Time-dependent traffic equilibria

JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS Vol 103. No 3. pp. 543-555. DECEMBER 1999

Time-Dependent Traffic Equilibria

P. DANIELE,1 A. MAUGERI,2 AND W. OETTLI3

Communicated by F. Giannessi

Abstract. We consider the existence, characterization, and calculationof equilibria in transportation networks, when the route capacities anddemand requirements depend on time. The problem is situated in aBanach space setting and formulated in terms of a variational inequality.

Key Words. Transportation networks, equilibrium solutions, Wardropcondition, time-dependent requirements.

1. Introduction

This paper deals with traffic equilibria in the case where the numericaldata of the traffic network depend on time. We shall discuss the followingissues: (i) characterization of equilibria by a suitably generalized form ofthe Wardrop condition; (ii) existence of equilibria; (iii) calculation ofequilibria by means of gap functions.

We shall consider also the case where the route flows have to satisfy,besides the usual demand requirements between origin-destination pairs androute-capacity restrictions, some additional constraints. The treatmentextends previous results of Refs. 1-4.

2. Basic Model

In order to introduce the problem, let us first consider the basic modelwithout time dependence.

1Assistant, Dipartimento di Matematica, Universita di Catania, Catania, Italy.2Professor, Dipartimento di Matematica, Universita di Catania, Catania, Italy.3Professor, Fakultat fur Mathematik und Informatik, Universitat Mannheim, Mannheini,Germany.

5430022-3239/99/1200-0543$16.00/0 © 1999 Plenum Publishing Corporation

544 JOTA: VOL. 103, NO. 3, DECEMBER 1999

In the traffic network, one has a set 'W of origin-destination pairs anda set R of routes. Each route rg^ links exactly one origin-destination pairwei^. The set of all re@ which link a given w&'W is denoted by &(w).

We consider flow vectors FeR*, where Fr, re@, denotes the flow inroute r. A feasible flow has to satisfy capacity restrictions,

and demand requirements,

where X < p are given in R* and p > 0 is given in R*.Introducing the pair-route incidence matrix d> = (<!>w,r), with weW,

rt3l, namely,

the demand requirements can be written in matrix-vector notation as

Thus, the set of all feasible flows is given by

Furthermore, we are given a cost function C: K_>R*. Then, to everyfeasible flow FeK, there corresponds a cost vector C(F)eR*; Cr(F) givesthe marginal cost of sending one additional unit of flow through route r,when the flow F is already present. We do not specify how C(F) dependson F; for this, a more detailed physical model is needed; see Ref. 5.

Definition 2.1. A flow HeU* is called an equilibrium flow iff

In order to characterize equilibrium flows by means of the Wardropcondition, we need the following result from linear programming.

Lemma 2.1. Let K be given by (1), let CeR* and HeK be arbitrary.Then, the following statements are equivalent:

JOTA: VOL. 103, NO. 3, DECEMBER 1999 545

Proof.

(a) Assume that (ii) does not hold. Select weif and q, set(w) suchthat Cq<Cs Hq<nq, Hs>ks. Let

and define

Then,

So, (i) does not hold.(b) Now, assume that (ii) holds. Let we if and

From (ii),

So, there exists 7weR such that

Let FeK be arbitrary. Then, for every re&l(w), Cr<yw implies r$A, henceHr = nr, hence Fr-Hr<0, hence (Cr-yw) ( F r - H r ) > 0 . Likewise, CP>yM .implies (Cr~ yw)(Fr-Hr)>0. Thus,

Hence,

and (i) holds. D

We observe that condition (ii) in Lemma 2.1 is equivalent to:

(ii') for every weW, there exists 7weR such that, for all reJ?(w),

In fact, the implication (ii) => (ii') has been shown in the proof ofLemma 2.1, and the reverse implication is obvious.


If Ur=+oo for all re^, then condition (ii) in Lemma 2.1 can berendered as follows:

for every we if, if Yw= min Cr,thenre3t(w)

From Lemma 2.1, we see that HeK is an equilibrium flow in the senseof (2) if, and only if,

for all we -W and all q, se@(w),

Condition (3) is called the Wardrop condition. Due to its decomposed form,it is more amenable to the user than the equivalent definition (2). One speaksof (3) as a user-oriented equilibrium.

3. Time-Dependent Equilibria

Now, we consider the dynamic case. The traffic network, whosegeometry remains fixed, is considered at all times t&3~, where I:=[ '•= [0, T}\the case where 3~ '•= (0, 1 , . . . , T] will be included trivially in the following.

