Constrained Newton Methods for Transport Network Equilibrium Analysis

TSINGHUA SCIENCE AND TECHNOLOGY ISSNll1007-0214ll14/16llpp765-775 Volume 14, Number 6, December 2009

Constrained Newton Methods for Transport Network Equilibrium Analysis* CHENG Lin (ё ॾ)**, XU Xiangdong (༘ົՊ), QIU Songlin (௎ഞॿ) School of Transportation, Southeast University, Nanjing 210096, China Abstract: A set of constrained Newton methods were developed for static traffic assignment problems. The Newton formula uses the gradient of the objective function to determine an improved feasible direction scaled by the second-order derivatives of the objective function. The column generation produces the active paths necessary for each origin-destination pair. These methods then select an optimal step size or make an orthogonal projection to achieve fast, accurate convergence. These Newton methods based on the constrained Newton formula utilize path information to explicitly implement Wardrop’s principle in the transport network modelling and complement the traffic assignment algorithms. Numerical examples are presented to compare the performance with all possible Newton methods. The computational results show that the optimal-step Newton methods have much better convergence than the fixed-step ones, while the Newton method with the unit step size is not always efficient for traffic assignment problems. Furthermore, the optimal-step Newton methods are relatively robust for all three of the tested benchmark networks of traffic assignment problems. Key words: traffic assignment; network flow; Newton method; column generation; line search

Introduction Traffic assignment models deal with the problem of assigning vehicular trips to a transport network, given an origin-destination (OD) trip matrix and the network characteristics. The transport network is represented by a directed graph where each link is associated with a link performance function, typically the travel time. The traffic assignment model is based on the behavioural assumption that each user travels on the path that minimizes the travel time from origin to Received: 2008-10-20; revised: 2009-06-09

* Supported by the National Natural Science Foundation of China (No. 50678037), the National Key Basic Research and Development (973) Program of China (No. 2006CB705500), and the National High-Tech Research and Development (863) Program of China (No. 2007AA11Z205)

** To whom correspondence should be addressed. E-mail: [email protected]; Tel: 86-25-83795649

destination. This rule implies that at equilibrium the path flow patterns are such that the travel times on all the used paths connecting any given OD pair will be equal and that the travel times on all the used paths will be less than or equal to the travel times of any unused paths. The network is then in user equilibrium (UE) with no user able to reduce their travel time by unilaterally changing paths[1]. The state of the art in traffic assignment solutions has evolved around two major directions, “link-based” and “path-based” methods. With the link-based methods, the paths are not conserved at each iteration, such as in the Frank-Wolfe algorithm[2] and restricted simplicial decomposition[3]. The path-based methods are path-enumeration algorithms where all the used paths are stored, including the gradient projection[4,5], disaggregate simplicial decomposition[6,7], and gap function[8,9] methods. The main difference between these two kinds of methods is that the optimization in the

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Tsinghua Science and Technology, December 2009, 14(6): 765-775

link-based methods is performed in the link-flow domain, while that of the path-based methods is performed in the path-flow domain. The by-product of path-based methods is the abundant information on the path flow patterns at user equilibrium. As is well known, the path flow pattern is not unique at user equilibrium[1], but it is used in various applications. For example, a path flow solution is typically used for real-time assignments, which is not possible with the link-based Frank-Wolfe algorithm, in advanced traveller information systems or advanced traffic management systems[10]. Sensitivity analyses[11] require a set of reference path flows, from which a particular path flow pattern can be extracted. The Newton methods can be used to provide reference path sets for such flow sensitivity analyses. The path solution is related to a class of bilevel problems[12] which uses the sensitivity of the link flow to the OD demand. When path travel costs are nonadditive or nonlinear[13], link flows can not be used alone to formulate and solve the traffic assignment. Thus all of these types of problems need a method to explicitly formulate and solve the problem in the path flow domain. This paper presents constrained Newton methods for solving path-formulated static traffic assignment problems. Column generation, orthogonal projection[4] or line search[14] can be incorporated into the solution procedure. With column generation, the active paths joining each OD pair are generated only when needed. The line search or orthogonal projection methods are used to avoid infeasible flows on the used paths and to equilibrate the demand among the active paths as well as the newly found shortest path for each OD pair. These methods are called constrained Newton methods because they extend the Newton formula over the restriction condition of the origin-destination demand conservation. The methods are fast and accurate but need to save and utilize the path information.

