Quadratic approximation and convergence of some bush-based algorithms for the traffic assignment...

Transportation Research Part B 56 (2013) 15–30

Contents lists available at ScienceDirect

Transportation Research Part B

journal homepage: www.elsevier .com/ locate/ t rb

Quadratic approximation and convergence of some bush-basedalgorithms for the traffic assignment problem

0191-2615/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.trb.2013.06.015

⇑ Corresponding author. Tel.: +1 847 467 0502; fax: +1 847 491 4011.E-mail address: y-nie@northwestern.edu (Yu (Marco) Nie).

Jun Xie a, Yu (Marco) Nie b,⇑, Xiaoguang Yang a

a School of Traffic and Transportation Engineering, Tongji University, 4800 Cao’an Road, Shanghai 201804, People’s Republic of Chinab Department of Civil and Environmental Engineering, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, United States

a r t i c l e i n f o

Article history:Received 4 December 2012Received in revised form 24 June 2013Accepted 24 June 2013

Keywords:Bush-based algorithmTraffic assignmentQuadratic approximationGreedy methodLine search

a b s t r a c t

This paper first shows that LUCE (Gentile, 2012), a recent addition to the family of bush-based algorithms, is closely related to OBA (Bar-Gera, 2002). LUCE’s promise comes mainlyfrom its use of the greedy method for solving the quadratic approximation of node-basedsubproblems, which determines the search direction. While the greedy algorithm acceler-ates the solution of the subproblems and reduces the cost of line search, it unexpectedlydisrupts the overall convergence performance in our experiments, which consistentlyshow that LUCE failed to converge beyond certain threshold of relative gap. Our analysissuggests that the root cause to this interesting behavior is the inaccurate quadratic approx-imation constructed on faulty information of second-order derivatives. Because the qua-dratic approximations themselves are inaccurate, the search directions generated fromthem are sub-optimal. Unlike OBA, however, LUCE does not have a mechanism to correctthese search directions through line search, which explains why its convergence perfor-mance suffers the observed breakdowns. We also attempt to improve LUCE using the ideasthat have been experimented for the improvement of OBA. While these improvements dowork, their effects are not enough to counteract the inability to adjust sub-optimal searchdirections. Importantly, the fact that the search direction has to be corrected in line searchto ensure smooth convergence attests to the limitation of origin-based flow aggregationshared by both OBA and LUCE. These findings offer guidelines for the design of high perfor-mance traffic assignment algorithms.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

The traffic assignment problem (TAP) is used to predict the route choices for travelers on urban road networks under theuser equilibrium (UE) principle (Wardrop, 1952), which was first formulated as a convex optimization problem with linearconstraints by Beckmann et al. (1956). Since then, finding efficient algorithms to solve this problem has attracted muchattention, especially following the initial success of the Frank–Wolfe algorithm (Frank and Wolfe, 1956; LeBlanc et al.,1975) and the later recognition of its painfully slow convergence behavior (LeBlanc et al., 1985; Hearn et al., 1985; Fukushi-ma, 1985; Larsson and Patriksson, 1992; Jayakrishnan et al., 1994; Patriksson, 1994; Florian and Hearn, 1995). The past dec-ade has witnessed the development and experiments of a class of bush-based algorithms (Bar-Gera, 2002; Dial, 2006; Nie,2010; Bar-Gera, 2010; Gentile, 2012; Nie, 2012; Inoue and Maruyama, 2012; Boyles, 2012). The promises of these algorithms,especially their ability to solve large-scale UE-TAPs at a high level of precision and efficiency, makes them a new focus intraffic assignment research.

16 J. Xie et al. / Transportation Research Part B 56 (2013) 15–30

All bush-based algorithms (BAs) known so far share a few important features. They construct and maintain a bush foreach origin (or destination) and restrict the assignment operation only to these bushes. The concept of bush was originallycoined by Dial in his celebrated work on logit traffic assignment (Dial, 1971). Simply speaking, a bush is an acyclic subnet-work that intends to encompass all UE shortest paths. Acyclicity promises great efficiency for many network operations.Moreover, when all bushes are equilibrated, the entire network is equilibrated. That is, the use of bush not only offers effi-ciency, but also assures optimality, in the sense that UE flows can be represented by equilibrated bushes (Nie, 2010). Bush-based algorithms construct optimal bushes at UE by iterating between two sub-problems: bush construction and bushequilibration.

According to the flow aggregation level in bush equilibration, bush-based algorithms can be classified as route flow based ororigin flow based. Dial’s Algorithm B (Dial, 2006) is in the former category; it equilibrates bushes by swapping flows betweenthe longest and shortest routes. In contrast, the origin-based algorithm, or OBA (Bar-Gera, 2002), operates in the space oforigin flows, which are represented by proportions of traffic arriving at each node from its predecessor links. OBA is nowwidely used as a benchmark to gauge the performance of newer traffic assignment algorithms such as Algorithm B andthe Local User Cost Equilibrium algorithm, or LUCE (Gentile, 2012). This study was initiated by an effort to document theconvergence performance of LUCE in relation to OBA. Limited computational evidence in the literature does not tell a con-sistent story about the performance of LUCE. Gentile (2012) observed that LUCE ‘‘compares favorably with the most ad-vanced methods recently proposed in the literature’’. Yet, his experiments with LUCE and other algorithms useexecutable codes, which makes it difficult to determine if the observed performance differences come from implementationor the algorithm itself. A recent computational study (Inoue and Maruyama, 2012), which independently implements OBAand LUCE, found that LUCE was unable to achieve very high levels of precision.

This paper will first show that LUCE and OBA are closely related. First of all, they solve the same restricted master problem(RMP) in the same space of origin flows.1 Second, they employ the same method to approximate the first-and second-orderderivatives of origin flows for the RMP. Thirdly, in bush equilibration stage, both OBA and LUCE transform the RMP into a se-quence of small approximated quadratic programs associated with each node in the bush, referred to as node-based subprob-lems (Gentile, 2012). The solution to these quadratic programs provide a descent direction. Finally, a line search has to be usedto find a proper step size in both algorithms. We will also show that the only major algorithmic difference between OBA andLUCE lies in the method to solve the node-based subproblem. LUCE applies a greedy method, inspired by the hyper-path algo-rithm for transit assignment. In contrast, OBA employs a one-step gradient projection (one-step GP) method, which is similar tothose used in Jayakrishnan et al. (1994) and Dial (2006). The first question we ask in this paper is how much does the solutionmethod to the quadratic approximation contributes to the relative convergence performance of the two algorithms. Intuitively, thealgorithm that can solve the node-based subproblem more quickly and precisely is likely to have an advantage. If this hypoth-esis holds true, then one should definitely prefer the greedy method to the one-step GP method, because the former can getexact solutions by directly solving the KKT system, whereas one-step GP is designed to achieve just an approximation (see Sec-tion 4 for more details).