For each time fe^~, we have a route-flow vector F(t)eU*. The feasibleflows have to satisfy the time-dependent capacity constraints and demandrequirements; namely, almost everywhere on -~,

where A (•) ^ M') as well as p(•) are given, and $ is again the pair-routeincidence matrix. F( •): ^"->R* is the flow trajectory over time. For technicalreasons, the functional setting for the flow trajectories is the reflexive Banachspace Lp(^',Ra') with p<1 We abbreviate it by X. The dual spaceLq(F, R*), 1 / p + 1 / q = 1 , will be denoted by J&?*. On ^* x <£, we definethe canonical bilinear form by

Instead of flow trajectories, we simply speak of flows Fe^f. The set offeasible flows is given by


We assume that A, n are in 5f, A</i, and that, for all weW, pw>0 is inLp(9~, R). We assume also that

Then, K is nonempty. It is seen easily that K is convex, closed, and bounded,hence weakly compact. Furthermore, we are given a mapping C: K-*y*,which assigns to each flow trajectory F(-)eK the cost trajectoryC(F)(-)eSe*.

Definition 3.1. He££ is an equilibrium flow iff

Theorem 3.1. HeK is an equilibrium flow in the sense of (5) if andonly if

Proof.

(a) Assume that (6) holds. Let FeK. Since the union of finitely manynull sets is a null set, it follows from Lemma 2.1 that

Hence «C(H), F-H» >0, and H is an equilibrium.(b) Assume that (6) does not hold. Then, there exist we if and

q, se3l(w) together with a set E<=,3~ having positive measure such that

Cq(t)<Cs(H)(t), H q ( t ) < U q ( t ) , Hs(t) >A s(t) a.e.on£.

For teE, let

Then, S(t) > 0 a.e. on E, and as in the proof of Lemma 2.1, we can constructFeK such that F= H outside E and

Thus, H is not an equilibrium. D

Condition (6) is the Wardrop condition for the time-dependent case.We observe that condition (6) is equivalent to the following: for every we'W,


there exists a real-valued function Y w ( • ) on y such that, for all re@(w)and a.e. on &~,

4. Additional Constraints

Now we introduce, as in Ref. 4, additional constraints on the flowFe^f. We assume that these constraints can be described collectively by therequirement

where D^Hf. We assume that D is convex and

K remains the same as in (4). The definition of an equilibrium flow has tobe modified as follows:

Theorem 4.1. HeKnD is a solution of (8) if and only if there existsS e f * such that

Proof. If (9) and (10) are satisfied, then it is clear that

and H is an equilibrium in the sense of (8).Conversely, let H satisfy (8). We apply the separation theorem to the

sets

Then, A and B are nonempty, convex subsets of Z£ x R; they are disjointbecause of (8). Moreover, A has nonempty interior, since D has nonemptyinterior. So, from the separation theorem for convex sets (Ref. 6, p. 58),


there exist (S, k)e^* x R, (S,k)/(0, 0), and creR such that

It follows from (11) or (12) that k>0. For contradiction, assume that k =0. Choose F0eKn int D), which is possible from (7). Then from (11) and(12), it follows that «s, F0» = a; and since now S^O, there exists Feint Dwith <SF a, which contradicts (11). Thus k>0, and we may normalize(S,k,a) such that k=1, without affecting the validity of (11) and (12).Then, by choosing y arbitrarily close to zero in (11), we obtain

and by choosing y := «C(H), F-H» in (12), we obtain

In particular, choosing F := H in (13) and (14), we obtain

Substituting for a in (13) and (14) gives (9) and (10).

Roughly speaking, the additional constraints produce a worsening ofthe costs, which are shifted from C(H) to C(H) + S.

From the proof of Theorem 3.1, replacing there C(H) by C :=C(H) + S, it follows that (10) is equivalent to

Thus for C := C(H) + S, (9) and (15) constitute the generalized Wardropcondition corresponding to problem (8). It is necessary and sufficient forHeKr\D to be an equilibrium in the sense of (8).

As an example, the requirement FeD may express linear inequalityconstraints imposed upon the vector of link flows (see Refs. 7, 8). For eachte&~, the vector of link flows f (t) depends on the vector of route flows F(t)by means of


where A is the link-route incidence matrix. The linear constraints on the linkflows may be written as

where the coefficient matrix G and right-hand vector g vary with time. Then,D takes the form

If D is defined by convex inequalities, then every SeJzf* satisfying (9)admits an explicit representation in terms of Lagrange multipliers; for detailsabout this and sensitivity analysis, see Ref. 4, p. 190.