1

Path-Formulated User Equilibrium Assignment

Given a transport network G(A, N) where A is the set of links and N is the set of nodes, each directed link aA is associated with a positive travel time ta(xa) which is an increasing function of the link flow xa. W is the set of origin destination pairs and each pair wW

has a given traffic demand qw. The user equilibrium assignment problem is stated as follows.

¦³

min z ( x )

a

s.t.

¦

k K w

xa

wW k K w

f kw . 0,

ta ( s )ds

(1a)

qw , w  W

(1b)

f kw ŽG akw , a  A

(1c)

 k  K w , w W

(1d)

f kw

¦¦

xa 0

where xa is the link flow. ta is the travel time on link a. G akw =1 if link a belongs to path k; otherwise, G akw =0. fkw is the flow on path k within OD pair w. Kw is the set of paths within OD pair w. qw is the origin-destination demand between pair w. The user equilibrium assignment problem is pathformulated due to the constraints of Eqs. (1b)-(1d). Alternatively, the same problem can be formulated using link variables indexed by origin or destination[7]. This paper focuses on the formulation using path variables. Consistent with the necessary and sufficient conditions of optimality, the equilibrium condition for each OD pair is stated as ta ŽG akw W w* if f kw ! 0 ½ ¦ ° a A (2) ¾ k , w w * w ta · G ak . W w if f k 0 ° ¦ a A ¿ * w where W w min{ta ŽG ak } is the shortest path travel a A

time within OD pair w, equal to the Lagrange multiplier at this point of Eq. (1b) at the optimal point. The solution of the UE assignment model can be characterized by Wardrop’s principle. Specifically, for a given path k connecting origin-destination pair w, the condition holds two possible combinations of path flow and travel time. Either the flow on the k-th path is zero, where the travel time on the path,

¦

t ŽG akw ,

a a

must be greater than or equal to the OD pair specific Lagrange multiplier, Ww, or the flow on the k-th path is positive, where

¦

t ŽG akw

a a

W w . In any event, the La-

grange multiplier of a given OD pair in Eq. (1b) is less than or equal to the travel time on all paths connecting this pair; hence Ww equals the minimum path travel time within OD pair w.

2

Constrained Newton Methods

In unconstrained minimization, the optimization can

CHENG Lin (ё ॾ) et al.ġConstrained Newton Methods for Transport Network Equilibrium Analysis

reach the bottom of the objective function where the gradient vector has all zero components. In a constrained minimization problem such as traffic assignments, however, all of the gradient components do not vanish at the minimum. Figure 1 shows a traffic assignment objective function defined over a feasible region in the path flow domain. The shadow area in Fig. 1 denotes the feasible region. When the minimum is on the boundary of the feasible region, the gradient is either equal to or greater than a constant for each OD pair, compatible with Wardrop’s principle. Since the gradient of the objective represents the vector of path travel times in the path flow domain, then this constant for each OD pair is probably equivalent to the shortest path time within the pair. The non-uniqueness of the path flows make the feasible region changeable with respect to the path flow vector. However, the uniqueness of the path time vector could be used to find a descent direction in path-based methods, therefore, improving the solution efficiency through utilization of second-order information in the objective function.

767

For path flows, the objective function, z(x), in Eq. (1a) is approximated by the following second-order Taylor series: z ( f ) # z ( f ( n ) )  ( f  f ( n ) )T ŽIJ ( f ( n ) )  1 ( f  f ( n ) )T ŽIJ' ( f ( n ) )Ž( f  f ( n ) ) (4) 2 where IJ ( f ( n ) ) denotes the vector of path costs and IJ' ( f ( n ) ) denotes their derivatives, which are the first

and second order derivatives of the objective function with respect to the path flows. f ( n ) is the vector of current path flow solutions. The minimization is then performed on an approximate function. The first derivative of the approximate function is n IJ ( f ) IJ ( f ( n ) )  ( f  f ( n ) )T ŽIJ' ( f  ) IJ ( f ( n ) )  'f ( n ) ŽIJ' ( f ( n ) )

(5)

The improved increment along the path flow domain is then: 'f ( n ) [IJ ( f ( n ) )  IJ ( f )]Ž[IJ' ( f ( n ) )]1 (6) For convenience of expression, the active paths are partitioned into the shortest path, k w* , and the non-shortest paths, kKw, for any pair wW. Near the optimum, the flows on all the non-shortest paths will decrease according to the direction expressed by Eq. (6). The decreased flows on the non-shortest paths within each OD pair will then shift to the corresponding shortest path. The shortest path, k w* , if not included in the current path set, will join the path set K w( n ) at the next iteration, i.e., K w( n 1) : K w( n ) ‰ kw* .