To verify the above hypothesis, we first implement LUCE and OBA on the same programming platform that allows us touse the same code wherever possible. Effectively, our LUCE and OBA codes share almost all components except for those re-lated to solving the node-based subproblem. To our surprise, we found that the computational performance of the two algo-rithms are quite similar for the convergence range that is of practical interest. Perhaps more important, the results show thatLUCE is consistently outperformed by OBA to achieve very precise solutions, which generally agrees with the findings of In-oue and Maruyama (2012). The disappointing convergence performance for LUCE appears counter-intuitive at first glance,considering that its greedy method is able to quickly find exact solutions to the quadratic subproblems. Therefore, our secondquestion is why does a better subproblem solver degrades the overall convergence performance. As it turns out, our attempt toanswer this question not only offers a plausible explanation to a seemingly mysterious phenomenon, but also leads to betterunderstanding on the fundamental limitation of operating in the space of aggregated origin flows. These findings suggestthat decomposing flows to the route or even segment level is the key to better performance.

The rest of this paper is organized as follows. Section 2 provides the formulation of the single-origin traffic assignmentproblem and the overall algorithmic framework shared by both LUCE and OBA. Section 3 presents the quadratic approxima-tion of the node-based subproblems, consistent with the interpretation based on linear user cost equilibrium. Section 4 pre-sents the gradient projection algorithm and greedy algorithm for the quadratic approximation, with theoretical justificationthat emphasizes their connections. After the results of numerical experiments are presented in Sections 5 and 6 launches athorough investigation on LUCE’s unexpected performance, which leads to an explanation and discussions on the limitationsof both algorithms in Section 7. The last section concludes the paper with comments on basic features of high performancetraffic assignment algorithms.

2. Formulation and solution framework

The static traffic assignment problem (TAP) can be formulated as an optimization problem in many different ways (Pat-riksson, 1994). Central to the development of the bush-based algorithms is a formulation based on decomposing assignment

1 LUCE was originally presented as destination-based. Since this difference is trivial from the algorithmic point of view, LUCE will be presented andimplemented as origin-based in this paper.

J. Xie et al. / Transportation Research Part B 56 (2013) 15–30 17

with respect to origin (or destination). As these origin-based subproblems are solved sequentially, it suffices for our purposeto focus on a single origin problem. Consider a network G(N,A) where N and A denote the sets of nodes and links, respec-tively. Let xij be the flow assigned on link ij 2 A associated with the origin and tij(�) be a strictly positive and increasing func-tion of xij. The single origin traffic assignment problem may be formulated as follows:

min zðxÞ ¼X

ij

Z xij

0tijðuÞdu ð1Þ

subject to :Xi2IðjÞ

xij �Xl2OðjÞ

xjl ¼ qj; 8j 2 N ð2Þ

qj ¼

qs j ¼ s 2 S

�Xs2S

qs j ¼ r

0 otherwise

8>><>>: ð3Þ

xij P 0; 8ij 2 A ð4Þ

where qs is traffic departing from the origin r destined for destination s 2 S, and S denotes the set of destination nodes. O(j)and I(j) are the sets of successor(s) and predecessor(s) of j, respectively. The salient feature shared by all bush-based algo-rithms is to restrict traffic assignment only to a child bush of the original network for each origin. In Nie (2010), such a childbush is defined as an acyclic subnetwork that has at least one path from the origin to every other node. When the underlyingnetwork is a bush associated with the origin r, Problem (1)–(4) can be transformed using routing variables to represent originflows (Gallager, 1977; Bar-Gera, 2002). Let gj be the total flow arriving at node j, and the routing variable /ij denote the pro-portion of gj on link ij. The problem is rewritten as

min zð/Þ ¼X

ij

Z xij

0tijðuÞdu ð5Þ

subject to :xij ¼ gj/ij; 8j ð6Þgj ¼ qj þ

Xl2OðjÞ

gl/jl; 8j ð7ÞXi2IðjÞ

/ij ¼ 1; 8j; /ij P 0; 8ij ð8Þ

Bush-based algorithms usually iterate between two main operations: bush construction and bush equilibration. The for-mer involves initializing, expanding and trimming bushes; and the latter assigns flows on the current bush to satisfy the userequilibrium conditions (Wardrop, 1952). Because the bush equilibration effectively solves the traffic assignment problem ona sub-network, it is also known as the restricted master problem (RMP). For the reader’s convenience, a conceptual frame-work of bush-based algorithms as applied in a single origin case is given below.

2.1. Bush-based algorithms

� Step 0: Initialize the bush as a shortest path tree rooted at the origin. Assign all flows to links on the tree.� Step 1: Expand the bush by adding new links that have the potential to reduce travel times on the bush, while keeping

acyclicity.� Step 2: Bush equilibration: Assign flows on the current bush to satisfy user equilibrium conditions.� Step 3: Trim the bush by removing unused links.� Step 4: If the convergence requirement is satisfied, stop; otherwise, go to Step 1.

The reader is referred to Nie (2010) for the details regarding Steps 1 and 3 above. The focus of the present paper is on Step2, namely performing a user equilibrium assignment on a given bush. Specifically, we are interested in exploring two algo-rithms that make use of the routing variables as the main solution variables: OBA and LUCE. To explain the working mech-anism of these algorithms, we first recall that the derivative of z(/) with respect to /ij can be calculated recursively using thefollowing formula, thanks to acyclicity (Bar-Gera, 2002):

v ij ¼ tij þ Vi; Vi ¼Xl2IðiÞ

/liv li; Vr ¼ 0 ð9Þ

where r is the origin, and define vij as follows:

gjv ij ¼@zð/Þ@/ij

ð10Þ

18 J. Xie et al. / Transportation Research Part B 56 (2013) 15–30

Theorem 1. A feasible solution of the optimization problem (5)–(8) satisfies the user equilibrium conditions if one can find wj, "jsuch that

2 Nodnode j iin term

v ij �wj P 0; /ijðv ij �wjÞ ¼ 0; 8i; j – r ð11Þ

Proof 1. cf. Theorem 3 (Gallager, 1977), also Lemma 5 (Bar-Gera, 2002). h

It is clear that wj above should be interpreted as the minimum travel cost from the origin to node j. Accordingly, vij can beinterpreted as the average travel cost from the origin to node j through link ij. In other words, at user equilibrium a link ijwould only receive positive flow if the average cost to j via link ij is minimum among all incoming links to j.

Because the above UE condition is defined with respect to each node on the bush, we can further decompose the bushassignment problem to a series of node assignment problem. Essentially, the following question should be answered for eachnode assignment problem: What proportions of flows ought to be assigned to each incoming links such that their averagecosts vij is equilibrated. The node assignment problems is typically solved in a descending topological order2, as the potentialflow changes created from solving a node assignment problem propagate in that direction. More specifically, the flow changeson the incoming links to node j could potentially alter the total incoming flows to nodes upstream of j, which has to be takeninto account when the node assignment problem is solved for those nodes.

Once an initial solution uij for the bush is given, one can try to improve it along a descent direction. A commonly adoptedapproach to finding such a descent direction is to construct and solve a quadratic approximation of the original problem atthe given solution. We will discuss how this can be done in greater detail in the next section, which is a focus of the paper.For now, suffice it to say that the quadratic approximation has to be solved for each node and that the collection of thesenode-based solutions forms a descent direction. Finally, a line search has to be performed to determine the step size. Theabove bush equilibration procedure is adopted in both OBA and LUCE, and is summarized below.