In the remaining parts, we shall return to the standard problem (5).

5. Existence of Equilibria

There are two standard approaches to the existence of equilibria,namely, with and without a monotonicity requirement (Stampacchia, Fan).We shall employ the following definitions.

Let Ebe a real topological vector space, K^Econvex. Then, C: K-+E*is said to be:

(i) pseudomonotone iff, for all x, yeK,

(ii) hemicontinuous iff, for all yeK, the function C-><C(^) ,y-^>is upper semicontinuous on K;

(iii) hemicontinuous along line segments iff, for all x, yeK, the func-tion n-»< C(£),}>-.*> is upper semicontinuous on the line seg-ment [x, y].

The following result is a special case of Theorems 2 and 3 in Ref. 9.

Theorem 5.1. Let E be a real topological vector space, and let K<=Ebe convex and nonempty. Let C: K-*E* be given such that:

(i) there exist A =K K nonempty, compact, and B s K compact, convexsuch that, for every xeK\A, there exists yeB with<C(x),y-x><0;

either (ii) or (iii) below holds:

(ii) C is hemicontinuous;(iii) C is pseudomonotone and hemicontinuous along line segments.

Then, there exists xeA such that <C(x),y-x0)) , for all yeK.


We may apply this result with E := £ and K given by (4). Then, K isconvex, closed, and bounded, hence weakly compact. So, if we endow Jz?with the weak topology, then K is compact, and condition (i) in Theorem5.1 is automatically satisfied by choosing A'-=K,B-=0.

If we endow Z£ with the strong topology, then K is no longer compact,and condition (i) must be used. In this case, since K is closed, the convexityof B in condition (i) is not needed. In fact, the convexity of B in the generalcase is used only to ensure that the convex hull of B u G is compact forevery finite subset G^K. But we can work also with the closed convex hullof B u G, and in a Banach space the closed convex hull of any compact setis compact (Ref. 10, p. 180).

Of course, requirement (ii) is weakened, if we pass from the weaktopology on <£ to the strong topology. For requirement (iii), it does notmatter which of the two topologies we use, since they coincide on linesegments.

Altogether, we obtain from Theorem 5.1 the following corollary.

Corollary 5.1. For K^tf given by (4) and C: K->y*, each of thefollowing conditions is sufficient for the existence of a solution to (5):

(i) C is hemicontinuous with respect to the strong topology on K,and there exist A^K nonempty, compact and B^K compactsuch that, for every HeK\A, there exists FeB with«C(H),F-H><0.

(ii) C is hemicontinuous with respect to the weak topology on A:.(iii) C is pseudomonotone and hemicontinuous along line segments.

We shall employ the following result: under condition (iii) of Corollary5.1, HeJ? is a solution of (5) if and only if

This is a slightly generalized form of a classical result of Minty; see Ref. 4.

6. Calculation of Equilibria

Given an abstract equilibrium problem of the form

where K<BBand (p: B*b B->R, a gap function (see Ref. 11) for problem (17)is a function T: B->R u {+00} such that ¥(H)>0, for all HeB, and


moreover,

Then, HeB solves (17) if and only if ¥(H)<0.In connection with the equilibrium problem (5), we choose

since this set is easy to handle (for instance, projections onto B are readilyavailable). Furthermore, we have to specialize J^ = L2(^", R*) = :Sf*, a Hil-bert space. Concerning the mapping C: K-Jzf, we make the followingassumptions:

(C 1) C is pseudomonotone;(C2) C is hemicontinuous with respect to the weak topology on K;(C3) the set (C(F) \ FeK} is norm-bounded.

All other data are as introduced in Section 3. Recall that K is weakly com-pact. From (C1) and (C2), it follows that the solution set of (5) is nonemptyand coincides with the solution set of (16). For all HeB, let

Then, the real-valued function T(H) := max{Y1(H), y2(H)} is seen easilyto be a gap function for (16). The function y 1 ( , being the maximumof a family of continuous, affine functions, is convex and weakly lowersemicontinuous. Then, *F is convex and weakly lower semicontinuous, too;in fact, the needed convexity of 4* is the main reason for considering (16),instead of attacking (5) directly. Using (C2), we shall see below that thesubdifferential

is nonempty for all HeB. Let

Because of the equivalence of (5) and (16), Hey is a solution of (5) if andonly if He F. The subgradient method for finding an element of T runs asfollows (Ref. 12): Choose H°eB arbitrary. Given HneB, Hn£F, let

where


ProjB denotes the Euclidean projection onto B. Using Assumption (C3), weshall see below that the Tn can be chosen in such a way that || rn || remainsbounded. If ||rn|| remains bounded and Hn$T for all n, then we have thefollowing result.

