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The dual of the traffic assignment problem with elastic demands
Tronspn Res:B Vol. 19B. No. 3, pp 227-237. Pnnted m the U.S.A.
THE DUAL
0191-2615/85 $3.00+ .oO 0 1985 Pergamon Press Ltd.
1985
OF THE TRAFFIC WITH ELASTIC
ASSIGNMENT DEMANDS
MALACHY CAREY School of Urban and Public Affairs, Carnegie-Mellon University, (Received 6 October
1983; in revisedform
Pittsburgh,
PROBLEM
PA 15213, U.S.A.
15 June 1984)
Abstract-This article is concerned with the dual of the traffic assignment problem, and of the combined generation, distribution, and assignment problem. The duals, and duality relations, for the arc-chain and node-arc formulations of the problem are derived using only the Kuhn-Tucker conditions for convex programs. This has the advantage of being more familiar to most readers than the conjugate function presentation which has been used elsewhere.
1. INTRODUCTION
In recent years, duality has played an increasingly important role in the interpretation, exposition, and solution of optimization models throughout the social and physical sciences. However, in the extensive literature on the traffic assignment problem, the dual problem is seldom mentioned, despite the fact that the dual problem has a simple and interesting form and interpretation. For example, recent texts, including Potts and Oliver (1972), Steenbrink (1974), Newell (1980), and Mandl (1979) give an extensive discussion of the primal traffic assignment problem, but do not mention the dual. Similarly, recent articles reviewing the formulation and interpretation of the traffic assignment problem (Gartner, 1980; Akcelik, 1980; Fernandez and Friesz, 1983) do not mention the dual problem. On the other hand, Hall and Peterson (1976) and Evans (1976) give a lengthy presentation of duality theory as applied to the traffic assignment problem. These excellent papers rely on the elegant mathematics of conjugate function, Fenchel-Rockafellar duality, etc. Much of this mathematics is, however, still unfamiliar to most researchers concerned with traffic assignment, despite the careful presentation of the above authors. Also, Evans assumes that the demand for trips is described by a gravity model in which the number of trips per unit time which begin and end in each zone are known. Here we instead assume that the demand for origin to destination trips is given by general demand functions, so that the number of trips which begin or end in each zone can vary. When such demand functions are combined with a traffic assignment model, the resulting model is described as traffic assignment with elastic demands, or as a combined trip generation, distribution and assignment model. The purpose of the present study is to give simple separate presentations of the dual and duality for the two standard equivalent formulations (the arc-chain and node-arc formulations) of the traffic assignment problem with elastic demands, and of the relationship between these primals and duals.
2.
The arc-chain stated as P 1:
THE
ARC-CHAIN
formulation
PRIMAL
AND
DUAL
of the traffic assignment
FORMULATION
problem with elastic demands can be
maximize I%s+h,) ZPI
Oq C
= I
dGl
(4)
d4rs
r.r
-
1’
C
f,(G)
dcj
(1)
0 ,a
subject to
(2) 227
M. CAREY
228
(3)
h[ 2 0
Vl EL
(4)
where? set of arcs in the network set of nodes in the network set of origin nodes set of destination nodes L= set of all path flows L = set of path flows from origin r to destination s h = [h,] = vector of flows on paths 1 E L v = [Vj] = vector of flows on arcs j E A s E S 4 = [%,I = , vector of travel demands from origins r E R to destinations 6, = 1 if arc j lies on path 1 J/ 0 otherwise I f,(v) = travel time/cost on arc j when the arc flows are v. If program Pl is to yield a user equilibrium solution then fJ(v) is the perceived private cost, i.e. the cost experienced by a trip maker traversing arc j. If program P is to yield a system optimal solution then f,(v) is the social or system marginal cost of traversing arc j, i.e. fJ(v) = a(vrf*(v))l av,, where f:(v) is the cost to an individual trip maker. v = f,-‘(c) is the inverse of c = f,(v). d,‘(q) = the marginal benefit, or marginal willingness-to-pay, for a trip from r to s when the travel volumes are q. p = [d,;‘(q)] = d-‘(q) is the inverse of the vector valued travel demand functions q = d(p). A=
N= R= S=
The symbol I denotes a line integral. In order to ensure that a line integral J-e g(x)rdx has a unique value it is well-known (Apostol, 1969) that the Jacobian [aglax] must be symmetric. Thus for the formulation Pl to be meaningful it is required that the matrices [af,(v)l &,!I and [ad,, (q)/aq,cst] be symmetric. The latter matrix is symmetric if and only if its inverse [~d,,v(q)/8q,~,~] is symmetric. The significance of the above symmetry assumptions are considered in Dafermos (1972) and Femandez and Friesz (1983). Explicit constraints vj 2 0, qrs 2 0 are not needed in PI since these are already ensured by constraints (2t(4). It is convenient to restate program PI more compactly, in matrix notation, as Pi:
maximize (q.v.h)
zp, =
’ d-‘(q)r I0
dg I
0Uf(it)T dG
(5)
subject to Ah = v
(6)
Bh = q
(7)
hz0
where A and B are the arc-chain
and node-chain
(8)
matrices,
respectively.
tvectors in this article are assumed to be column vectors, unless they are accompanied by a transpose sign, i.e. a T superscript, or used as the argument of a function, e.g. f(v). In all matrix addition, subtraction or multiplication it is assumed that the vector and matrix dimensions are such as to permit these operations, even if the dimensions are not explicitly stated.
229
The dual of the traffic assignment problem with elastic demands
It will be shown below that the dual of program Pl can be stated as Dl:
minimize zn, = (P,&,} (9)
subject to
c 4,c, 2 Prs It is convenient
VI E L,, r E R, s E S
to restate program Dl more compactly,
in matrix notation,
(10) as
-
Dl :
minimize fP.4
zn, =
(11)
subject to Arc 2 BTp
(12)
where p = d-‘(O), C = f(O), and - A and B are the arc-chain and node-chain matrices introduced earlier. The vectors c and p in Dl are the Lagrange multipliers (dual variables) corresponding to (6) and (7), respectively, in Pl. Similarly, the vector h (variable h,) in program m is the Lagrange multiplier (dual variable) corresponding to (the Ith member of) (10) in Dl. Constraints (10) and (12) can be interpreted as follows: the left-hand side represents the cost of traversing any path 1 from origin r to destination s, and the right-hand side represents the cost of getting from r to s via a utilized path, hence the constraints state that for each origin-destination pair the travel cost on a utilized path can not exceed the travel cost for any other path. Let J(d) = [dd@)/dp] and J(f) = [af(v)l&] denote the Jacobian matrices formed from the demand functions q = d(p) and the cost functions c = f(v), respectively. We will assume that J(d) and J(f) are negative definite. This will ensure that the demand functions and cost functions are invertable, ensure that Kuhn-Tucker conditions are necessary and sufficient for solutions of Pl and Dl and allows us to prove the duality of Pl and Dl. More formally we have: LEMMA 0
If./(d) is negative definite in any convex region (e.g. forp 2 0) then the demand functions possess an inversep = d-‘(q) everywhere in that region. Similarly if --J(f) is negative definite in any convex region then the cost functions c = f(v) possess an inverse v = f-‘(c) everywhere in that region. Proof.
