Variational inequalities and time-dependent traffic equilibria

Transcription

1 Time-dependent traffic equilibria Analyse mathématique/mathematical analysis Variational inequalities and time-dependent traffic equilibria Patrizia DANIELE, Antonino MAUGERI and Werner OETTLI P. D. A. M. : Dipartimento di Matematica, Università di Catania, Viale A. Doria 6, I 95125, Italy; W. O. : Fakultät für Mathematik und Informatik, Universität Mannheim, D 68131, Mannheim, Germany. Abstract. We consider existence, characterization and calculation of equilibria in transportation network, when the route capacities and demand requirements depend on time. The problem is expressed in terms of a variational inequality and is situated in a Banach space setting. Inéquations variationnelles et équilibres du trafic dépendant du temps Résumé. On considère l existence, la caractérisation et le calcul des équilibres dans un résean de transport, quand les capacités des chemins et la demande dépende du temps. Le problème est exprès en termes d une inéquation variationnelles et il est placé dans le contexte d un espace de Banach. Note présentée par... 1

2 2 Version française abrégée Le trafic de résean est considéré pour tous les instants t T = [0, T ]. Pour chaque instant t T on a le vecteur des fluxes sur les chemins F (t) R R, oǹ R est l ensemble des chemins. Les fluxes admissibles doivent satisfaire les contraintes de capacité λ(t), µ(t) qui dépendent du temps aussi bien que, pour chaque couple origine-destination w W, les demandes des usagers ρ(t). Par conséquent, en choisissant L p (T, R R ), p > 1 (noté L pour abréger) comme espace fonctionnel des trajectoires des fluxes, l ensemble des fluxes admissibles est donné par K = {F L λ(t) F (t) µ(t) p.p. sur T, ΦF (t) = ρ(t) p.p.}. En posant L = L q (T, R R ), 1/p + 1/q = 1 et G, F = G(t), F (t), G L, F L, T et en notant C une application C: K L qui assigne à chaque trajectoire de flux F ( ) K le coût de la trajectoire C(F )( ) L, on peut donner la définition suivante de trajectoire d équilibre (voir [1] [5]): Définition H L est une trajectoire d équilibre si et seulement si (1) H K et C(H), F H 0 F K. On a le théorème suivant (voir [5]): Théorèm 1. H est une trajectoire d équilibre ou sens (1) si et seulement si pour chaque couple origine-destination w W, pour chaque couple de chemins q, s joignant w et presque partour sur T on a C q (H(t)) < C s (H(t)) = H q (t) = µ q (t) ou bien H s (t) = λ(t). Le problème (1) est étudié sous quelques point de vue (voir [5]): En evaluant comme l équilibre est influencé par le présence de contraintes additionnelles dûes à la congestion. En prouvant quelques théorèmes généraux d existence sans ou bien avec hypothèses de monotonie. En indiquant une méthode pour le calcul de la solution.

3 3 In the traffic network one has a set W of origin-destination pairs and a set R of routes. Each route r R links exactly one origin-destination pair w W. The set of all r R which link a given w W is denoted by R(w). The traffic network whose geometry remains fixed is considered at all times t T = [0, T ] (the case where T = {0, 1,..., T } is trivially included). For each time t T we have a route-flow vector F (t) R R. The feasible flow have to satisfy the time-dependent capacity constraints and demand requirements, namely that, almost everywhere on T, λ(t) F (t) µ(t) and ΦF (t) = ρ(t), where λ( ) µ( ) as well as ρ( ) are given and Φ = (Φ w,r ) with w W, r R, is the pair-route incidence matrix, namely 1 if route r connects the pair w, Φ w,r = 0 else. F ( ): T R R is the flow trajectory over time. The functional setting for the flow trajectory is the reflexive Banach space L p (T, R R ) with p > 1. We abbreviate it by L and the dual space L q (T, R R ), 1/p + 1/q = 1, will be denoted by L. On L L we define the canonical bilinear form by G, F = G(t), F (t), G L, F L, and the set of feasible flows is given by T K = {F L λ(t) F (t) µ(t) a.e. on T, ΦF (t) = ρ(t) a.e. on T }. Assuming that Φλ(t) ρ(t) Φµ(t) a.e. on T, it is easily seen that K is non empty, convex, closed and bounded set. Furthermore, we are given a mapping C: K L, which assigns to each flow trajectory F ( ) K the cost trajectory C(F )( ) L. We may give the definition of equilibrium flow: Definition. H L is an equilibrium flow if and only if (1) H K and C(H), F H 0 F K. The following theorem holds (see [4], [5]): Theorem 1. H K is an equilibrium flow in the sense of (1) if and only if for all w W, all q, s R(w), and a.e. on T there holds: C q (H)(t) < C s (H)(t) = H q (t) = µ q (t) or H s (t) = λ s (t).