Theorem 6.1. There holds that V(H n)->0. The sequence {Hn} hasweak cluster points, and every weak cluster point is in F. If the sequence{Hn} has a strong cluster point H, then H is unique, and {Hn} convergesstrongly to H.

Proof. For arbitrary yeF, using the nonexpansivity of the projectionmapping and the support inequality

we obtain

Since the sequence {\\Hn- y||2} is decreasing and bounded from below, andsince the || rn \\ are bounded from above, it follows that Y(Hn) ->0.

In particular,

This shows that {Hn} is bounded and therefore has a weak cluster point.Since *P(Hn)-»0, and since ¥ is weakly lower semicontinuous, it followsthat, if H is a weak cluster point, ¥(H) <0, hence HeT.

Now, let H be a strong cluster point of { H n } . Choosing y := //in (18),we obtain

in addition since \\Hn(J) — H\\->0 for some subsequence {Hn(i)}, we obtain\\Hn- H|| -0 for the entire sequence {Hn}. D


If <£ is finite-dimensional [the case 9~ '•= {0, 1,. . ., T] ], then weak andstrong cluster points coincide. Hence, in this situation, the entire sequence{Hn converges to some Her.

It remains to discuss the choice of Ted*¥(H). For given HeB, definet1'eJ by

where F is a solution of max{«C(F), H-F^\FeK}. Such an F existsbecause of Assumption (C2). Then, 1e<3y1(H) . Define r2e^? by

Then, T 2 e d W 2 ( H ) . Now, define

Then T€d*¥(H), and this T remains bounded, if H varies in B, due toAssumption (C3). In particular, if we select r n ed*¥(H n ) according to thisrule, then \ \ t n \ \ remains bounded, as stipulated for Theorem 6.1.

References

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2. MAUGERI, A., Monotone and Nonmonotone Variational Inequalities, Rendicontidel Circolo Matematico di Palermo, Serie 2, Supplemento, Vol. 48, pp. 179-184,1997.

3. MAUGERI, A., Dynamic Models and Generalized Equilibrium Problems, NewTrends in Mathematical Programming, Edited by F. Giannessi et al., Kluwer,Dordrecht, Holland, pp. 191-202, 1998.

4. MAUGERI, A., OETTLI, W., and SCHLAGER, D., A Flexible Form of Wardrop'sPrinciple for Traffic Equilibria with Side Constraints, Rendiconti del CircoloMatematico di Palermo, Serie 2, Supplemento, Vol. 48, pp. 185-193, 1997.

5. SMITH, M. J., A New Dynamic Traffic Model and the Existence and Calculationof Dynamic User Equilibria on Congested Capacity-Constrained Road Networks,Transportation Research, Vol. 27B, pp. 49-63, 1993.

6. SCHAEFER, H. H., Topological Vector Spaces, Springer, New York, New York,1971.

7. FERRARI, P., Equilibrium in Asymmetric Multimodal Transport Networks withCapacity Constraints, Le Matematiche, Vol. 49, pp. 223-241, 1994.

8. LARSSON, T., and PATRIKSSON, M., Equilibrium Characterizations of Solutionsto Side-Constrained Asymmetric Traffic Assignment Models, Le Matematiche,Vol. 49, pp. 249-280, 1994.

9. OETTLI, W., and SCHLAGER, D., Generalized Vectorial Equilibria and GeneralizedMonotonicity, Functional Analysis with Current Applications, Edited by M.Brokate and A. H. Siddiqi, Longman, Harlow, England, pp. 145-154, 1998.

10. CONWAY, J. B., A Course in Functional Analysis, 2nd Edition, Springer, NewYork, New York, 1990.

11. AUSLENDER, A., Optimisation: Methodes Numeriques, Masson, Paris, France,1976.

12. OETTLI, W., An Iterative Method, Having Linear Rate of Convergence, for Solv-ing a Pair of Dual Linear Programs, Mathematical Programming, Vol. 3, pp. 302-311, 1972.


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Time-dependent traffic equilibria