This is a special case of, e.g. theorem 6 in Gale and Nikaido (1959). LEMMA 1
Let J(d) and -J(f) be symmetric negative definite. Then program D 1 is a dual of program Pl and vice versa. That is: (a) The optimal solution (q”, v”) of Pl equals the optimal solution (q = &‘(p’), v = f-‘(P)) of Dl , hence conversely, the optimal solution Q”,co) of Dl equals the optimal solution (p = d(q’), c = f(9)) of Pl. (b) The set of optimal solutions {ho} of Pl equals the set of optimal Lagrange multipliers associated with constraints (9) in Dl. (c) For any feasible solution of Pl and any feasible solution of Dl , zp, 5 zDI, with zp, = zn, if and only if the feasible solutions of P 1 and D 1 are also optimal solutions.
230
M. CAREY
Proof.
J(d) and -J(f) are negative definite, hence the objective function of Pl is (strictly) convex. It is well known (see Bazaraa and Shetty (1979), Chapter 4) that, for a minimization problem with a convex objective function and linear constraints, the Kuhn-Tucker (K-T) conditions are necessary and sufficient for an optimum, hence the (set of) optimal solution(s) of Pl is equivalent to the solution set of the corresponding K-T conditions. Also, since Pl has a convex constraint set and a strictly convex objective function, it has a unique optimal solution (SO, v”) in the variables (q, v) which appear in the objective function, but not necessarily a unique solution in the variables h which appear only in the constraints (Bazaraa and Shetty (1979), Chapter 3). By an argument similar to the above, program Dl has a unique optimal solution @‘, co), and the optimal solution of Dl is equivalent to the solution of the corresponding K-T conditions. Thus, to compare the solutions of Pl and Dl we need only compare their K-T conditions. We now show that the K-T conditions for Pl are identical to those for Dl: (a) and (b) follow almost immediately. The K-T conditions for PI can be written as (see eqns (A19)--(A21) below) Ah = v, Bh = q, h 2 0 hT(ATa, - BTa2) = 0
ATa, 2 B’a,, aI =
(13)
f(v),
a2 =
(14)
(1-v
d-‘(q),
where a, and a2 are vectors of Lagrange multipliers associated with the constraints (6) and (7), respectively. Similarly, the K-T conditions associated with program Dl can be written as (see eqns (A22)-(A24))
AP, = f-‘(c), ATc2
BTp,
BB, = d(p), P, 2 0 (-ATc
+ BTprP,
= 0
Where /I, is a vector of Lagrange multipliers associated with constraints Now introduce two new definitional vectors, p2 and /$, thus
P* = f-Ye),
p3 = 4P)
(16) (17) (10) of program Dl .
(18)
Using eqn (18) to substitute for f-‘(c) and d(p) in eqn (16) and then comparing eqns (16)(18) with eqns (13t( 15), respectively, shows that these two systems are formally mathematically identical, with {h, v, q, a,, az} in eqns (13)-(15) corresponding to @,, a, a, c, p} in eqns (16)-( 18). The two systems therefore have identical solution sets, i.e. {h, v, q, a,, al} = @,, a, /&, c, p}. This completes the proof of (a) and (b). (c) Consider any optimal solution of Pl and any optimal solution of Dl . These solutions also satisfy eqns ( 13)-( 15) and eqns (16)-( 18)) respectively. Integrating-by-parts the objective function of Pl yields
ZPI
=
PT4 +
’ d($)T dp I lI
(3 v -
jl.f-‘(#dr)
(19)
where, from (15) and (18), p = d-‘(q) and c = f(v). Subtracting zn,, the objective function of Dl , from (19) gives zp, - zn, = pTq - c’v. Substituting in this for q and v, using (7) and (6), yields zp, - z,,, = pTBh - cTAh. Adding the latter to the right-hand equation in (14) or (16) yields zp, - zn, = 0, i.e. the optimal values of programs Pl and Dl are equal. Clearly, for any feasible non-optimal solution to Pl and Dl, respectively, zp, < (optimal zp,) and (optimal z,,,) < zn,. But optimal zp, = optimal zu,, hence zp, < zu, for all feasible n solutions.
The dual of the traffic assignment 3.
The node-arc stated as
THE
NODE-ARC
formulation
P2:
PRIMAL
AND
zp2 = +
c
(srd,,,~
FORMULATION
problem with elastic demands
&’ (q) d& -
1”o cEA f,(c) dc,
rs
subject to
c
vjA -
c
where in addition to the notation
Vi E N, s E S, i # s
vjjA,sES
in the previous
Vi = set of arcs terminating W, = set of arcs originating v = [v,J = vector of flows -qzJ if i is an origin e, = 1 0 otherwise.
(21)
section,
in matrix notation
I
(20)
(22)
at node i. at node i. on arcs j E N to destinations node, i E R
to restate P2 more compactly
can be
/
v,~ = e,?
v,,rO
It is convenient
DUAL
of the traffic assignment
maximize
231
problem with elastic demands
s E S.
as
9
E:
maximize
zpZ =
(9,“)
~(4)~ dq
-
0
OV f(C)T dC
(23)
I
subject to Ev, = e, v,LO
Vs E S
(24)
VSES
(25)
where the notation in E is apparent from comparing I% with P2, so that E is the node-arc incidence matrix and e, = [e,,, Vi E N]. It will be shown below that a dual of program P2 can be written as D2:
minimize zn2 = (P&.%1
subject to uky -
(27)
urs= prs Vr E R, s E S
(28)
h,s -
where i, and k, are neighboring nodes such that arc j points out of node i and into node k, and uiF is the time/cost incurred by unit flow from i to s. Also, u,, can be interpreted as the Lagrange multiplier associated with the isth member of the constraints (21) in P2. Similarly, the variables vjs and qrs in P2 are the Lagrange multipliers associated with constraints (27) and (28), respectively, in D2. Note, of course, that (28) need not appear as an explicit constraint in D2: it can be used to substitute for p = [p,,] in (26). Thus, using matrix notation D2:
minimize zn2 = fC.U)
d- '(0) d(p)r d/j u*
I
L’ f-‘(r) I f(O)
de
(2%
M.
232
CAREY
subject to ETu, 2 c
Vs E S
(30)
where u, = [uil, Vi E N], u* = [u,,., Vr E R, s E S] and II = [u,,, Vi E N, s E S]. LEMMA
2
Program D2 is a dual of program P2 and vice versa. That is: (a) and (c), as for lemma 1, with P2 and D2 substituted for Pl and Dl. (b) The optimal solution set [up,] of D2 equals the set of optimal Lagrange multipliers associated with the constraints (21) of P2. Proof.
There are at least two alternative ways of proving this. (1) We have shown that D 1 is a dual of PI and we show in Section 4 below that programs Dl and D2 are equivalent, hence program D2 can be treated as a dual of program Pl . But the primal programs Pl and P2 are equivalent hence D2 can also be treated as a dual of P2. (2) Lemma 2 can alternatively be proven in a manner which is closely analogous to the H proof of lemma 1. In view of the similarity of the proof we do not set it out here.
