Mathematical Problems in Engineering

Volume 2019, Article ID 7313808, 15 pages

https://doi.org/10.1155/2019/7313808

## Improving Truncated Newton Method for the Logit-Based Stochastic User Equilibrium Problem

Correspondence should be addressed to Bojian Zhou; moc.liamg@783jbz

Received 16 March 2019; Revised 5 August 2019; Accepted 3 September 2019; Published 9 October 2019

Academic Editor: Roberta Di Pace

Copyright © 2019 Min Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 1. Introduction

The main role of traffic assignment models is to forecast equilibrium link or path flows in a transportation network. These models are widely used in the field of transportation planning and network design. Traditionally, traffic assignment models are formulated as the user equilibrium (UE) or stochastic user equilibrium (SUE) problems, in which no traveler can reduce his/her actual or perceived travel time by unilaterally changing routes at equilibria [1, 2]. Among various types of traffic assignment models in the literature, the logit-based stochastic user equilibrium traffic assignment problem is most widely adopted and extensively studied [3]. This problem incorporates a random error term in the route cost function to simulate travelers’ imperfect perceptions, which follows Gumbel distribution [4]. The logit-based SUE problem can be equivalently formulated as a mathematical programming problem with a unique solution. This feature facilitates its usage in both theoretical and practical studies [5–8].

The widespread application of the logit-based SUE model makes its solution approach also receive considerable interest in recent years. In general, there are two classes of solution algorithms for the logit-based SUE problem. The first class is link-based algorithms. This type of algorithms uses link flow as its variable. Since link flow is an aggregate variable of different path flows, link-based algorithms do not require explicit path enumeration. 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]). The most well-known link-based algorithm is the method of successive averages (MSAs) proposed in Sheffi and Powell’s study [4]. This 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. The step size for MSA is a predetermined sequence that is decreasing towards zero, such as where is the iteration index. Maher [10] made further modifications to the method of successive averages. In his study, the Davidon–Fletcher–Powell (DFP) method was used to generate a search direction and the cubic (or quadratic) interpolation was applied to estimate the optimal step sizes.

The other class is path-based algorithms. This kind of algorithm is built on path flow variables. It requires an explicit choice of a subset of feasible paths prior to or during the assignment. Unlike the link flow variable, path flow variable is a disaggregate variable which cannot be further decomposed. Therefore, different nonlinear programming methods can be utilized in a more flexible way. For example, Damberg et al. [13] extended the disaggregate simplicial decomposition method of Larsson and Patriksson [14] to solve the logit-based SUE problem. This path-based method iteratively solves subproblems that are generated through partial linearization of the objective function. The search direction is obtained by the difference between the solution of the subproblem and the current iteration point. Bekhor and Toledo [15] proposed using the gradient projection (GP) method to solve this problem. In their study, the gradient of the objective function was projected on a linear manifold of the equality constraints, with the scaling matrix being diagonal elements of the Hessian.

This study focuses on path-based algorithms for the logit SUE problem. To the best of our knowledge, almost all existing path-based algorithms have a linear or sublinear convergence rate, which is relatively slow when the iteration point is approaching the optimal solution. In order to improve the convergence, it is desirable to develop an algorithm with a superlinear convergence rate. Recently, Zhou et al. [16] proposed a modified truncated Newton (MTN) method to solve the logit-based SUE problem. This method consists of two phases. The major iteration phase is performed in the original space, while the minor iteration phase is performed in the reduced space. At each major iteration, a reduced Newton equation is approximately solved using the preconditioned conjugate gradient (PCG) method. The reduced variables in the reduced Newton equation can be changed dynamically, which facilitates the usage of the PCG method. Zhou et al. proved that the convergence rate of the MTN method is superlinear. It works very fast once the iteration point gets near to the optimal SUE solution. However, there are two important problems that are not resolved in Zhou et al.’s research. First, when the iteration point is far from the optimal SUE solution, the truncated Newton type methods are relatively slow. The reason for this phenomenon is not clear. Second, in Zhou et al.’s research, a dynamic principle on how to choose the basic route is proposed. This is only an intuitive principle. The rationale behind this principle is not explained.