Fig. 1 Optimum of capacitated UE assignment

The current method relies on only the used elements with path flow vector with an iterative solution to the problem in Eq. (1) to move from one feasible point to an improved feasible point. Given a feasible point, path flow vector f ( n ) , a moving flow 'f ( n ) and a step size O are determined such that the following two properties are true: x f ( n )  O Ž'f ( n ) is feasible. x The value of the objective function f ( n )  O Ž'f ( n ) is better than that at f ( n ) .

at

Since the flow must be conserved for every OD pair, the sum of the reduced flows will be equal to the increased flow on the shortest path or will be equal to the path flow on the shortest path if the shortest path had no flow since it was just found. The flow change in Eq. (6) in the used paths can also be expressed by 'f (n ) [IJ ( f ( n ) )  W ( f kw* )Ž I ]Ž[IJ c( f ( n ) )]1 (7a) w

where I is a vector having the appropriate dimension with the elements all being unity and W ( f kw* ) reprew

senting the cost of the shortest path k

* w

for pair w.

(3)

The flow or flow change on the shortest path can then be calculated by f kw* , (n 1) qw  ¦ kK w f kw, (n 1) , w  W (7b)

The process is repeated until the objective function cannot be further improved.

If the flow changes are mainly between paths belonging to a currently used path set, the flows on some

This leads to a new point, f ( n 1) f ( n )  OŽ'f ( n )

w

k z k w*

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Tsinghua Science and Technology, December 2009, 14(6): 765-775

paths may become negative, which is not allowed in traffic assignment problems. The first feasibility in Eq. (3) for a new point can be satisfied in two ways as follows. (1) If the fixed step strategy results in infeasible path flows, an orthogonal projection is made on the constraint boundaries of Eq. (1d) by setting the negative path flow to zero. This fixed-step strategy is also referred to as the gradient projection (GP) method[4,5] when the step size O equals one. (2) In contrast to the fixed step strategy, the current strategy employs an optimized step size, subject to various constraints. For example, the step could be limited by the non-negativity of the link or path flows or the optimal link flows. Both strategies can be viewed as constrained Newton methods because both use the extended Newton formula in Eq. (6), where the feasible decent direction is not only modified by information for the latest shortest path but also scaled by second-order information in the objective function (called the scaling matrix). Both make successive moves towards the minimum of the quadratic approximation of the objective function by shifting flows from the used paths on the particular OD pair to its shortest path. The flow increment guarantees that the second property that the objective function at f ( n )  O Ž'f ( n ) is better than that at f (n) .

The movement direction expressed by the first term in the right hand side (RHS) of Eq. (6) is determined according to Eq. (7), the critical steps become how to calculate the scaling matrix expressed by the second term in the RHS of Eq. (6) and how to decide the step size expressed in Eq. (3). 2.1

Newton methods with fixed steps

The directional step provided by the gradient in Eq. (7a) has to be scaled to coordinate the directions of the path flows within the OD pair, as well as the directions among different OD pairs expressed by the second term on the RHS of Eq. (7a). The scaling is determined by the derivative of the path costs, which is the diagonal Hessian matrix of the objective function in Eq. (1a) due to the assumption that the path selection for different OD pairs is not related. The Hessian is defined in the domain of the used paths not including the

shortest paths with its diagonal elements being w2 z( f ) ¦ tac Ž(G akw  G akw w* )2 , k z kw* , w (wf kw ) 2 aA

(8)

where tac denotes the derivative of the travel time function for link a. The path flows on the non-shortest paths will always decrease as the flow decreases on the non-shortest paths in each OD pair shifting to the corresponding shortest path. However, the amount of flow for each path has to be controlled to avoid negative path flows. In the path flow domain, if the flow movement step size is predetermined, a simple scheme is to set negative path flows to zero whenever the movement results in an infeasible solution. This operation is usually called an orthogonal projection expressed as f kw, ( n 1) [ f kw, ( n )  'f kw, ( n ) ] , k  K w , k z k w* (9) where [Ž] is the orthogonal projection operator and