4. EQUIVALENCE
OF THE
DUAL
FORMULATIONS
We now show that programs Dl and D2 are equivalent to each other. This reflects the well-known equivalence of the associated primal prroblems Pl and P2. 3 Program Dl and D2 have the same optimal solution, that is, the solution (co, p”) of D2 equals the solution (co, p”) of Dl. Further, the solution set {u”} of D2 can be computed from the solution co of Dl . LEMMA
Proof
The objective functions of Dl and D2 are identical, hence we need only show that their constraints are equivalent. Recursive substitution of (27) of D2 along any path from node r E R to s E S yields U, 5 C, 6,,c, for all paths from r to s. Substituting this in (28) of D2 yields (10) of Dl, hence the solution of D2 solves Dl. To show the converse, i.e. to construct a solution of D2 from a solution of D 1, first define II,, = pFCfor all r E R and s E S, where the superscript ’ denotes optimal values for Dl . To construct uks for non-origin nodes (k $!! R), consider any node k E N, any node s E S and any path 1 from k to s. From the solution cy’s from Dl define uks as the minimum of (E, 6,,c,) over all utilized paths 1 from k to s. These uI.,‘s satisfy D2. Corollary.
Program Dl is a dual of program P2, and vice versa. Similarly, program Pl, and vice versa.
program D2 is a dual of
Proof.
This follows from lemmas
n
1-3.
5. SOME
SPECIAL
CASES
In Sections 2-4 above it is assumed that the demand functions 4 = d(p) and cost functions c = f(v) are invertable. But inverses will not exist if travel demands, prices or costs are given as constants rather than as functions. Inserting these constants into programs Pl and P2 and the dual programs Dl and D2 above reduces these programs to special forms which are noted below. With appropriate notational changes, lemmas 1-3 above, and their proofs, will continue to apply.
The dual of the traffic assignment
Case 1 Let constant objective the duals
problem with elastic demands
233
all demands be taken as fixed (price inelastic), thus 9 = S = [&I, where 4 is a vector. The effect of this on Pl and P2 is to eliminate the term JS d-‘(4) d4 in the functions and change the variables q to constants S in the constraints. The effect on Dl and D2 is to replace the objective function term Ji d@)* dp with -qmTp.
Case 2 Let the prices which travellers are willing to pay for trips be fixed, in which case travel demands are said to be completely elastic. Thus let p = p = bkS], where p is a constant vector. This reduces the first term in the objective function of Pl and P2 to 4’3. The effect on the d@) dp and change p to p in the constraints. duals Dl and D2 is to eliminate the term spdm’(o) Case 3 Letthetravelcostsbeconstant(i.e. f(v) = C = [Z,,l)uptoanarccapacitylimitv 5 V = [\i,,]. In programs Pl and P2 this reduces the second term in the objective function to vT-c and imposes the constraint v 5 V. In the dual programs Dl and D2, this eliminates the second term in the objective function and adds constraints c + 3. = ?, J 2 0, where 1. = [&I is the Lagrange multiplier corresponding to v 5 V in Pl and P2, and v is the Lagrange multiplier corresponding to c + 1 = C in Dl and D2. Note that the equation c = (C - E.) can be used to eliminate c from Dl and D2 by substitution. If the arc capacity constraints v 5 V in Pl and P2 are removed, or are assumed to be not binding, then the 1. drops out of the duals Dl and D2, so that c + 1. = ? reduces c = C, thus Dl ‘:
minimize
P(O) d@) d9
I P
subject to
which reduces Dl and D2 to the problem origins and destinations.
6.
GRAPHICAL
of finding
the shortest paths between
all pairs of
INTERPRETATION
A graphical interpretation of the primal programs, Pl and P2, for the case of a single arc, is discussed in Gartner (1980). A graphical interpretation of the dual program is presented below, and may assist in the intuitive understanding of the dual. It should be remembered, of course, that the ultimate justification for both the primal and dual programs, in the case of user equilibrium, is that an optimal solution of either of these programs yields a user equilibrium traffic assignment, as described by Wardrop’s first principle. The latter problem is formulated initially as a complementarity problem, and can be reformulated as an optimization problem only if certain symmetry or integrability conditions are satisfied (see Femandez and Friesz (1983)). Consider a network consisting of a single arc joining a single O-D pair. In this case, both Dl and D2 reduce to D:
minimize P.C subject to
c 2 p
This is illustrated in Fig. 1. The horizontally and vertically shaded areas represent the first and second integral terms, respectively, in D. It is easy to see that if p and c are chosen so as to
M. CAREY
234
f (VI
v=d 0
A
(VI
(P)
(~1
A
V
Fig. l(b).
Fig. l(a).
minimize the sum of the two shaded areas then p = c = AB and v = d(p) = f’(c), as in Fig. l(b). If f(v) is the social marginal cost function, so that D is a system optimizing model, rather than a user equilibrium model, then the triangle cBp 1s usually interpreted as the maximum net user benefit or social or system surplus which accrues to trip makers. Then program D can be interpreted as minimizing the unattainable net benefit pEBCc. At the optimum this unattainable net benefit is zero. An alternative description of program D in the system optimum case is as follows. The first term in the objective function (si d(p) dp = @!Zp) represents consumer surplus and the second term in the objective function (pi f-r(E) di: = CCC) can be shown to equal the toll revenue collected when the marginal social cost is c (see, e.g. Gartner (1980)). Then program D minimizes (consumer surplus plus toll revenues), subject to the constraint that at the optimum the perceived marginal benefit p can not exceed the marginal social cost c. When f(v) is the cost of traversing an arc, as perceived by the typical individual trip maker, then program D yields a user equilibrium rather than a system optimum. For this case, most authors, dating back to Beckman et al. ( 1956), agree that the area ?B~J does not have any useful or meaningful economic interpretation, though some authors (Gartner, 1980) have sought to construct some such interpretation.
7.
CONCLUDING
REMARKS
The primal formulations Pl and P2 of the traffic assignment problem assume that the inverse p = d-‘(q) of the travel demand functions q = d@) is available, whereas the dual formulations Dl and D2 assume only that the direct demand functions q = d(p) are available. Thus, if the demand functions are noninvertable, it will still be possible to construct the dual formulations, though the corresponding primal formulations will not exist. Demand functions are noninvertable if there is more than one vector of O-D travel costs/prices which will generate the same vector of O-D travel demands: for example, if multiplying all O-D travel costs by a scale factor k, or adding a scale factor to all O-D travel costs, does not affect the resulting vector of travel demands, i.e. if d(p) = d(kp) or d@) = d(p + k). Noninvertable demand systems are much more common than might be thought, since many commonly used demand systems, including market-share type, gravity type and logit type demand systems are noninvertable. A more general test for noninvertability is that a vector valued function is noninvertable at all points at which its matrix of first derivatives (Jacobian) is singular. Turning from the travel demand functions to the travel time/cost functions, we note that the primal formulations Pl and P2 above assume that the arc travel cost functions c = f(v) are available, whereas the dual formulations assume only that the functions v = f-‘(c) are available. In general, when inverse demand functions are available, there does not appear to be any computational advantage in seeking to solve the dual problem rather than the primal problem.