2.2. Bush equilibration

Step 2.0: Forward pass: First update link travel time tij and its derivative gij, for each ij on the bush. Then following anascending order of topological distance, calculate vij and its derivative sij (defined in Section 3).Step 2.1: Backward pass: Following a descending order of topological distance, compute a descent direction /ij by decom-posing the bush assignment problem into a series of node assignment problems, and solving their quadratic approxima-tions built using vij and sij.Step 2.2: Line search.Step 2.3: Update bush flows based on the steps size and descent direction obtained from previous steps. If the conver-gence requirement is satisfied for the current bush or the maximum number of iteration is reached, stop; otherwise,go to Step 2.0.

3. Quadratic approximation for the node assignment problem

Now, we are ready to tackle the node problem, the basic element of the bush-based traffic assignment algorithms dis-cussed in this paper. If the node problems are solved following a descending order of the topological distance in Step 2.1above, the total flow arriving at the node j is known as gj by the time the node is visited. The node problem is to determine/ij, i 2 I(j) that satisfy the user equilibrium condition (11). Because the main purpose of solving the node problem is to pro-vide a descent direction, a simple and effective approach is to construct a quadratic approximation for the original problem.To this end, we first need to evaluate the second order derivative of the objective function with respect to /ij. Unfortunately,this derivative is not available in exact close-form, to the best of our knowledge. Upper and lower bounds for these deriv-atives have been found as follows (Bertsekas and Gafni, 1983; Bar-Gera, 2002):

@z2ð/Þ@2/ij

¼ g2j gij þ

@2zð/Þ@2qi

!ð12Þ

Xl2IðiÞ

/2li gli þ

@2zð/Þ@2ql

!" #6@2zð/Þ@2qi

6

Xl2IðiÞ

/2ligli þ

Xl2IðiÞ

/li

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi@z2ð/Þ@2ql

s0@ 1A2

ð13Þ

@z2ð/Þ@2qr

¼ 0 ð14Þ

es in a bush can be ordered according to their topological distances. If we set the length of every link in a bush to be 1, then the topological distance of as the maximum distance from the origin to j. Visiting the nodes in a descending topological order means starting from the node furthest from the origins of topological distance. The problems may also be solved in an ascending order, see Eqs. 2.67–2.68 (Bar-Gera, 1999).

J. Xie et al. / Transportation Research Part B 56 (2013) 15–30 19

If the second order derivatives are approximated using their lower bound, they can be evaluated following an ascendingorder of topological distance

Si �Xl2IðiÞ

/2lisli

� �; sli ¼ gli þ Sl; Sr ¼ 0 ð15Þ

If the upper bound is used instead, the recursive formula becomes

Si ¼Xl2IðiÞ

/2li�sli þ

Xm2IðiÞ; m–l

/li/mi

ffiffiffiffiffiffiffiffiffiSmSl

q0@ 1A; �sli ¼ gli þ Sl; Sr ¼ 0 ð16Þ

To simplify notation, we shall define

g2j sij ¼ gj

@v ij

@/ij¼ @z2ð/Þ

@2/ij

in what follows, with the understanding that sij must be approximated by its lower or upper bounds, or some average of thetwo bounds. We note that both Bar-Gera (2002) and Gentile (2012) make use of the lower bound approximation.

At the current solution uij,"i 2 I(j),vij and sij can be evaluated and denoted as vuij and su

ij , respectively. The quadraticapproximation of the node assignment problem can then be formulated as follows:

min zj–rð/Þ ¼Xi2IðjÞ

vuij gj � su

ij g2j uij

� �/ij þ 0:5su

ij g2j /

2ij

� �ð17Þ

subject to :Xi2IðjÞ

/ij ¼ 1 ð18Þ

/ij P 0; 8i 2 IðjÞ ð19Þ

Note that the objective function (17) is obtained from the second-order Taylor expansion

~zjð/Þ ’zjðuÞ þXi2IðjÞ

@zðuÞ@/ij

ð/ij �uijÞ þ 0:5@z2ð/Þ@2/ij

ð/ij �uijÞ2

!’zjðuÞ þ

Xi2IðjÞ

vuij gjð/ij �uijÞ þ 0:5su

ij g2j ð/ij �uijÞ

2� �

¼zjðuÞ þXi2IðjÞ

0:5suij g

2j u

2ij � vu

ij gjuij þ vuij gj � su

ij g2j uij

� �/ij þ 0:5su

ij g2j /

2ij

� �

ij g2j u2

ij and vuij gjuij are constants, minimizing ~zjð/Þ is equivalent to minimizing zj as defined in (17). It should

be noted that the above approximation only considers the diagonal elements in the Hessian r2z, as obtaining the off-diag-onal elements in closed form is even more challenging.

Note that the above problem is not properly defined when gj = 0. For gj > 0, its KKT conditions can be stated as follows:

vuij þ su

ij gjð/ij �uijÞ �wj P 0; /ij vuij þ su

ij gjð/ij �uijÞ �wj

� �¼ 0; ð20Þ

where wj is the multiplier associated with the conservation condition (18). Comparing this condition to (11), it is easy toconclude that the quadratic assignment problem actually approximates vij with the following linear function,

v ij ¼ vuij þ su

ij gjð/ij �uijÞ; ð21Þ

and then attempts to equilibrate flows on all the incoming links. Indeed, this interpretation appears to be how LUCE derivesits name, i.e. an Equilibrium based on Linear User Cost. It is clear from the above demonstration that the method has its rootin quadratic approximation.

4. Solution algorithms for the quadratic approximation

In this section we discuss how to solve the quadratic program (17)–(19), which is what really distinguishes LUCE fromOBA. Before we discuss the algorithms, let us further simplify our notation by introducing the following constants:

ci � vuij � su

ij gjuij; bi � suij gj ð22Þ

With this definition, the approximated average approach travel time v ij can be simplified as

v ij ¼ ci þ bi/ij

OBA uses a gradient projection (GP) algorithm similar to Bertsekas (1976) and Jayakrishnan et al. (1994). To apply thismethod, we first choose the reference link mj such that

20 J. Xie et al. / Transportation Research Part B 56 (2013) 15–30

m ¼ argmin vuij ; i 2 IðjÞ

n o

/mj ¼ 1�X

i2IðjÞ;i–m

/ij ð23Þ

Using the routing variables on the non-reference links as the solution variables, the problem (17)–(19) can be reformu-lated as:

min zj–rð/Þ ¼gj cm 1�X

i2IðjÞ;i–m

/ij

!þ 0:5bm 1�

Xi2IðjÞ;i–m

/ij

!2

þX

i2IðjÞ;i–m

ci/ij þ 0:5bi/2ij

� �0@ 1A ð24Þ

subject to :/ij P 0; 8i 2 IðiÞi – m ð25Þ

The first-and second-order derivative of the objective function (24) with respect to /ij, i – m are

@zjð/Þ@/ij

¼ ci þ bi/ij � cm � bm 1�X

i2IðjÞ;i–m

/ij

!¼ v ij � vmj ð26Þ

@2zjð/Þ@2/ij

¼ bi þ bm ð27Þ

The GP algorithm finds a new solution for the quadratic program along a projected Newton direction as

/ij ¼max 0;uij � kv ij � vmj

bi þ bm

� �; 8i 2 IðiÞ; i – m; /mj ¼ 1�

Xi2IðjÞ;i–m

/ij; ð28Þ

where k is a step size. If there is only one alternative i other than m then (28) actually solves the quadratic problem preciselyin just one step for k = 1. To see why this is the case, the reader can verify the following equation:

v�ij ¼ ci þ bi uij �v ij � vmj

bi þ bm

� �¼ v�mj ¼ cm þ bm umj þ

v ij � vmj

bi þ bm

� �ð29Þ

That is, if both links remain used after the flow shift, the average travel time to node j is the same via i or m if the cost can beevaluated using the linear function (21).