n is the number of iterations. The occurrence of orthogonal projection is determined by the size of the predetermined step. Previous algorithms[5,15] used a similar method but with a step of unity, called the GP algorithm. The GP algorithm is a special case of the Newton methods with fixed steps. Their experience showed that a step size, O, equal to 1 achieved a very good convergence rate when the flow update equations used automatic scaling based on the second derivative information. The effectiveness of the GP algorithm excellence is attributed to the manipulation of the orthogonal projection as examined in many numerical tests[5,16,17]. The GP algorithm has many advantages over other algorithms[16,17] due to the projection procedure. However, the orthogonal projection is not always guaranteed by the constant step size of unity. An example given later shows that the Newton method with a step size of unity yields poor convergence if no projection occurs. Thus the constant step size has to be carefully selected to guarantee the occurrence of the orthogonal projection. 2.2

Newton methods with optimal steps

The weakness of Newton methods with fixed steps can be overcome if the step is optimized. If an optimal step size is employed in the Newton methods, the elements of the Hessian matrix do not need to be calculated precisely as in Eq. (8). The diagonal elements of the

CHENG Lin (ё ॾ) et al.ġConstrained Newton Methods for Transport Network Equilibrium Analysis

Hessian matrix can be approximated by the OD demand or path flow to reduce the overhead associated with calculating the second derivatives of the objective function. The approximate diagonals of the Hessian matrix can be expressed as ­(qw )1 , k , w, if approximated ° by the OD demand; w2 z( f ) ° # ® w 1 w 2 (wf k ) °( f k ) , k , w, if approximated by ° the path flow ¯ (10) Either a fixed or optimal step can be applied if the flow update equations use automatic scaling based on the second derivative information. However, if the Hessian matrix is approximated, a step size of unity might not be appropriate, but should be modified based on decision variables not being negative or on a line search. The non-negativity of the path flows requires f kw  Opath Ž'f kw . 0, k  K w , w  W (11a) which is rewritten as fw Opath -  k w , k  K w , w  W 'f k

(11b)

The step sizes for the flows on every path must be coordinated so the optimal step size for all paths must satisfy the following relationship. § ­ fw ½ · Opath min ¨¨ min ® k w 'f kw  0 ¾ , 1¸¸ (11c) k K ¿ ¹ © w ¯ 'f k Then, the step size required by the link flows can be determined using mathematical programming. The second-order approximation of the objective function can be expressed as z ( x ) # z ( x ( n ) )  O Ž'x T Ž z c( x ( n ) )  1 2 O Ž'x T Ž z cc( x ( n ) )Ž'x ( n ) (12a) 2 where O is a feasible step size, 'x is the vector of the increased link flows, and zƍ and zƎ are the first and second order derivatives of the objective function (5a). Substituting 'x ( n ) { AŽ'f ( n ) (Incidence relationship for moving flows between a link and a path) and z cc( x )

t'

(Second derivatives of the function)

into Eq. (12a) gives

z ( x ) # z ( x ( n ) )  OŽ¦ ['xa Žta ] a A

x( n )

769

1  O 2 ¦ ['xa2 Žtac ] 2 aA

x( n )

(12b) The optimal step size for link flows may be calculated by solving the equation z' (O ) 0 . The step sizes for each link flow must be coordinated and not larger than Opath. Therefore, the feasible link flow step should satisfy °­ ¦ 'xa Žta °½ O * # min ® aA 2 , Opath ¾ (12c) ¯° ¦ aA 'xa Žtac ¿° 2.3

Discussion of Newton methods

Two strategies with fixed or optimal steps are presented for Newton methods in Eqs. (3) and (7). The first strategy uses a fixed step and requires the calculation of the Hessian matrix. When the move towards the minimum in the negative gradient direction results in an infeasible solution, the direction has to be projected onto the feasible region. The projection operation may result in rapid changes to the demand conservation conditions and allow moves towards the minimum along the face of the feasible region of the path flow variables. However, although experience[5,16,17] has shown that a step size of unity works very well, the predetermination of the proper step size is not easy for all types of the networks. Traffic practitioners sometimes have to try various steps to realize an acceptable convergence ratio. Also there is no theoretical assurance that projection always occurs with a unit step size, since moves towards the minimum may not result in the infeasible solutions so projection will not occur which will hurt the convergent ratio. To overcome the weakness of the Newton methods with fixed steps, the second strategy introduces an optimal step size with the requirement of non-negativity of the decision variables or line search. The optimal step size gives fast convergence as the gradient projection method[4]. In most situations, traffic practitioners are accustomed to calculating the gradient ’z( f ) but not the Hessian ’2 z( f ) of the objective function. However, the Hessian can usually be approximated using the origin-destination demand or path flow without increasing the number of iterations. The approximation of the Hessian matrix then compensates for the expense of calculating the optimal step. The approximate Hessian is not appropriate for