_---
The dual of the traffic assignment
problem with elastic demands
235
Indeed, if we let A, N, D, S, and p denote the number of arcs, nodes, O-D pairs, destinations, and pat&, respectively, then the numbers of constraints in the primal problems Pl and P2 are -(x + D) and (E*S), respectively, whereas in the dual problems Dl and D2 the numbers of constraints are P and A*S, respectively: in-- general p is very much greater than (A + o), and -(A*S) is usually a few times greater than (N*S). Since the dual formulation D2 does not require path enumeration it in general has many times fewer constraints than the dual formulation Dl . It is noteworthy that some of the algorithms which have been developed for solving the traffic assignment problem (PI or P2 above) can also be derived and analyzed by considering instead the dual problem (Dl or D2 above). An example is the algorithm of LeBlanc et al. (1975), which is perhaps the most popular algorithm for solving the traffic assignment problem. This same algorithm can be derived by starting with the dual problem Dl and proceeding in a manner very similar to that used by LeBlanc et al. Acknowledgement--I
wish to thank an anonymous
reviewer for various helpful comments
and suggestions.
REFERENCES Akcelik R. (1979) A graphical explanation of the two principles and two techniques of traffic assignment. Transpn. Res. 13A, 179-184. Apostol T. 0. M. (1969) Calculus, Vol. II. Blaisdell, Waltham, Massachusetts. Bazaraa M. S. and Shetty C. M. (1979) Nonlinear Programming: Theory and Algorithms. Wiley, New York. Beckman M. J., McGuire C. B. and Winsten C. B. (1956) Studies in the Economics ofTransportation. Yale University Press, New Haven, Connecticut. Dafermos S. C. (1972) The traffic assignment problem for a multiclass-user transportation network. Transpn. Sci. 6, 73-87.
Evans S. P. (1976) Derivation and analysis of some models for combining trip distribution and assignment. Transpn. Res. 10, 37-57. Femandez J. E. and Friesz T. L. (1983) Equilibrium predictions in transportation markets: the state of the art. Transpn. Res. 17B,
155-172.
Gale D. and Nikaido H. (1959) The Jacobian matrix and the global univalence of mappings, Mathematische Annalen 159, 81-93. Reprinted in Readings in Mathematical Economics (Edited by P. Newman), Vol. 1. The Johns Hopkins Press, Baltimore, Maryland. Gartner N. H. (1980) Optimal traffic assignment with elastic demands: a review; Part 1. Analysis framework. Transpn. Sci. 14, 174-191.
Hall M. A. and Peterson E. L. (1974) Traffic equilibria analyzed via geometric programming, in TrasJic Equilibrium Methods, Proc. 1976, (Edited by M. A. Florian), pp. 53-105. Springer, New York. Kuhn H. W. and Tucker A. W. (195 1) Nonliner programming. In Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability. J. Neyman (editor), 481-492. University of California Press, Berkeley, CA. LeBlanc L. J., Morlok E. and Pierskalla W. (1975) An efficient approach to solving the road network equilibrium traffic assignment problem. Transpn. Sci. 9, 309-3 18. Mandl C. (1979) Applied Network Optimization. Academic Press, London. Newell G. F. (1980) Flows on Transporrafion Networks. M.I.T. Press, Boston, MA. Potts R. B. and Oliver R. M. (1972) Flows in Transportation Networks. Academic Press, New York, Steenbrink P. A. (1974) Optimization of Transportation Nehvorks. Wiley, New York.
APPENDIX Lemma 1 above applies the Kuhn-Tucker (K-T) conditions (Kuhn and Tucker, 1951) to Pl and Dl, and lemma 2 above applies the K-T conditions to programs P2 and D2. In each case the appropriate K-T conditions are stated without derivation. Thus, to make the exposition more complete, we will here set out the K-T conditions for a general nonlinear programming problem, and from these derive the particular form of the K-T conditions as they apply to programs Pl and Dl and as used in lemma 1 above. The K-T conditions for programs P2 and D2 can be derived in a similar manner. Consider the general nonlinear programming problem P: minimize
2 = g,(x)
(Al)
subject to
g,(x) = 0
(AT)
g*(x) 5 0
L43)
where go(x), g!(x) and g*(x) are vector valued functions of the vector x. Introduce a Lagrange multiplier for each constraint in P: let a and B be the vectors of multipliers associated with (A2) and (A3), respectively. Then the K-T conditions (see Bazaraa and Shetty (1979), Chap. 4) corresponding to problem P consists of (A2)-(A6), where
ago/ax+ (ag,iaga+ (ag,ia.qp = 0
(A4)
236
M. CAREY g&Y0
= 0
(As)
/J>O
(A6)
If the objective function (Al) of P is convex, or at least pseudo-convex, and the constraint functions g,(x) and g?(x) are linear, then the above K-T conditions are necessary and sufficient for an optimal solution of P (see Bazaraa and Shetty (1979), theorems 4.3.6 and 4.3.7). In applying the K-T conditions to any given program it is often very convenient to think of (A4) as follows: form a Lagrangian function by associating a multiplier with each of the constraints in (A2) and (A3) and sum, thus W, Then (A4) is equivalent
a, m = ‘?&) + a’g,(x)
to setting X/i)X
Now let us apply the above K-T conditions PI: minimize
= 0.
-zp,
’ d ‘(4)’ dy + ,,’ f(o)r dG I I0
= -
Ah
Bh
ph
-
programs
v, 9, a,, a?, /r, =
” 9.
hence the K-T conditions
(A2)-(A6)
11 = 0
(AlO)
(All)
5 0
(.412)
to constraint
sets (6)-(7),
and (8) respectively
and form
-
’ d-‘(q)’ dy + ’ f(C)7di i I + t;; (Ah - ~b) + rr;“(Bh - 9) - /Ph
Pl and P, noting that constraints
h’
(A9)
9 = 0
First, introduce multiplier vectors a,, (I?, and b corresponding the Lagrangian L(h,
(A8)
Pl , which can be written as,
to program
subject to
Next, compare (A3), and
(A7)
+ $K2(X).
I ,
(6))(7) correspond
(A13)
to (A2). constraints
(8) correspond
hence ?&/ax =
yield
(61-U)
(A14)
(8)
(AlS
aLlah
= 0 = ATa,
+ B’u?
dL/dv
= 0 = f(v)
-
aLid
= 0 = -d-‘(y) -h7/J
=
~ /j
(I,
(Ale) -
(1:
0
(Al7)
BZO Using the first equation set in (Al6) reduces these K-T conditions to
to
to substitute
(Al8)
for /I in (Al7) and (Al8),
and renaming
(I~ as -a2
throughout,
(Al91 (A20) (A21)
constraints (6)-(8) f(v) = CI,, d-‘(q) = ~(1 h’(Aru,
-
Bra,)
= 0, ATa
2 Bra,
which are identical to the K-T conditions for program PI set out in the proof of Lemma To obtain the K-T conditions for program Dl , form the Lagrangian UC, p, 8,) =
’ dOi) dp + I i’
’ ,f ‘(P)di +
(-Arc
+ B’p)‘/i,
1 in section 2 above.
The dual of the traffic assignment hence applying
the K-T conditions
(A2)-(A6)
to program
problem with elastic demands Dl yields,
b-3
(A12) aLi+
237
= 0 = -d(p)
+ B/9,
a.uac= 0 = f-l(~) - ,4/r',
1
(A23
(~24)
which are the K-T conditions
(16)-(17)
used in Lemma
1 above.