When there are more than two incoming links to node j, however, performing the flow update (28) once cannot guaranteefinding the optimal solution to the quadratic program. One can certainly perform as many iterations of the flow update (28) asneeded to solve the quadratic approximation with a satisfactory accuracy. Yet, most applications of the GP algorithm, includ-ing OBA, actually conduct the flow update only once and then accept the resulting approximation solution. Such a simplifi-cation seems reasonable because the gains from obtaining a more precise solution to the quadratic program may notjustify the computational expense, since the quadratic program itself is only an approximation after all. The quadratic approx-imation (17) of our node assignment problem is subject to even more approximation errors, because not only the off-diagonalelements in the Hessian have to be ignored, but the diagonal elements are merely lower bounds of the true values.

An alternative and perhaps more efficient method to solve the quadratic program (17)–(19) is to directly tackle the KKTconditions (20), as pointed out by Gentile (2012). Suppose we know the set of all used incoming links Xj (defined by the tailnode of the link); then (20) can be translated to the following equation system

ci þ bi/ij ¼ w; i 2 Xj ð30ÞXi2Xj

/ij ¼ 1 ð31Þ

From the above system, one can solve the minimum average approach travel time w and /ij as follows

w ¼1þ

Pi2Xj

ci=biPi2Xj

1=bið32Þ

/ij ¼ w=bi � ci=bi; 8i 2 Xj ð33Þ

It is clear that all the incoming links not included in Xj should receive zero proportion. To determine Xj, one can use a greedyalgorithm as described below:

4.1. Greedy algorithm

Step 0: Sort all incoming links in I(j) according to the increasing order of ci. Relabel I(j) = {1,2,3, . . .}, with c1 6 c2 6 � � �. Ini-tialize w = c1 + b1, B = 1/b1, C = c1/b1. Set k = 1, Xj = {1}.

J. Xie et al. / Transportation Research Part B 56 (2013) 15–30 21

Step 1: Set k = k + 1. If ck P w, terminate; otherwise, set C = C + ck/bk, B = B + 1/bk, Xj = Xj [ k, go to Step 2.Step 2: Update w = (1 + C)/B, /ij = (w � ci)/bi, i = 1, . . . ,k. Go to Step 1.

The above algorithm bears a similar structure as the hyper-path search algorithm described by Spiess and Florian (1989)and other authors, as noticed by Gentile (2012).3 In the worst case, the algorithm will find the exact solution to the quadraticprogram after all incoming links are visited.

In addition, the one-step GP algorithm provides a mechanism to adjust the search direction in accord with line search. InOBA, specifically, the step size k used in the flow update (28) is actually the same step size from the line search in BushEquilibration (Bar-Gera, 2002). To clarify this subtle yet important issue, we now provide details of line search in the Step2 of Bush Equilibration

4.2. Line search in bush equilibration

Step 2.2.1: Set step size k = 1, compute the social pressure (Bar-Gera, 2002)

r ¼

Xij

tðxij þ kgjð/ij �uijÞÞð/ij �uijÞgj LUCEXij

tðxij þ gjð/ij �uijÞÞð/ij �uijÞgj OBA

8>><>>: ð34Þ

Step 2.2.2: if r < 0; go to Step 2.3 in Bush Equilibration; otherwise, set k = k/2, go to Step 2.2.1 (LUCE) or Step 2.1 in BushEquilibration (OBA).

Two remarks are in order here about the calculation of the social pressure in Eq. (34). First, the total node arriving flow gj

depends on the choice of step size, because the flows at upstream nodes change with the new downstream bush flows, whichdepends on the step size. Thus, the calculation should be performed following a descending order of the topological distance.Second, the formula for OBA does not use the step size because /ij already embeds the step size information through Eq. (28).

It is important to notice that each line search in OBA actually invokes Step 2.1 in Bush Equilibration, which solves allnode problems in a backward pass. In contrast, there is no need to resolve the node problems at the current step size in LUCE,because the exact solutions have been obtained for their quadratic approximation. Consequently, OBA’s line search is likelymore time consuming than LUCE’s.

Having presented the algorithms, two questions naturally arise: first, which algorithm, one-step GP or Greedy, is a betteroption for solving the quadratic approximation; and second, which one is better for solving the whole traffic assignmentproblem? The answer to the first question seems to be affirmative in favor of Greedy. Note that the Greedy algorithm canfind an exact solution within a finite and often very small (considering the number of incoming links to a node rarely exceeds3–4 in a transportation network) number of steps. In contrast, the GP algorithm can quickly find an exact solution only whenthere are two incoming links. In fact, the GP algorithm aims at a coarse approximation to the quadratic program in mostcases, as mentioned before. The answer to the second question is less obvious, however. On one hand, it seems that the abil-ity to solve subproblems quickly and more precisely would likely help the overall convergence. On the other hand, one has torecognize that the subproblems themselves are merely approximations, and hence a more precise solution to them is notnecessarily better for the original problem. Since the analysis cannot provide a compelling answer, we proceed to performnumerical experiments in order to gain some insights.

5. Numerical experiments

In this section, we will compare the convergence performance of OBA and LUCE using numerical experiments. To providea performance benchmark, the Frank–Wolfe (FW) algorithm is also included in the experiments. Note that the FW algorithmis well known for its inability to achieve a precise equilibrium solution. All three algorithms are coded using TNM, a C++ classlibrary specialized in modelling transportation networks (Nie, 2006). All numerical results reported in this section were pro-duced on a Windows XP-64 Workstation with two Xeno 3.0 GHz CPUs and 8 GB RAM.

The algorithms are applied to perform traffic assignment on six networks, of which two are regional scale. The main con-vergence indicator used in this study is the so-called relative gap, which measures how close the current solution is to thetrue user equilibrium. In our notation, the relative gap is calculated by

3 It isusing (3

Ggap ¼ 1�X

rs

ursqrs

Xij2A

,xijtij; ð35Þ

where urs is the minimum travel time between the O–D pair rs based on the current link travel time tij, and qrs is the traveldemand between the O–D pair rs. Finally, the BPR-function is used to calculate link travel times.

worth noting that the greedy algorithm implemented in LUCE is slightly different from what is presented above. Specifically, it first solves w and /ij

2) with all links included in Xj, and then iteratively removes the links whose ck > v.