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Tsinghua Science and Technology, December 2009, 14(6): 765-775

Newton methods with fixed steps, but either a precise or approximate Hessian can be used in the optimal-step strategy. The combination of a scaling matrix and a variable step size gives constrained Newton methods for traffic assignment problems for a systematic framework as illustrated in Fig. 2. Additionally, unlike the Frank-Wolfe method[2] or the simplicial decomposition method[3], the Newton methods take advantage of the availability of information in the second-order derivative of the objective function and column generation[18] to achieve superior convergence and accuracy. The following shows the superiority of the gradient projection (or the Newton method with a step size of unity)[5,16].

methods. The actual nature of the objective functions and the constraints in the network assignment can be quite different.

Fig. 3 Graphical image of the Newton methods’ convergence

3 Solution Procedures Fig. 2 Framework for Newton methods with a negative gradient direction

The differences between the various Newton methods can be illustrated by the qualitative graphical comparison in Fig. 3. If the predetermined step is appropriate, orthogonal projection will occur and the Newton methods will move toward the minimum along the face of the path-flow region. Computational experience indicates that the Newton method with a unity step size outperforms other types of algorithms whenever projection occurs once or twice. However, if the projection does not occur, the fixed-step Newton methods move in directions that are almost orthogonal to the negative gradient direction. In such cases, the iteration always moves to the optimum along the inside region in a zigzag pattern, similar to the successive iterations of the Frank-Wolfe method[1,2]. In contrast to the fixed-step Newton methods, the optimal-step Newton methods optimize the step size in the descent direction when they are close to the optimum. The performance of the optimal-step ones is competitive with the GP algorithm as the optimal-step Newton methods move along the scaled descent direction within the inside region. This example illustrates the qualitative reasons behind the performance of Newton

Various Newton methods have been developed to solve the user equilibrium traffic assignment. The Newton algorithms generally consist of initialization, column generation, equilibration, and convergence procedures. Column generation includes finding the shortest paths and augmenting the path set, while equilibration includes implementation of the constrained Newton formula to update the traffic flows. The differences between various Newton methods lie only in the equilibration with strategies using fixed or optimal step sizes combined with the actual Hessian matrix or an approximate matrix. The algorithm steps are as follows. Initialization x Set iteration counter n=0. x Solve the shortest path problem and create an initial path set K w(0) , w  W . x Perform All-or-Nothing assignment and obtain the initial path and link flow. Column generation x Increment the iteration counter: n:=n+1. x Update the cost and solve the shortest path problem. x Record the set of shortest paths kw( n )* and augment the path set.

CHENG Lin (ё ॾ) et al.ġConstrained Newton Methods for Transport Network Equilibrium Analysis

K w( n 1) ‰ k w( n )* , if k w( n )*  K w( n 1) ; other-

Set K w( n ) wise, set K

(n) w

K

( n 1) w

.

Equilibration x Calculate the move direction for the path flow vector. x Update the used path flows using the Newton methods with fixed or optimal step sizes. x If f kw 0 , drop path k: K w( n ) K w( n ) \ k . Here K w( n ) \ k denotes that path k is excluded from

path set K w( n ) . x Update the path and link flow patterns. Convergence w, ( n )  W kw*, ( n ) · f kw, ( n ) § W k w ¨ ¸ - H , terminate; othIf max ¦ w, ( n ) w ¨ ¸ qw Wk k © ¹ erwise, go to Column generation.

4

Computational Considerations

An important issue in path-based Newton methods is how to store the paths to limit memory usage due to the large networks used in the Newton methods. One shortest-path tree is built at each iteration from each origin and each path is part of the shortest-path trees from the corresponding origin node. The shortest path tree should not be stored as a node list to avoid repeated storage of the common portions of different paths from the same origin. If one node number is stored for each node, each tree in a network of N nodes requires N storage locations which results in NouN storage locations for each iteration, where No is the number of origins. Thus, the main memory requirement for the Newton methods becomes NouNuNi , where Ni is the number of iterations to reach convergence. Note that No and N are fixed by the network topology, while Ni depends on the convergence of the methods. Thus, the methods must converge well to be of practical use. Careful implementation is absolutely essential for Newton methods to perform well. Almost all of the time in the Frank-Wolfe algorithm is spent in the shortest-path routine for larger networks. Apart from the computational intensity associated with the shortest-path routine, Newton methods also have other procedures that consume excessive computational time. The four key operations that must be carefully implemented are the assigning of path flows to each link to