THE DUAL
0191-2615/85 $3.00+ .oO 0 1985 Pergamon Press Ltd.
1985
OF THE TRAFFIC WITH ELASTIC
ASSIGNMENT DEMANDS
MALACHY CAREY School of Urban and Public Affairs, Carnegie-Mellon University, (Received 6 October
1983; in revisedform
Pittsburgh,
PROBLEM
PA 15213, U.S.A.
15 June 1984)
Abstract-This article is concerned with the dual of the traffic assignment problem, and of the combined generation, distribution, and assignment problem. The duals, and duality relations, for the arc-chain and node-arc formulations of the problem are derived using only the Kuhn-Tucker conditions for convex programs. This has the advantage of being more familiar to most readers than the conjugate function presentation which has been used elsewhere.
1. INTRODUCTION
In recent years, duality has played an increasingly important role in the interpretation, exposition, and solution of optimization models throughout the social and physical sciences. However, in the extensive literature on the traffic assignment problem, the dual problem is seldom mentioned, despite the fact that the dual problem has a simple and interesting form and interpretation. For example, recent texts, including Potts and Oliver (1972), Steenbrink (1974), Newell (1980), and Mandl (1979) give an extensive discussion of the primal traffic assignment problem, but do not mention the dual. Similarly, recent articles reviewing the formulation and interpretation of the traffic assignment problem (Gartner, 1980; Akcelik, 1980; Fernandez and Friesz, 1983) do not mention the dual problem. On the other hand, Hall and Peterson (1976) and Evans (1976) give a lengthy presentation of duality theory as applied to the traffic assignment problem. These excellent papers rely on the elegant mathematics of conjugate function, Fenchel-Rockafellar duality, etc. Much of this mathematics is, however, still unfamiliar to most researchers concerned with traffic assignment, despite the careful presentation of the above authors. Also, Evans assumes that the demand for trips is described by a gravity model in which the number of trips per unit time which begin and end in each zone are known. Here we instead assume that the demand for origin to destination trips is given by general demand functions, so that the number of trips which begin or end in each zone can vary. When such demand functions are combined with a traffic assignment model, the resulting model is described as traffic assignment with elastic demands, or as a combined trip generation, distribution and assignment model. The purpose of the present study is to give simple separate presentations of the dual and duality for the two standard equivalent formulations (the arc-chain and node-arc formulations) of the traffic assignment problem with elastic demands, and of the relationship between these primals and duals.
2.
The arc-chain stated as P 1:
THE
ARC-CHAIN
formulation
PRIMAL
AND
DUAL
of the traffic assignment
FORMULATION
problem with elastic demands can be
maximize I%s+h,) ZPI
Oq C
= I
dGl
(4)
d4rs
r.r
-
1’
C
f,(G)
dcj
(1)
0 ,a
subject to
(2) 227
M. CAREY
228
(3)
h[ 2 0
Vl EL
(4)
where? set of arcs in the network set of nodes in the network set of origin nodes set of destination nodes L= set of all path flows L = set of path flows from origin r to destination s h = [h,] = vector of flows on paths 1 E L v = [Vj] = vector of flows on arcs j E A s E S 4 = [%,I = , vector of travel demands from origins r E R to destinations 6, = 1 if arc j lies on path 1 J/ 0 otherwise I f,(v) = travel time/cost on arc j when the arc flows are v. If program Pl is to yield a user equilibrium solution then fJ(v) is the perceived private cost, i.e. the cost experienced by a trip maker traversing arc j. If program P is to yield a system optimal solution then f,(v) is the social or system marginal cost of traversing arc j, i.e. fJ(v) = a(vrf*(v))l av,, where f:(v) is the cost to an individual trip maker. v = f,-‘(c) is the inverse of c = f,(v). d,‘(q) = the marginal benefit, or marginal willingness-to-pay, for a trip from r to s when the travel volumes are q. p = [d,;‘(q)] = d-‘(q) is the inverse of the vector valued travel demand functions q = d(p). A=
N= R= S=
The symbol I denotes a line integral. In order to ensure that a line integral J-e g(x)rdx has a unique value it is well-known (Apostol, 1969) that the Jacobian [aglax] must be symmetric. Thus for the formulation Pl to be meaningful it is required that the matrices [af,(v)l &,!I and [ad,, (q)/aq,cst] be symmetric. The latter matrix is symmetric if and only if its inverse [~d,,v(q)/8q,~,~] is symmetric. The significance of the above symmetry assumptions are considered in Dafermos (1972) and Femandez and Friesz (1983). Explicit constraints vj 2 0, qrs 2 0 are not needed in PI since these are already ensured by constraints (2t(4). It is convenient to restate program PI more compactly, in matrix notation, as Pi:
maximize (q.v.h)
zp, =
’ d-‘(q)r I0
dg I
0Uf(it)T dG
(5)
subject to Ah = v
(6)
Bh = q
(7)
hz0
where A and B are the arc-chain
and node-chain
(8)
matrices,
respectively.
tvectors in this article are assumed to be column vectors, unless they are accompanied by a transpose sign, i.e. a T superscript, or used as the argument of a function, e.g. f(v). In all matrix addition, subtraction or multiplication it is assumed that the vector and matrix dimensions are such as to permit these operations, even if the dimensions are not explicitly stated.
229
The dual of the traffic assignment problem with elastic demands
It will be shown below that the dual of program Pl can be stated as Dl:
minimize zn, = (P,&,} (9)
subject to
c 4,c, 2 Prs It is convenient
VI E L,, r E R, s E S
to restate program Dl more compactly,
in matrix notation,
(10) as
-
Dl :
minimize fP.4
zn, =
(11)
subject to Arc 2 BTp
(12)
where p = d-‘(O), C = f(O), and - A and B are the arc-chain and node-chain matrices introduced earlier. The vectors c and p in Dl are the Lagrange multipliers (dual variables) corresponding to (6) and (7), respectively, in Pl. Similarly, the vector h (variable h,) in program m is the Lagrange multiplier (dual variable) corresponding to (the Ith member of) (10) in Dl. Constraints (10) and (12) can be interpreted as follows: the left-hand side represents the cost of traversing any path 1 from origin r to destination s, and the right-hand side represents the cost of getting from r to s via a utilized path, hence the constraints state that for each origin-destination pair the travel cost on a utilized path can not exceed the travel cost for any other path. Let J(d) = [dd@)/dp] and J(f) = [af(v)l&] denote the Jacobian matrices formed from the demand functions q = d(p) and the cost functions c = f(v), respectively. We will assume that J(d) and J(f) are negative definite. This will ensure that the demand functions and cost functions are invertable, ensure that Kuhn-Tucker conditions are necessary and sufficient for solutions of Pl and Dl and allows us to prove the duality of Pl and Dl. More formally we have: LEMMA 0
If./(d) is negative definite in any convex region (e.g. forp 2 0) then the demand functions possess an inversep = d-‘(q) everywhere in that region. Similarly if --J(f) is negative definite in any convex region then the cost functions c = f(v) possess an inverse v = f-‘(c) everywhere in that region. Proof.