22 J. Xie et al. / Transportation Research Part B 56 (2013) 15–30

For the four small networks (see Table 1), the convergence criterion is set to 10�14 in order to examine what levels ofprecision the two bush-based algorithms are capable of achieving. The convergence curves of all three algorithms are re-ported in Fig. 1. In these plots, x axis represents CPU times, and y axis represents the relative gap. As expected, the conver-gence of the FW algorithm almost becomes a standstill after it reaches a relative gap of about 10�5. Fig. 1 also depicts a muchsharper convergence behavior for both OBA and LUCE up to certain relative gap. In fact, the convergence curves of OBA andLUCE almost overlap completely, before they diverge after reaching certain critical relative gap. In most cases, LUCE stopsconverging after it attains a critical relative gap, which seems problem specific. For the Anaheim network, that gap is about10�9; for Barcelona, it is 10�10; for Chicago Sketch, it is around 10�12. For the Winnipeg network, the gap obtained by LUCEcan reach as small as 10�14, although LUCE’s convergence curve lagged behind OBA’s for most parts.

For the larger regional scale networks (see Table 1), PRISM and Chicago regional networks, we set the convergence cri-terion to 10�12 and 10�10 respectively to avoid excessive computation time. The results presented in Fig. 2.a basically tells

0 2 4 6 8 10 12−16

−14

−12

−10

−8

−6

−4

−2

0

2

4

(a) Anaheim

rela

tive

equi

libriu

m g

ap (1

0e)

Time [seconds]

OBALUCEFW

0 0.5 1 1.5 2−16

−14

−12

−10

−8

−6

−4

−2

0

2

4

(b) BarcelonaTime [minutes]

OBALUCEFW

0 1 2 3 4 5 6 7 8−16

−14

−12

−10

−8

−6

−4

−2

0

2

4

(c) WinnipegTime [minutes]

OBALUCEFW

0 1 2 3 4 5 6−16

−14

−12

−10

−8

−6

−4

−2

0

2

4

(d) Chicago SketchTime [minutes]

OBALUCEFW

rela

tive

equi

libriu

m g

ap (1

0e)

rela

tive

equi

libriu

m g

ap (1

0e)

rela

tive

equi

libriu

m g

ap (1

0e)

Fig. 1. Convergence performance on small networks.

Table 1Details of test networks.

Scale Network Node Link Zone Trip

Small Anaheim 416 914 38 104,694Barcelona 1020 2522 97 184,680Winnipeg 1052 2836 135 64,784Chicago Sketch 933 2950 386 1,260,910

Large PRISM 14,639 33,937 898 609,670Chicago Regional 12,982 39,018 1771 1,360,430

0 2 4 6 8 10 12 14−12

−10

−8

−6

−4

−2

0

2

4

(a) PRISM

rela

tive

equi

libriu

m g

ap (1

0e)

Time [hours]

OBALUCEFW

0 10 20 30 40 50 60−10

−8

−6

−4

−2

0

2

4

(b) Chicago Regional

rela

tive

equi

libriu

m g

ap (1

0e)

Time [hours]

OBALUCEFW

Fig. 2. Convergence performance on regional scale networks.

Fig. 3. Topology of the simple network.

J. Xie et al. / Transportation Research Part B 56 (2013) 15–30 23

the same story as in Fig. 1. That is, OBA and LUCE both reached a relative gap of 10�6 comfortably along a very similar con-vergence trajectory. Yet, after that point, LUCE’s convergence made a sharp turn for worse and never recovered from it. Thesituation is slightly different for the Chicago regional network (see Fig. 2b), where LUCE had not yet reached its critical gapobserved for other networks within the committed computation time. Yet, the performance of LUCE lagged behind that ofOBA after they passed the relative gap of 10�4.

That the two bush-based algorithms would perform similarly is not a big surprise given how similar they are in the over-all structure. What is surprising is that LUCE seems to often get into a similar standstill as FW do in most cases, even thoughit usually gets stuck at a much smaller relative gap. In contrast, OBA was able to converge well into the vicinity of the floatnumber limit (i.e. 10�15) for the small networks. The trend revealed from Fig. 2 suggests that it would likely do the same forthe larger networks should enough computation time be committed. One may argue that the level of precision beyond 10�8

is of little practical significance. Yet, it is intriguing and useful to understand why LUCE delivers such a disappointing per-formance, because it could potentially shed lights on some underlying algorithmic design issues. As mentioned before, oneplausible theory is that a better solution to the quadratic approximation is not necessarily better for the original problem.While this theory may explain why LUCE fails to reach high precision, it does not seem to explain why OBA, with a coarsersolution scheme to the quadratic approximation, consistently succeeds to achieve highly precise solutions. The analysis inthe next sections will substantiate the theory and provide explanations to both phenomena.

6. Analysis of LUCE’s unexpected performance

The performance of bush-based algorithms is affected by many factors. It is usually difficult to pinpoint which factor is toblame for unexpected performance such as observed in the last section. However, in our implementation, most of these fac-tors are tightly controlled. LUCE and OBA share most codes except for Step 2.1 in Bush Equilibration and Step 2.2.2 in LineSearch. It is thus reasonable to attribute the discrepancy in their performance to those steps.

To gain insights, we consider a traffic assignment problem on a toy network described in Fig. 3. Although the networktopology is extremely simple, it resembles the elementary node assignment problem quite well. For most transportation net-works, the number of incoming links to any node on a bush rarely exceeds 2 or 3. The performance function of the three linksare given by

0 5 10 15 20 25 30−15

−10

−5

0

Rea

tive

gap

Iteration

LUCELUCE-GPOBA

Fig. 4. Convergence performance of LUCE, LUCE-GP and OBA for the three-link network.

Fig. 5. Illustration of the impacts of inaccurate second-order derivatives on OBA and LUCE.

24 J. Xie et al. / Transportation Research Part B 56 (2013) 15–30

t1 ¼ 10 1þ 0:15x1

2

� �4� �

; t2 ¼ 20 1þ 0:15x2

4

� �4� �

; t3 ¼ 25 1þ 0:15x3

3

� �4� �

ð36Þ

and the total demand between o to d is q = 10. The reader who wish to repeat the calculation presented in this section mayfind the OBA and LUCE solution procedures adapted to this special case in Appendix A.

It is worth noting that the greedy method is not the only choice for solving the quadratic approximation (17). For in-stance, one can solve (17) to any desired precision by performing as many GP iterations as necessary. The procedure wouldinvolve updating the linear cost and the reference link �a after each GP iteration. Yet, replacing the greedy method with otheralgorithms would not change LUCE’s overall convergence behavior as long as (17) is solved precisely (although the compu-tation time required to achieve a practically ‘‘precise’’ solution may be different). To demonstrate this point, we test a variantof LUCE in which (17) is solved by GP, denoted as LUCE-GP.