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find the total flow on each link, calculating the firstorder derivatives for all the paths between each OD pair, finding the second-order derivatives for each pair, and finding the optimal step size for both path and link flow movement for each iteration. With the optimal-step Newton methods, the second-order derivatives can be readily approximated by the origin-destination demand or path flow, so the computational intensity of the Hessian matrix is reduced to a simple line search in the negative gradient direction.

5

Numerical Results

Two numerical examples are presented to illustrate the performance of the Newton methods for traffic assignment problems. The first example illustrates the differences between the various Newton methods summarized in Fig. 3, while the second example compares the performance of Newton methods with fixed or optimal step sizes and the Frank-Wolfe algorithm for various networks. The assignments used a Bureau of Public Roads link-cost function, t t0 [1  0.15( x /c)4 ] ,

where t is the link travel time, t0 is the free-flow cost, x is the flow, and c is the link capacity. The convergence tolerance was set as H = 1×103 for all the assignment cases. 5.1

Example 1

Consider a small network consisting of 4 links and 3 nodes as shown in Fig. 4. The origin-destination demand and link performances are listed in Table 1.

Fig. 4 Table 1

Network

Input data for Example 1

Link

ta0

ca

1

10

600

2

17

500

3

9

800

4

60

400

q qOD(1,2)=600 qOD(1,3)=400 qOD(2,3)=600

The fixed-step Newton methods are affected by the magnitude of the step size. Nonlinear programming experience has shown that a step size of 1 usually works well with Newton methods. However, this example provides quite different insight into choosing the

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Tsinghua Science and Technology, December 2009, 14(6): 765-775

step size. Table 2 lists sequences of the objective function obtained for different step sizes (O { 0.2, 0.4, 0.5, 0.6, 0.8, 0.9, 1.0). The bold numbers represent where the objective function has reached the minimum but the calculation continues for more iterations while reaching the accuracy level of H = 1×103. The results in iterations after the bold number have the same values as the iterations of the Newton methods equilibrate the flows among the existing paths to satisfy the convergence tolerance. For a step size of O { 1.0, the convergence requires at least 1767 iterations, the slowest Table 2 Iteration

of all the step sizes. The convergence with a step size of unity is quite slow in this case because orthogonal projection does not occur in this case. Even with an adaptive step size that is larger but avoids any negative path flows, i.e., avoids orthogonal projection; the adaptive step size is always unity and the computational result is the same as for a step size of unity. Thus, the adaptive step size calculation verifies that projection never occurs in this example with a unit step size. Therefore, projection would also not occur with any step sizes less than 1.

Computational results for Newton methods with various step sizes Objective function value

ȜŁ0.2

ȜŁ0.4

ȜŁ0.5

ȜŁ0.6

ȜŁ0.8

ȜŁ0.9

ȜŁ1.0

1

0

0

0

0

0

0

0

2

21 974

21 974

21 974

21 974

21 974

21 974

21 974

3

21 828

21 746

21 727

21 721

21 746

21 775

21 815

4

21 764

21 721

21 721

21 734

21 779

21 900

5

21 738

21 721

21 721

21 724

21 743

21 798

6

21 727

21 721

21 722

21 740

21 859

7

21 723

21 721

21 721

21 730

21 786

8

21 722

21 721

21 728

21 834

9

21 721 21 721

21 721

10 11 12

21 725

21 778

21 724

21 816

21 721

21 723

21 772

21 721

21 722

21 803

13

21 722

21 767

14

21 793

15

21 721 21 721

17

21 721

21 785

21 763

…

…

1767

21 721

However, step sizes less than 1 seem to work quite well even when projection does not occur, especially for O { 0.4, 0.5, 0.6. Thus, O close to 1 is not always the best choice for network problems, with other constant step sizes yielding very rapid convergence even without orthogonal projection. Further work is needed to clarify why the step size O { 0.6 is better than O { 0.5, since O { 0.5, is equivalent to the optimal step size, as described in the following. Table 3 shows sequences of the step sizes and the objective function obtained with optimal step sizes which depend on the feasibility of the flows and the optimality condition. The scaling matrices were the exact Hessian matrix or the Hessian matrix approximated by the OD demand or the path flow. All