This is a special case of, e.g. theorem 6 in Gale and Nikaido (1959). LEMMA 1
Let J(d) and -J(f) be symmetric negative definite. Then program D 1 is a dual of program Pl and vice versa. That is: (a) The optimal solution (q”, v”) of Pl equals the optimal solution (q = &‘(p’), v = f-‘(P)) of Dl , hence conversely, the optimal solution Q”,co) of Dl equals the optimal solution (p = d(q’), c = f(9)) of Pl. (b) The set of optimal solutions {ho} of Pl equals the set of optimal Lagrange multipliers associated with constraints (9) in Dl. (c) For any feasible solution of Pl and any feasible solution of Dl , zp, 5 zDI, with zp, = zn, if and only if the feasible solutions of P 1 and D 1 are also optimal solutions.
230
M. CAREY
Proof.
J(d) and -J(f) are negative definite, hence the objective function of Pl is (strictly) convex. It is well known (see Bazaraa and Shetty (1979), Chapter 4) that, for a minimization problem with a convex objective function and linear constraints, the Kuhn-Tucker (K-T) conditions are necessary and sufficient for an optimum, hence the (set of) optimal solution(s) of Pl is equivalent to the solution set of the corresponding K-T conditions. Also, since Pl has a convex constraint set and a strictly convex objective function, it has a unique optimal solution (SO, v”) in the variables (q, v) which appear in the objective function, but not necessarily a unique solution in the variables h which appear only in the constraints (Bazaraa and Shetty (1979), Chapter 3). By an argument similar to the above, program Dl has a unique optimal solution @‘, co), and the optimal solution of Dl is equivalent to the solution of the corresponding K-T conditions. Thus, to compare the solutions of Pl and Dl we need only compare their K-T conditions. We now show that the K-T conditions for Pl are identical to those for Dl: (a) and (b) follow almost immediately. The K-T conditions for PI can be written as (see eqns (A19)--(A21) below) Ah = v, Bh = q, h 2 0 hT(ATa, - BTa2) = 0
ATa, 2 B’a,, aI =
(13)
f(v),
a2 =
(14)
(1-v
d-‘(q),
where a, and a2 are vectors of Lagrange multipliers associated with the constraints (6) and (7), respectively. Similarly, the K-T conditions associated with program Dl can be written as (see eqns (A22)-(A24))
AP, = f-‘(c), ATc2
BTp,
BB, = d(p), P, 2 0 (-ATc
+ BTprP,
= 0
Where /I, is a vector of Lagrange multipliers associated with constraints Now introduce two new definitional vectors, p2 and /$, thus
P* = f-Ye),
p3 = 4P)
(16) (17) (10) of program Dl .
(18)
Using eqn (18) to substitute for f-‘(c) and d(p) in eqn (16) and then comparing eqns (16)(18) with eqns (13t( 15), respectively, shows that these two systems are formally mathematically identical, with {h, v, q, a,, az} in eqns (13)-(15) corresponding to @,, a, a, c, p} in eqns (16)-( 18). The two systems therefore have identical solution sets, i.e. {h, v, q, a,, al} = @,, a, /&, c, p}. This completes the proof of (a) and (b). (c) Consider any optimal solution of Pl and any optimal solution of Dl . These solutions also satisfy eqns ( 13)-( 15) and eqns (16)-( 18)) respectively. Integrating-by-parts the objective function of Pl yields
ZPI
=
PT4 +
’ d($)T dp I lI
(3 v -
jl.f-‘(#dr)
(19)
where, from (15) and (18), p = d-‘(q) and c = f(v). Subtracting zn,, the objective function of Dl , from (19) gives zp, - zn, = pTq - c’v. Substituting in this for q and v, using (7) and (6), yields zp, - z,,, = pTBh - cTAh. Adding the latter to the right-hand equation in (14) or (16) yields zp, - zn, = 0, i.e. the optimal values of programs Pl and Dl are equal. Clearly, for any feasible non-optimal solution to Pl and Dl, respectively, zp, < (optimal zp,) and (optimal z,,,) < zn,. But optimal zp, = optimal zu,, hence zp, < zu, for all feasible n solutions.
The dual of the traffic assignment 3.
The node-arc stated as
THE
NODE-ARC
formulation
P2:
PRIMAL
AND
zp2 = +
c
(srd,,,~
FORMULATION
problem with elastic demands
&’ (q) d& -
1”o cEA f,(c) dc,
rs
subject to
c
vjA -
c
where in addition to the notation
Vi E N, s E S, i # s
vjjA,sES
in the previous
Vi = set of arcs terminating W, = set of arcs originating v = [v,J = vector of flows -qzJ if i is an origin e, = 1 0 otherwise.
(21)
section,
in matrix notation
I
(20)
(22)
at node i. at node i. on arcs j E N to destinations node, i E R
to restate P2 more compactly
can be
/
v,~ = e,?
v,,rO
It is convenient
DUAL
of the traffic assignment
maximize
231
problem with elastic demands
s E S.
as
9
E:
maximize
zpZ =
(9,“)
~(4)~ dq
-
0
OV f(C)T dC
(23)
I
subject to Ev, = e, v,LO
Vs E S
(24)
VSES
(25)
where the notation in E is apparent from comparing I% with P2, so that E is the node-arc incidence matrix and e, = [e,,, Vi E N]. It will be shown below that a dual of program P2 can be written as D2:
minimize zn2 = (P&.%1
subject to uky -
(27)
urs= prs Vr E R, s E S
(28)
h,s -
where i, and k, are neighboring nodes such that arc j points out of node i and into node k, and uiF is the time/cost incurred by unit flow from i to s. Also, u,, can be interpreted as the Lagrange multiplier associated with the isth member of the constraints (21) in P2. Similarly, the variables vjs and qrs in P2 are the Lagrange multipliers associated with constraints (27) and (28), respectively, in D2. Note, of course, that (28) need not appear as an explicit constraint in D2: it can be used to substitute for p = [p,,] in (26). Thus, using matrix notation D2:
minimize zn2 = fC.U)
d- '(0) d(p)r d/j u*
I
L’ f-‘(r) I f(O)
de
(2%
M.
232
CAREY
subject to ETu, 2 c
Vs E S
(30)
where u, = [uil, Vi E N], u* = [u,,., Vr E R, s E S] and II = [u,,, Vi E N, s E S]. LEMMA
2
Program D2 is a dual of program P2 and vice versa. That is: (a) and (c), as for lemma 1, with P2 and D2 substituted for Pl and Dl. (b) The optimal solution set [up,] of D2 equals the set of optimal Lagrange multipliers associated with the constraints (21) of P2. Proof.
There are at least two alternative ways of proving this. (1) We have shown that D 1 is a dual of PI and we show in Section 4 below that programs Dl and D2 are equivalent, hence program D2 can be treated as a dual of program Pl . But the primal programs Pl and P2 are equivalent hence D2 can also be treated as a dual of P2. (2) Lemma 2 can alternatively be proven in a manner which is closely analogous to the H proof of lemma 1. In view of the similarity of the proof we do not set it out here.