With the aim of addressing the above two problems, in this study, we propose an improved truncated Newton (ITN) method for the logit-based SUE problem. The ITN method makes two improvements over the traditional truncated Newton method. First, a preprocessing procedure is introduced. This procedure utilizes the partial linearization method (Patriksson [17]) to generate a good initial point in the original space. It can largely replace the early iteration stage of the traditional truncated Newton method. Second, on the basis of the generated initial point, a static principle on how to partition the coefficient matrix and the variables is developed. With this principle, the computational efficiency of the truncated Newton method in the late iteration stage can be enhanced. Furthermore, the rationale behind these two improvements is analyzed theoretically, which broadens the theoretical significance of this study.

The remainder of the paper is organized as follows. Section 2 outlines the traditional truncated Newton method for a linear equality constrained optimization problem. Section 3 discusses some implementation issues when applying the traditional truncated Newton method to the logit-based SUE problem. Section 4 proposes a preprocessing procedure to determine a good initial point. Section 5 develops a maximal flow principle for the choice of the basic/nonbasic variable. Numerical experiments are conducted in Section 6. Section 7 wraps up the paper with conclusions and future research directions.

#### 2. The Truncated Newton Method for a Linear Equality Constrained Optimization Problem

Consider the following convex problem:where is a strictly convex function that is twice continuously differentiable, is an matrix of full row rank, and . [P1] can be viewed as a general formulation of the logit-based stochastic user equilibrium problem that will be investigated in this study. We will first show how to solve [P1] by the truncated Newton method.

Since [P1] only involves linear equality constraints, it can be transformed into an unconstrained optimization problem using variable reduction technique. Specifically, the matrix and variable are partitioned as follows:where is a nonsingular matrix and , , and . is called the basic matrix and its columns correspond to the basic variables . is called the nonbasic matrix. The columns of correspond to the nonbasic variables .

Therefore, the constraints can be rewritten as

By rearranging the above equation, the basic variables can be expressed as follows:

Substituting equation (4) into [P1], we obtain the following reduced unconstrained problem:where is referred to as the reduced objective function.

Let be a feasible point for [P1-RED]. By approximating using a second-order Taylor series around , the following subproblem can be obtained:where and are the reduced gradient and reduced Hessian of , and is the difference between the nonbasic variable and the feasible point .

Clearly, [SUB-1] is a quadratic programming problem. A typical method for this problem is the preconditioned conjugate gradient (PCG) method. This method constructs a sequence of conjugate directions using the objective gradient and minimizes the objective function along each of the directions. Interested readers may refer to Chapter 5 in Nocedal and Wright [18] for a detailed description of this method. It is commonly known that for large-scale optimization problems, finding the exact solution of [SUB-1] is computationally intensive. The truncated Newton method is thus designed to alleviate this drawback by solving [SUB-1] approximately if is far from the optimal solution of [P1] and solving [SUB-1] more accurately when the optimal solution is approached.

Let be an approximate solution of [SUB-1] generated by the PCG method. According to Lemma A.2 in Dembo and Steihaug [19], defines a descent direction with respect to the reduced objective function . Hence, by finding an appropriate step size in this direction, the new solution point for the next iteration can be obtained.

In what follows, we will give a detailed description of the truncated Newton method for the linear equality constrained optimization problem [P1]. As elaborated above, this method consists of two phases. The major iteration phase transforms the original problem into an unconstrained one and applies the truncated Newton framework to solve it. The minor iteration phase uses the PCG method to solve a quadratic programming subproblem approximately.

The detailed steps of major iteration are described in Algorithm 1 below. It is performed in the reduced variable space.