three schemes converge to an accuracy level of H =1×103 within five iterations, but with different step sizes. The optimal-step Newton methods have robust convergence without the instabilities of the fixed-step Newton methods, regardless of the combination of the scaling matrix, and step size. Comparison of Tables 2 and 3 shows that the Newton method scaled by the Hessian matrix with the optimal step has the same result as that of the Newton method of a fixed step size of O { 0.5. The same result occurs because the two methods have the same direction, scaling matrix, and step size. The approximation of the scaling matrix is strongly suggested for the optimal-step Newton methods to reduce the overhead associated with computing the Hessian matrix.

CHENG Lin (ё ॾ) et al.ġConstrained Newton Methods for Transport Network Equilibrium Analysis Table 3 Iteration

Computational results for Newton methods with optimal step sizes

Scaled by Hessian matrix Step size

Objective

2

0.5

3 4 5

Scaled by OD demand matrix Step size

Objective

21 974

0.021 60

0.5

21 727

0.5

21 721

0.5

21 721

1

773

Scaled by path flow matrix Step size

Objective

21 974

0.021 60

21 974

0.029 37

21 727

0.032 60

21 727

0.031 15

21 721

0.035 29

21 721

0.031 21

21 721

0.035 38

21 721

0

0

0

As shown in Fig. 5, the optimal-step Newton methods are superior to the fixed-step Newton method with unit step size[4,5], because the latter does not experience an orthogonal projection with zigzagging of the result. The phenomenon is also shown by the path flow variations shown in Fig. 6. The optimal-step Newton methods have much better convergence in this case showing that the Newton method with the unit step size is not always efficient for traffic assignment problems.

(a) In OD (1, 2)

Fig. 5 Comparison of Newton methods’ convergence for Example 1

5.2

Example 2

In the second example, Newton methods are used to solve various traffic assignment problems to test their robustness. The three traffic networks are described in Table 4, with the computational results in Table 5. For the first two networks, the Newton method with step size O { 1 (GP) exhibits more rapid convergence than the optimal-step Newton methods, but the fixed-step Newton method does not converge as well for the third case. Since projection does not occur the solution zigzags for the Sioux Falls network as in Example 1. The optimal-step Newton methods are relatively robust and convergent for all types of problems. The link-based Frank-Wolfe algorithm is not compared with the Newton methods here. However, published results in Jayakrishnan et al.[5], Chen and Lee[16],

(b) In OD (1, 3) Fig. 6 Path flows for Example 1

and Lee et al.[17] indicate that the GP algorithm (a special case of the Newton methods) is typically about twice as fast as the ordinary Frank-Wolfe algorithm, in terms of the number of iterations, in finding solutions.

774

Tsinghua Science and Technology, December 2009, 14(6): 765-775 Table 4 Three traffic networks for Example 2

Networks

No. of nodes

No. of links

13

19

4

8

24

12

24

76

528

Nguyen Inoue Sioux Falls Table 5 Networks

No. of OD pairs

References Nguyen and Dupuis[19] Inoue[20] OD demand: LeBlanc et al.[2]; Link capacity: Yang and Meng[21]

Computational results with the Newton methods for the three traffic networks Optimal step size

Fixed step size ȜŁ1

Optimal step size

scaled by Hessian

scaled by OD demand

Optimal step size scaled by path flow

No. of Iter.

Objective

No. of Iter.

Objective

No. of Iter.

Objective

No. of Iter.

Objective

8

67 249

9

67 249

10

67 249

12

67 249

Inoue

6

92 469

9

92 469

11

92 469

9

92 469

Sioux Falls*

ü

ü

15

2810

15

2490

5

2422

Nguyen

*Accuracy level of 0.001 was not used due to the measurement unit problem.