4. EQUIVALENCE
OF THE
DUAL
FORMULATIONS
We now show that programs Dl and D2 are equivalent to each other. This reflects the well-known equivalence of the associated primal prroblems Pl and P2. 3 Program Dl and D2 have the same optimal solution, that is, the solution (co, p”) of D2 equals the solution (co, p”) of Dl. Further, the solution set {u”} of D2 can be computed from the solution co of Dl . LEMMA
Proof
The objective functions of Dl and D2 are identical, hence we need only show that their constraints are equivalent. Recursive substitution of (27) of D2 along any path from node r E R to s E S yields U, 5 C, 6,,c, for all paths from r to s. Substituting this in (28) of D2 yields (10) of Dl, hence the solution of D2 solves Dl. To show the converse, i.e. to construct a solution of D2 from a solution of D 1, first define II,, = pFCfor all r E R and s E S, where the superscript ’ denotes optimal values for Dl . To construct uks for non-origin nodes (k $!! R), consider any node k E N, any node s E S and any path 1 from k to s. From the solution cy’s from Dl define uks as the minimum of (E, 6,,c,) over all utilized paths 1 from k to s. These uI.,‘s satisfy D2. Corollary.
Program Dl is a dual of program P2, and vice versa. Similarly, program Pl, and vice versa.
program D2 is a dual of
Proof.
This follows from lemmas
n
1-3.
5. SOME
SPECIAL
CASES
In Sections 2-4 above it is assumed that the demand functions 4 = d(p) and cost functions c = f(v) are invertable. But inverses will not exist if travel demands, prices or costs are given as constants rather than as functions. Inserting these constants into programs Pl and P2 and the dual programs Dl and D2 above reduces these programs to special forms which are noted below. With appropriate notational changes, lemmas 1-3 above, and their proofs, will continue to apply.
The dual of the traffic assignment
Case 1 Let constant objective the duals
problem with elastic demands
233
all demands be taken as fixed (price inelastic), thus 9 = S = [&I, where 4 is a vector. The effect of this on Pl and P2 is to eliminate the term JS d-‘(4) d4 in the functions and change the variables q to constants S in the constraints. The effect on Dl and D2 is to replace the objective function term Ji d@)* dp with -qmTp.
Case 2 Let the prices which travellers are willing to pay for trips be fixed, in which case travel demands are said to be completely elastic. Thus let p = p = bkS], where p is a constant vector. This reduces the first term in the objective function of Pl and P2 to 4’3. The effect on the d@) dp and change p to p in the constraints. duals Dl and D2 is to eliminate the term spdm’(o) Case 3 Letthetravelcostsbeconstant(i.e. f(v) = C = [Z,,l)uptoanarccapacitylimitv 5 V = [\i,,]. In programs Pl and P2 this reduces the second term in the objective function to vT-c and imposes the constraint v 5 V. In the dual programs Dl and D2, this eliminates the second term in the objective function and adds constraints c + 3. = ?, J 2 0, where 1. = [&I is the Lagrange multiplier corresponding to v 5 V in Pl and P2, and v is the Lagrange multiplier corresponding to c + 1 = C in Dl and D2. Note that the equation c = (C - E.) can be used to eliminate c from Dl and D2 by substitution. If the arc capacity constraints v 5 V in Pl and P2 are removed, or are assumed to be not binding, then the 1. drops out of the duals Dl and D2, so that c + 1. = ? reduces c = C, thus Dl ‘:
minimize
P(O) d@) d9
I P
subject to
which reduces Dl and D2 to the problem origins and destinations.
6.
GRAPHICAL
of finding
the shortest paths between
all pairs of
INTERPRETATION
A graphical interpretation of the primal programs, Pl and P2, for the case of a single arc, is discussed in Gartner (1980). A graphical interpretation of the dual program is presented below, and may assist in the intuitive understanding of the dual. It should be remembered, of course, that the ultimate justification for both the primal and dual programs, in the case of user equilibrium, is that an optimal solution of either of these programs yields a user equilibrium traffic assignment, as described by Wardrop’s first principle. The latter problem is formulated initially as a complementarity problem, and can be reformulated as an optimization problem only if certain symmetry or integrability conditions are satisfied (see Femandez and Friesz (1983)). Consider a network consisting of a single arc joining a single O-D pair. In this case, both Dl and D2 reduce to D:
minimize P.C subject to
c 2 p
This is illustrated in Fig. 1. The horizontally and vertically shaded areas represent the first and second integral terms, respectively, in D. It is easy to see that if p and c are chosen so as to
M. CAREY
234
f (VI
v=d 0
A
(VI
(P)
(~1
A
V
Fig. l(b).
Fig. l(a).
minimize the sum of the two shaded areas then p = c = AB and v = d(p) = f’(c), as in Fig. l(b). If f(v) is the social marginal cost function, so that D is a system optimizing model, rather than a user equilibrium model, then the triangle cBp 1s usually interpreted as the maximum net user benefit or social or system surplus which accrues to trip makers. Then program D can be interpreted as minimizing the unattainable net benefit pEBCc. At the optimum this unattainable net benefit is zero. An alternative description of program D in the system optimum case is as follows. The first term in the objective function (si d(p) dp = @!Zp) represents consumer surplus and the second term in the objective function (pi f-r(E) di: = CCC) can be shown to equal the toll revenue collected when the marginal social cost is c (see, e.g. Gartner (1980)). Then program D minimizes (consumer surplus plus toll revenues), subject to the constraint that at the optimum the perceived marginal benefit p can not exceed the marginal social cost c. When f(v) is the cost of traversing an arc, as perceived by the typical individual trip maker, then program D yields a user equilibrium rather than a system optimum. For this case, most authors, dating back to Beckman et al. ( 1956), agree that the area ?B~J does not have any useful or meaningful economic interpretation, though some authors (Gartner, 1980) have sought to construct some such interpretation.
7.
CONCLUDING
REMARKS
The primal formulations Pl and P2 of the traffic assignment problem assume that the inverse p = d-‘(q) of the travel demand functions q = d@) is available, whereas the dual formulations Dl and D2 assume only that the direct demand functions q = d(p) are available. Thus, if the demand functions are noninvertable, it will still be possible to construct the dual formulations, though the corresponding primal formulations will not exist. Demand functions are noninvertable if there is more than one vector of O-D travel costs/prices which will generate the same vector of O-D travel demands: for example, if multiplying all O-D travel costs by a scale factor k, or adding a scale factor to all O-D travel costs, does not affect the resulting vector of travel demands, i.e. if d(p) = d(kp) or d@) = d(p + k). Noninvertable demand systems are much more common than might be thought, since many commonly used demand systems, including market-share type, gravity type and logit type demand systems are noninvertable. A more general test for noninvertability is that a vector valued function is noninvertable at all points at which its matrix of first derivatives (Jacobian) is singular. Turning from the travel demand functions to the travel time/cost functions, we note that the primal formulations Pl and P2 above assume that the arc travel cost functions c = f(v) are available, whereas the dual formulations assume only that the functions v = f-‘(c) are available. In general, when inverse demand functions are available, there does not appear to be any computational advantage in seeking to solve the dual problem rather than the primal problem.