Fig. 4 reports the convergence curves of OBA, LUCE and LUCE-GP. As expected, the convergence pattern of LUCE-GP is thesame as that of LUCE because they both solve the quadratic program exactly in each main iteration and use that solution togenerate the search direction. In contrast, OBA only solves the quadratic problem approximately, so it requires more mainiterations to reach the same level of convergence. Hence, the greedy method indeed outperforms one-step GP in this

0 10 20 30 40−14

−12

−10

−8

−6

−4

−2

0

Iteration

LUCEOBA

(a) θ= 0.7

0 20 40 60 80 100−14

−12

−10

−8

−6

−4

−2

0

Iteration

LUCEOBA

(b) θ = 0.58

0 50 100 150 200−14

−12

−10

−8

−6

−4

−2

0

Iteration

LUCEOBA

(c) θ= 0.56

0 0.5 1 1.5 20

50

100

150

200

Num

ber o

f ite

ratio

ns re

quire

d to

reac

h a

rela

tive

gap

1e−1

2

θ

LUCEOBA

(d) Overall trend

Fig. 6. Impacts of second-order derivatives on the convergence performance of LUCE and OBA for the three-link network. (h 2 [0,2] is used as a multiplier toperturb the second-order derivative from the correct value.)

J. Xie et al. / Transportation Research Part B 56 (2013) 15–30 25

example. Then why does LUCE in the real tests fail to demonstrate such superiority? We have noted that the quadratic prob-lem itself is an approximation to the original problem. When it is a good approximation, solving it precisely undoubtedlyhelps the overall convergence. Yet, how well the quadratic problem approximates the original problem depends on manyfactors: the form of link performance function, network topology, congestion level, just to name a few. If the quadraticapproximation is of poor quality due to, for example, the inaccurate estimation of the second order derivatives, it may becounter-productive to solve it precisely.

To elaborate this point, consider a two-link example whose equilibrium solution may be obtained through a graphicalsolution as shown in Fig. 5 (A similar example was used in Gentile (2012) to illustrate the underlying mechanism of LUCE).In the plot, ti(/i) denotes the travel time function on link i = 1, 2, and Ti(/i), i = 1, 2 denotes the linear approximation at thecurrent solution u. Note that the slopes of Ti are derivatives of link travel time, or the second-order derivative of the objectivefunction. The solid dot in the plot marks the true equilibrium point /⁄, where t1 and t2 cross each other. The intersection of T1

and T2, marked by the empty dot in the plot and denoted using /a, is the exact solution to the quadratic approximation. Asshown in the plot, when the second order derivatives are correctly computed, /a is much closer to /⁄ than u.

Suppose now that the derivative of the second link g2 cannot be estimated correctly.4 In this case, the estimated linkcost is no longer the tangent line of t2. If the estimated derivative is smaller than the true value, then the linear approxima-tion may be depicted by a line similar to bT 2ð/2Þ in Fig. 5. It is evident from the plot that now the solution to the quadraticapproximation, denoted by /b (the solid square), no longer improves the current solution. As a result, the descent directionobtained from /b could deviate significantly from the steepest descent direction of a quadratic approximation. Although the

4 For this example, the derivative can always be computed correctly. Yet, in the context of solving the traffic assignment using OBA or LUCE, the second-orderderivatives are difficult to estimate with high accuracy, see (Nie, 2012) for discussions.

0 5 10 15 20 25 30 350

0.2

0.4

0.6

0.8

1

1.2

(a) Anaheim

Step

Siz

e

Iteration

OBALUCE

0 5 10 15 20 25 30 35 400

0.2

0.4

0.6

0.8

1

1.2

(b) Barcelona

Step

Siz

e

Iteration

OBALUCE

0 10 20 30 40 50 60 70 800

0.2

0.4

0.6

0.8

1

1.2

(c) Winnipeg

Step

Siz

e

Iteration

OBALUCE

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1

1.2

(d) Chicago Sketch

Step

Siz

e

Iteration

OBALUCE

Fig. 7. Average step sizes in main iterations for OBA and LUCE.

26 J. Xie et al. / Transportation Research Part B 56 (2013) 15–30

line search will eventually ensure admitting only solutions that improve the current solution, an ineffective descent directionis bound to produce smaller improvements. We can also explain now why OBA can outperform LUCE in such a circumstance.The key is that, in the line search, OBA not only adjusts the step size, but also refines the search direction itself. /c (the emptysquare in the plot) represents a new search direction obtained in line search, which must lie between u and /b. When /b isas seriously defective as portrayed in Fig. 5, such a refinement could help move it closer to the correct solution of the qua-dratic approximation.

To confirm the above analysis, another experiment is performed in which errors are deliberately introduced into the sec-ond-order derivative gi, i = 1, 2, 3 in our toy example. Specifically, we multiply {gi} with a coefficient h 2 [0,2] in every iter-ation. As reported in Fig. 6, when his reduced to 0.7, the performance of LUCE and OBA (in terms of the total number of mainiterations required to achieve a relative gap of 10�12) becomes comparable. When h is further reduced to 0.58, LUCE lagsbehind OBA. For even smaller values of h (h < 0.58), LUCE failed to converge beyond certain threshold, a behavior remarkablysimilar to what we have observed in large experiments.

Interestingly, OBA maintains a fairly reliable convergence performance even with the faulty second-order derivativeinformation. Note that for all h < 1.0, OBA required about 30 main iterations to reach the desired relative gap. Apparently,the ability to refine the search direction in line search plays a key role in this success. When h is larger than 1, Fig. 6(d) showsthat both OBA and LUCE can reach the required precision, although more iterations are required.

Finally, why would LUCE get stuck and fail to improve the relative gap beyond certain threshold, even with line search? Aplausible explanation is that as the solution reaches the close vicinity of the true equilibrium, the defective descent directionmakes it difficult to find a ‘‘global’’ step size that guarantees improvements when applying to all node subproblems. Whenthis happens, the algorithm designers may choose between two bad options. They can run the line search process until a stepsize that guarantees improvement is found, but the chances are that such a step size may be too small to make any

0 5 10 15 20 25 30 35 40−16

−14

−12

−10

−8

−6

−4

−2

0

2

4R

elat

ive

gap

(10e

)

Time [seconds]

OBALUCELUCE−UBLUCE−UD

(a) Anaheim

0 0.5 1 1.5 2 2.5 3−16

−14

−12

−10

−8

−6

−4

−2

0

2

4

Time [minutes]

OBALUCELUCE−UBLUCE−UD

(b) Barcelona

0 0.5 1 1.5 2 2.5 3 3.5−16

−14

−12

−10

−8

−6

−4

−2

0

2

4OBALUCELUCE−UBLUCE−UD

(c) Winnipeg

0 0.5 1 1.5 2 2.5 3 3.5−16

−14

−12

−10

−8

−6

−4

−2

0

2

4OBALUCELUCE−UBLUCE−UD

(d) Chicago sketch

Rel

ativ

e g

ap (1

0e)

Rel

ativ

e g

ap (1

0e)

Rel

ativ

e g

ap (1

0e)

Time [minutes] Time [minutes]

Fig. 8. Convergence performance of different variants of LUCE.

J. Xie et al. / Transportation Research Part B 56 (2013) 15–30 27

noticeable change. Another option is to stop the search process whenever the step size is smaller than a pre-determined min-imum. Yet, using such a predetermined minimum could produce oscillations. To support the above conjecture, Fig. 7 plotsthe average step sizes used by LUCE and OBA for solving the four small networks tested in Section 4. Note that the step size ofa main iteration is averaged for all origins and inner iterations. The plots demonstrate that LUCE consistently employs muchsmaller step sizes compared to OBA, which agrees with the above explanation and likely an important contributor to itsworse-than-expectation performance.