6

Conclusions

This paper presents a general framework for constrained Newton methods for traffic assignments. The techniques of column generation, orthogonal projection, and line search can be incorporated into the solution procedures. The column generation allows the active paths joining each OD pair to be generated internally only when needed. The discussion shows that there are several schemes for the Newton methods. The methodology highlights the importance of determining the optimal step size to accelerate convergence of the Newton methods for traffic assignment, with the optimal-step Newton methods being robust for all the various networks analyzed. The conventional gradient projection algorithm is then a special case of the Newton methods. The performance of the alternative Newton methods is improved by the combination of the descent direction calculation, scaling matrix, the occurrence of orthogonal projection, and the selection of the optimal step size. References

[4] Bertsekas D P. On the Goldstein-Levin-Poljak gradient projection method. IEEE Transactions on Automatic Control, 1976, 21: 174-183. [5] Jayakrishnan R, Tsai W K, Prashker J N, et al. A faster path-based algorithm for traffic assignment. Transportation Research Record, 1994, 1443: 75-83. [6] Larsson T, Patriksson M. Simplicial decomposition with disaggregated representation for the traffic assignment problem. Transportation Science, 1992, 26: 4-17. [7] Patriksson M. The Traffic Assignment Problem: Models and Methods. Utrecht, The Netherlands: VSP BV, 1994. [8] Hearn D W, Lawphongpanich S. A dual algorithm for traffic assignment problems. Transportation Research, 1990, 24: 423-430. [9] Lo H K, Chen A. Traffic equilibrium problem with route-specific costs: Formulation and algorithms. Transportation Research, 2000, 34B: 493-513. [10] Kaysi I, Ben-Akiva M, Koutsopoulos H. An integrated approach to vehicle routing and congestion prediction for real-time driver guidance. Transportation Research Record, 1993, 1408: 66-74. [11] Tobin R L, Friesz T L. Sensitivity analysis for equilibrium network flow. Transportation Science, 1988, 22: 242-249. [12] Bell M G H, Iida Y. Transportation Network Analysis. New

[1] Sheffi Y. Urban Transportation Networks. Englewood Cliffs, New Jersey: Prentice-Hall, 1985. [2] LeBlanc L J, Morlok E K, Pierskalla W P. An efficient approach to solving equilibrium traffic assignment problem. Transportation Research, 1975, 9: 309-318. [3] Hearn D W, Lawphongpanich S, Vebtura J A. Restricted simplicial decomposition: Computation and extensions. Mathematical Programming Study, 1987, 31: 99-118.

York: John Wiley & Sons, 1997: 114-148. [13] Gabriel S, Berstein D. The traffic equilibrium problem with nonadditive path costs. Transportation Science, 1997, 31: 337-348. [14] Cheng L, Iida Y, Uno N, et al. Alternative quasi-Newton methods for capacitated UE assignment. Transportation Research Record, 2003, 1857: 109-116. [15] Bertsekas D P, Gallager. Data Networks. Englewood Cliffs,

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[16] Chen A, Lee D H. Path-based algorithms for large-scale

[19] Nguyen S, Dupuis C. An efficient method for computing

traffic equilibrium problems: A comparison between DSD

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[20] Inoue H. Traffic equilibrium and its solution in congested

[17] Lee D H, Yu N, Chen A, et al. Link and path based traffic

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Former Harvard Business School Senior Associate Dean F. Warren McFarlan Appointed Co-Director of Tsinghua School of Economics and Management China Business Case Center Tsinghua University’s School of Economics and Management (SEM) announced the appointment on August 18ˈ 2009 of former Senior Associate Dean of the Harvard Business School Professor F. Warren McFarlan as Co-Director of SEM’s China Business Case Center for the next three years. During his tenure, Professor McFarlan will design a Harvard Business School case study based teaching platform for SEM adapted to the current business environment in China. Professor McFarlan will also select a company as the basis for case study and research on the problems the company encounters in its growth process. He may also open further case development and teaching seminars in the future. SEM’s Dean Qian Yingyi, Associate Dean Yang Bin, and Associate Dean Xia Donglin, who is also Director of SEM’s China Business Case Center, as well as the Director of Tsinghua’s SEM EDP Center Xue Lei and Director of the School’s Office for Communication Zheng Yuhuang attended the ceremony at which Professor McFarlan’s appointment was announced. Professor McFarlan earned his AB from Harvard University in 1959, and his MBA and DBA from the Harvard Business School in 1961 and 1965 respectively. He has played a significant role in introducing materials on Management Information Systems to all major programs at the Harvard Business School since the first course on the subject was offered in 1962. He has been a long-time teacher in the Advanced Management Program, the International Senior Managers Program, the Delivering Information Services Program, and several of the Social Sector programs. He currently teaches the MBA first year Financial Reporting and Control course and the second year Doing Business in China in the Early 21st Century course. He also teaches in several Executive Education programs. (From http://news.tsinghua.edu.cn, 2009-08-21)