing the basic variables in the logit-based SUE problem. At a certain initial point, if the basic route for each OD pair is chosen as the one whose flow is nearly zero, the condition number of the reduced Hessian matrix defined in equation (53) will be very large. From Proposition 3, we know that at the initial point, if the basic route for each OD pair is chosen as the one whose flow is away from zero, we can avoid the case of very large condition numbers. The condition number for route 2 is 397.28, which is much larger than the condition number for any other routes
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