In a nutshell, the mysterious convergence behavior of LUCE seems to be rooted in its unconditional trust on the solutionfrom the quadratic approximation. When these approximations are constructed on faulty information, the sub-optimal des-cent direction generated by them slows down the overall convergence.

7. Limitation of origin-based flow aggregation

Having explained why LUCE’s performance is worse than expected, one naturally wonder if there is room for improve-ments. Because the inaccurate second-order derivatives are considered the main cause, one idea is to estimate these deriv-atives using their upper bounds instead of lower bounds (see (13)). Nie (2012) showed that the upper bound estimationhelps improve the convergence performance of OBA, but to what extent the strategy can help LUCE is unclear. Anotherimprovement is inspired by a shortcoming shared by OBA and LUCE that stems from the use of origin-based flow aggrega-tion. Note that both OBA and LUCE are designed to update the aggregated origin flows all at once. Accordingly, they generatea search direction by solving the node subproblems in a descending topological order. Each node sub-problem is constructedbased on flows on the portion of the bush upstream of the node. The problem is that, when a node subproblem is solved, the

28 J. Xie et al. / Transportation Research Part B 56 (2013) 15–30

flow distribution on the upstream portion will change, so will the problem itself. Consequently, the search direction gener-ated by downstream node subproblem is not consistent with the upstream flows after the potential changes are accountedfor. To overcome this problem, one would have to apply flow updates and re-solve the node problem until it converges, be-fore moving to the next node subproblem. However, because bush flow update is bound together in OBA and LUCE, such aninconsistency in the search direction is unavoidable. To alleviate this problem, Nie (2010) suggests applying flow updatesimmediately after each node subproblem is solved. This strategy makes line search unsuitable and unnecessary becausethe flow update is no longer bound at the bush level. Nie (2010) notices that the strategy leads to substantial improvementin OBA’s performance.

Two variants of LUCE are implemented to test how the above ideas might help its performance: one is based on the upperbound estimation of the second-order derivative (referred to as LUCE-UB), the other is based on the decomposed flow updatestrategy proposed in Nie (2010) (referred to as LUCE-UD). Fig. 8 compares these variants with the original LUCE and OBA onthe four small networks. Although the relative performance varies across the problems, two general trends can be observed.First, the use of the upper bound estimation does not seem to help LUCE much, if at all. In fact, only in the Chicago Sketchnetwork are noticeable differences between LUCE and LUCE-UB observed.5 Second, LUCE-UD does make consistent improve-ments in all problems. For the Winnipeg network, it even outperforms OBA with a very significant margin. For other networks, itis not as good as OBA, but is clearly better than LUCE and LUCE-UB. Importantly, even with the decomposed flow update strat-egy, LUCE-UD can still run into a deadlock on its convergence path. This indicates that the problem associated with the sub-opti-mal search direction still exists.

In summary, these additional experiments further confirm that OBA outperforms LUCE mainly because it is able to adjustthe search direction through line search. This is probably bad news for both algorithms because the heavy dependence online search means that the search direction is not properly guided by the information of second-order derivatives. Indeed,OBA and its variants are subject to two major limitations rooted in the use of origin-based flow aggregation. First and fore-most, the closed form second order derivatives are unknown due to the complex interactions between the origin-basedflows. Second, operating in an aggregated origin flow space is inefficient, because it restricts the ability to make localimprovements (those that could be committed to a small portion of a bush, but not admissible once considered aggregatelywith other parts), which could become vitally important at the later stage of convergence.

8. Concluding remarks

We have shown that LUCE’s inability to adjust sub-optimal search directions, which are obtained from solving inaccuratequadratic approximations of node subproblems, is the main reason why it is consistently outperformed by OBA. Our exper-iments and analysis also reveal how the use of origin flow aggregation has become a performance bottleneck for OBA andLUCE.

What can we learn from these findings about the design of high performance TAP algorithms? It seems that the key isto decompose the original problem so that the resulting subproblems can be solved quickly and precisely. Operating in ahighly aggregated link flow space is a main reason for the poor convergence performance of the Frank–Wolfe algorithm.Although the use of bushes significantly improves efficiency, the performance of OBA and LUCE is still limited by the bush-level assignment operations, as revealed in the experiments conducted in this study. However, ‘‘more’’ decomposition isnot always better. For example, a widely adopted decomposition scheme is to solve the assignment problem as a sequenceof route choice problems for each origin–destination pair (Bertsekas and Gafni, 1983; Larsson and Patriksson, 1992; Jaya-krishnan et al., 1994; Lee et al., 2002). In the context of standard traffic assignment, this scheme leads to the most disag-gregated subproblem because route flows cannot be further decomposed. However, even though the subproblem underthis decomposition scheme can be solved fairly easily (assuming a set of used routes is given), constructing, storingand manipulating all these routes raise efficiency concerns. Note that the number of used routes stored by these conven-tional route-based algorithms is proportional to the number of O–D pairs, which easily reaches several million in today’sregional networks. Operating such a sheer number of routes is neither efficient nor necessary because of significant over-lapping among routes.

A compromise that appears to hold promise is to keep the assignment at the route level while getting rid of the over-lapped portions as much as possible. This requires the assignment operations be focused on the segments where theroutes differ, which leads to a level of flow aggregation that falls between bush and route flows. Specifically, the flowsare aggregated on all routes that share one of these alternative segments. It is important to note that there is no need tostore these alternative route segments for each O–D pair. In fact, there is no need to store them at all. In Algorithm B(Dial, 2006), for example, such route segments are identified from a bush on the fly whenever flow shifting takes place.In contrast, Bar-Gera (2010) proposed to explicitly storing these route segments, termed ‘‘paired alternative segments’’(or PAS), in his TAPAS algorithm. Experiments conducted by Bar-Gera (2010) showed that the number of active PASs ismuch smaller than the number of O–D pairs. Consequently, it is easy to see that storing PAS could bring significantcomputational benefits, not only because it could save the time spent on repeatedly identifying same segments, but also

5 The upper bound estimation may help OBA more for a subtle reason that requires a lengthy explanation. Simply speaking, it has to do with a refinement inthe estimation of second order derivatives in the original OBA. The reader is referred to Nie (2012) for details.

J. Xie et al. / Transportation Research Part B 56 (2013) 15–30 29

because these PASs can be shared across different bushes. Yet, it is worth noting that operating with PASs further com-plicates the algorithmic design and implementation. Also, additional costs may be incurred in managing the set of PASs.The only computational study known to us that includes both TAPAS and Algorithm B is Inoue and Maruyama (2012),who ranked TAPAS and Algorithm B as the top two performers, with TAPAS being consistently ahead of Algorithm B. It isbeyond the scope of this paper to examine whether or not directly managing the shared route segments will pay off, andif so, what is the best way to do it. We leave the answers to these intriguing questions to future research.