4 4 To describe the congestion effects, we introduce additional constraints on the flow F L. We assume that these constraints can be described collectively by the requirements F D, where D L. We assume that D is convex and K int D. The definition of an equilibrium flow has to be modified as follows: (2) H K D and C(H), F H 0 F K D and the characterization of (2) is the following (see [5]): Theorem 2. H K D is a solution of (2) if and only if there exists S L such that (3) S, F H 0 F D (4) C(H) + S, F H 0 F K. The additional constraints produce a worsening of the costs, which are shifted from C(H) to C(H) + S. From the proof of Theorem 1, replacing these C(H) by C = C(H) + S, it follows that (4) is equivalent to (5) For all w W, all q, s R(w), and a.e. on T there holds: C q (t) < C s (t) = H q (t) = µ q (t) or H s (t) = λ s (t). Following the two standard approaches to the existence of solutions to Variational Inequalities, namely with and without a monotonicity requirement (Stampacchià, Fou), we shall employ the following definitions: C: K L is said to be: pseudomonotone if and only if for all F, H K C(F ), H F 0 = C(H), F H 0; hemicontinuous if and only if for all F K, the function H C(H), F H is upper semicontinuous on K; hemicontinuous along line segments if and only if for all F, H K, the function W C(W ), H F is upper semicontinuous on the line segment [F, H].

5 5 Then the following existence result holds: Theorem 3. Each one of the following conditions is sufficient for the existence of a solution of (1) : i) C is hemicontinuous with respect to the strong topology on K, and there exist A K nonempty, compact, and B K compact such that for every H K \ A, there exists F B with C(H), F H < 0; ii) C is hemicontinuous with respect to the weak topology on K; iii) C is pseudomonotone and hemicontinuous along line segments. (see [1] [5]). For what concerns calculation of equilibria, let us set B = {F L λ(t) F (t) µ(t) a.e. on T } and let us assume that C is pseudomonotone and hemicontinuous along line segments. Then, setting where ψ(h) = max {ψ 1 (H), ψ 2 (H)} H B, ψ 1 (H) = sup C(F ), H F, ψ 2 (H) = ΦH(t) ρ(t) p p dt, ψ(h) results a convex gap function for (1), namely ψ(h) 0 for all H B and ψ(h) = 0 if and only if H solves (1). Then we may apply the subgradient method for minimizing ψ. We choose L = L 2 (T, R R ) = L and assume that the convex function ψ is finite and continuous on some neighbourhood of B. We set Γ = {H B ψ(h) 0} and choose H 0 B. Given H n B, H n / Γ, let H n+1 = Proj B (H n ρ n τ n ) where τ n ψ(h n ) is arbitrary, and ρ n = ψ(hn ). If Hn / τ n 2 remains bounded, then we have: T Γ for all n, and τ n Theorem 4. There holds ψ(h n ) 0. The sequence {H n } has weak cluster point, and every weak cluster point is in Γ. If the sequence {H n } has a strong cluster point H, then H is unique and {H n } converges strongly to H. (see [5]). References

6 6 [1] Maugeri A., Le Matematiche, 49, pp [2] Maugeri A., Rend. Circ. Mat. Palermo, 48, pp [3] Maugeri A., New Trends in Mathematical Programming, Kluwer, Dordrecht, to appear. [4] Maugeri A., Oettli W. and Schläger, Rend. Circ. Mat. Palermo, 48, pp [5] Daniele P., Maugeri A. and Oettli W., to appear.