_---
The dual of the traffic assignment
problem with elastic demands
235
Indeed, if we let A, N, D, S, and p denote the number of arcs, nodes, O-D pairs, destinations, and pat&, respectively, then the numbers of constraints in the primal problems Pl and P2 are -(x + D) and (E*S), respectively, whereas in the dual problems Dl and D2 the numbers of constraints are P and A*S, respectively: in-- general p is very much greater than (A + o), and -(A*S) is usually a few times greater than (N*S). Since the dual formulation D2 does not require path enumeration it in general has many times fewer constraints than the dual formulation Dl . It is noteworthy that some of the algorithms which have been developed for solving the traffic assignment problem (PI or P2 above) can also be derived and analyzed by considering instead the dual problem (Dl or D2 above). An example is the algorithm of LeBlanc et al. (1975), which is perhaps the most popular algorithm for solving the traffic assignment problem. This same algorithm can be derived by starting with the dual problem Dl and proceeding in a manner very similar to that used by LeBlanc et al. Acknowledgement--I
wish to thank an anonymous
reviewer for various helpful comments
and suggestions.
REFERENCES Akcelik R. (1979) A graphical explanation of the two principles and two techniques of traffic assignment. Transpn. Res. 13A, 179-184. Apostol T. 0. M. (1969) Calculus, Vol. II. Blaisdell, Waltham, Massachusetts. Bazaraa M. S. and Shetty C. M. (1979) Nonlinear Programming: Theory and Algorithms. Wiley, New York. Beckman M. J., McGuire C. B. and Winsten C. B. (1956) Studies in the Economics ofTransportation. Yale University Press, New Haven, Connecticut. Dafermos S. C. (1972) The traffic assignment problem for a multiclass-user transportation network. Transpn. Sci. 6, 73-87.
Evans S. P. (1976) Derivation and analysis of some models for combining trip distribution and assignment. Transpn. Res. 10, 37-57. Femandez J. E. and Friesz T. L. (1983) Equilibrium predictions in transportation markets: the state of the art. Transpn. Res. 17B,
155-172.
Gale D. and Nikaido H. (1959) The Jacobian matrix and the global univalence of mappings, Mathematische Annalen 159, 81-93. Reprinted in Readings in Mathematical Economics (Edited by P. Newman), Vol. 1. The Johns Hopkins Press, Baltimore, Maryland. Gartner N. H. (1980) Optimal traffic assignment with elastic demands: a review; Part 1. Analysis framework. Transpn. Sci. 14, 174-191.
Hall M. A. and Peterson E. L. (1974) Traffic equilibria analyzed via geometric programming, in TrasJic Equilibrium Methods, Proc. 1976, (Edited by M. A. Florian), pp. 53-105. Springer, New York. Kuhn H. W. and Tucker A. W. (195 1) Nonliner programming. In Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability. J. Neyman (editor), 481-492. University of California Press, Berkeley, CA. LeBlanc L. J., Morlok E. and Pierskalla W. (1975) An efficient approach to solving the road network equilibrium traffic assignment problem. Transpn. Sci. 9, 309-3 18. Mandl C. (1979) Applied Network Optimization. Academic Press, London. Newell G. F. (1980) Flows on Transporrafion Networks. M.I.T. Press, Boston, MA. Potts R. B. and Oliver R. M. (1972) Flows in Transportation Networks. Academic Press, New York, Steenbrink P. A. (1974) Optimization of Transportation Nehvorks. Wiley, New York.
APPENDIX Lemma 1 above applies the Kuhn-Tucker (K-T) conditions (Kuhn and Tucker, 1951) to Pl and Dl, and lemma 2 above applies the K-T conditions to programs P2 and D2. In each case the appropriate K-T conditions are stated without derivation. Thus, to make the exposition more complete, we will here set out the K-T conditions for a general nonlinear programming problem, and from these derive the particular form of the K-T conditions as they apply to programs Pl and Dl and as used in lemma 1 above. The K-T conditions for programs P2 and D2 can be derived in a similar manner. Consider the general nonlinear programming problem P: minimize
2 = g,(x)
(Al)
subject to
g,(x) = 0
(AT)
g*(x) 5 0
L43)
where go(x), g!(x) and g*(x) are vector valued functions of the vector x. Introduce a Lagrange multiplier for each constraint in P: let a and B be the vectors of multipliers associated with (A2) and (A3), respectively. Then the K-T conditions (see Bazaraa and Shetty (1979), Chap. 4) corresponding to problem P consists of (A2)-(A6), where
ago/ax+ (ag,iaga+ (ag,ia.qp = 0
(A4)
236
M. CAREY g&Y0
= 0
(As)
/J>O
(A6)
If the objective function (Al) of P is convex, or at least pseudo-convex, and the constraint functions g,(x) and g?(x) are linear, then the above K-T conditions are necessary and sufficient for an optimal solution of P (see Bazaraa and Shetty (1979), theorems 4.3.6 and 4.3.7). In applying the K-T conditions to any given program it is often very convenient to think of (A4) as follows: form a Lagrangian function by associating a multiplier with each of the constraints in (A2) and (A3) and sum, thus W, Then (A4) is equivalent
a, m = ‘?&) + a’g,(x)
to setting X/i)X
Now let us apply the above K-T conditions PI: minimize
= 0.
-zp,
’ d ‘(4)’ dy + ,,’ f(o)r dG I I0
= -
Ah
Bh
ph
-
programs
v, 9, a,, a?, /r, =
” 9.
hence the K-T conditions
(A2)-(A6)
11 = 0
(AlO)
(All)
5 0
(.412)
to constraint
sets (6)-(7),
and (8) respectively
and form
-
’ d-‘(q)’ dy + ’ f(C)7di i I + t;; (Ah - ~b) + rr;“(Bh - 9) - /Ph
Pl and P, noting that constraints
h’
(A9)
9 = 0
First, introduce multiplier vectors a,, (I?, and b corresponding the Lagrangian L(h,
(A8)
Pl , which can be written as,
to program
subject to
Next, compare (A3), and
(A7)
+ $K2(X).
I ,
(6))(7) correspond
(A13)
to (A2). constraints
(8) correspond
hence ?&/ax =
yield
(61-U)
(A14)
(8)
(AlS
aLlah
= 0 = ATa,
+ B’u?
dL/dv
= 0 = f(v)
-
aLid
= 0 = -d-‘(y) -h7/J
=
~ /j
(I,
(Ale) -
(1:
0
(Al7)
BZO Using the first equation set in (Al6) reduces these K-T conditions to
to
to substitute
(Al8)
for /I in (Al7) and (Al8),
and renaming
(I~ as -a2
throughout,
(Al91 (A20) (A21)
constraints (6)-(8) f(v) = CI,, d-‘(q) = ~(1 h’(Aru,
-
Bra,)
= 0, ATa
2 Bra,
which are identical to the K-T conditions for program PI set out in the proof of Lemma To obtain the K-T conditions for program Dl , form the Lagrangian UC, p, 8,) =
’ dOi) dp + I i’
’ ,f ‘(P)di +
(-Arc
+ B’p)‘/i,
1 in section 2 above.
The dual of the traffic assignment hence applying
the K-T conditions
(A2)-(A6)
to program
problem with elastic demands Dl yields,
b-3
(A12) aLi+
237
= 0 = -d(p)
+ B/9,
a.uac= 0 = f-l(~) - ,4/r',
1
(A23
(~24)
which are the K-T conditions
(16)-(17)
used in Lemma
1 above.