Acknowledgements

We would like to thank Tom van Vuren of Mott MacDonald and Klaus Noekel of PTV Group for sharing the PRISM net-work. The China Scholarship Council (CSC) provided a scholarship that had supported the first author to visit NorthwesternUniversity between 2010 and 2012, during which this work was conducted. We are grateful to David Boyce of NorthwesternUniversity as well as two anonymous reviewers for their valuable comments. The remaining errors are the authors’ alone.

Appendix A

The origin-based formulation of the traffic assignment problem on the network in Fig. 3 can be stated as:

min zð/Þ ¼X3

a¼1

Z xa

0taðuÞdu; ð37Þ

subject to : xa ¼ q/a; 8a 2 f1;2;3g ð38ÞX3

a¼1

/a ¼ 1 ð39Þ

/a P 0; 8a 2 f1;2;3g ð40Þ

The first-and second-order derivatives with respect to /a are:

@zð/Þ@/a

¼ taq;@z2ð/Þ@/2

a

¼ gaq2: ð41Þ

When adapted to this small network, the general procedure for LUCE and OBA can be described as follows:

Step 0: Set iteration index k = 0. Initialize xka through an all-or-nothing assignment based on free flow travel time, and set

uka ¼ xk

a=q; 8a.Step 1: Update link travel time tk

a and its derivative gka; 8a 2 A. Set k = 1.

Step 2: Convergence test: Let u = min{ta, a = 1, 2, 3}, if 1� uqPa

tkaxk

a< �, stop; else go to Step 3.

Step 3: For LUCE, find search direction /ka by solving the following quadratic program

min zkð/Þ ¼X3

a¼1

tkaq� gk

aq2ua

� �/a þ 0:5gk

aq2/2a

� �; subject to constraints ð39Þ and ð40Þ; ð42Þ

then go to Step 4.1.For OBA, set �a ¼ argmin tk

a; a ¼ 1;2;3

, and the search direction

/ka ¼ uk

a �min uka; k

tka � tk

�a

q gka þ gk

�a

� �( ); 8a–�a; /k

�a ¼ 1�Xa–�a

/ka;

then go to Step 4.2.Step 4: Line search.

Step 4.1: Initialize k = 1.Step 4.2: Compute

r ¼

X3

a¼1

tka xa þ qk /k

a �uka

� �� �/k

a �uka

� �q LUCE

um3a¼1tk

a xa þ q /ka �uk

a

� �� �/k

a �uka

� �q OBA

8>><>>: ð43Þ

If r < 0, go to Step 5; otherwise set k = k/2. For LUCE, repeat Step 4.2; for OBA, go to Step 3.

Step 5: For LUCE, update /kþ1a ¼ uk

a þ k /ka �uk

a

� �; For OBA, update /kþ1

a ¼ /ka. For both algorithms, update xkþ1

a ¼ q/kþ1a . Set

k = k + 1, go to Step 1.

References

Bar-Gera, H., 1999. Origin-based algorithms for transportation network modeling. Ph.D. Thesis, Civil Engineering, University of Illinois at Chicago, Chicago,IL.

Bar-Gera, H., 2002. Origin-based algorithm for the traffic assignment problem. Transportation Science 36 (4), 398–417.Bar-Gera, H., 2010. Traffic assignment by paired alternative segments. Transportation Research Part B 44 (8–9), 1022–1046.Beckmann, M., McGuire, C.B., Winsten, C.B., 1956. Studies in the Economics of Transportation. Yale University Press, New Haven, Connecticut.Bertsekas, D.P., 1976. On the Goldstein–Lvitin–Poljak gradient projection method. IEEE Transactions on Automatic Control 21 (AC-2), 174–184.Bertsekas, D.P., Gafni, E.M., 1983. Projected newton methods and optimization of multicommodity flows. IEEE Transactions on Automatic Control 28 (AC-

12), 1090–1096.Boyles, S.D., 2012. Bush-based sensitivity analysis for approximating subnetwork diversion. Transportation Research Part B 46 (1), 139–155.Dial, R.B., 1971. A probabilistic multipath assignment model that obviates path enumeration. Transportation Research 5 (2), 83–111.Dial, R.B., 2006. A path-based user-equilibrium traffic assignment algorithm that obviates path storage and enumeration. Transportation Research Part B 40

(10), 917–936.Florian, M., Hearn, D., 1995. Network routing. Handbooks in OR & MS, vol. 8. Elsevier Science, Oxford, UK (Chapter Network Equilibrium Models and

Algorithms, pp. 485–549).Frank, M., Wolfe, P., 1956. An algorithm for quadratic programming. Naval Research Logistics Quarterly 3 (1–2), 95–110.Fukushima, M., 1985. A modified Frank–Wolfe algorithm for solving the traffic assignment problem. Transportation Research Part B 18 (2), 169–177.Gallager, R.G., 1977. A minimum delay routing algorithm using distributed computation. IEEE Transactions on Communications 25 (1), 73–85.Gentile, G., 2012. Local User Cost Equilibrium: A Bush-Based Algorithm for Traffic Assignment. Transportmetrica Available On-Line.Hearn, D.W., Lawphongranich, S., Ventura, J.A., 1985. Finiteness in restricted simplicial decomposition. Operations Research Letters 4 (3), 125–130.Inoue, S., Maruyama, T., 2012. Computational experience on advanced algorithms for user equilibrium traffic assignment problem and its convergence error.

In: 8th International Conference on Traffic and Transportation Studies Changsha, China, vol. 43, August 1–3, 2012, pp. 445–456.Jayakrishnan, R., Tsai, W.K., Prashker, J.N., Rajadhyaksha, S., 1994. A faster path-based algorithm for traffic assignment. Transportation Research Record

1443, 75–83.Larsson, T., Patriksson, M., 1992. Simplicial decomposition with disaggregated representation for the traffic assignment problem. Transportation Science 26

(1), 4–17.LeBlanc, L.J., Morlok, E.K., Pierskalla, W.P., 1975. An efficient approach to solving the road network equilibrium traffic assignment problem. Transportation

Research 9 (5), 309–318.LeBlanc, L.J., Helgason, R., Boyce, D.E., 1985. Improved efficiency of the Frank–Wolfe algorithm for convex network problems. Transportation Science 19 (4),

445–462.Lee, D.H., Nie, Y., Chen, A., 2002. A conjugate gradient projection algorithm for the traffic assignment problem. Journal of Mathematical and Computer

Modelling 37 (7-8), 863–878.Nie, Y., 2006. A Programmer’s Manual for Toolkit of Network Modeling (TNM). University of California, Davis.Nie, Y., 2010. A class of bush-based algorithms for the traffic assignment problem. Transportation Research Part B 44 (1), 73–89.Nie, Y., 2012. A note on Bar-Gera’s algorithm for the origin-based traffic assignment problem. Transportation Science 46 (1), 27–38.Patriksson, M., 1994. The Traffic Assignment Problem: Models and Methods. VSP, Utrecht, The Netherlands.Spiess, H., Florian, M., 1989. Optimal strategies: a new assignment model for transit networks. Transportation Research Part B 23 (2), 83–102.Wardrop, J.G., 1952. Some theoretical aspects of road traffic research. Proceedings of the Institution of Civil Engineers, Part II 1 (3), 325–378.

30 J. Xie et al. / Transportation Research Part B 56 (2013) 15–30