Variational Inequalities and Applications to a Continuum Model of Transportation Network with Capacity Constraints

Variational inequalities and applications to a continuum model of transportation network with capacity constraints

G. Idone D.I.M.E.T., Faculty of Engineering, University of Reggio Calabria, Italy

Abstract

In this paper the author considers a continuum model of a transportation network in the presence of capacity constraints on the ﬂow. The equilibrium conditions are expressed in terms of a Variational Inequality for which an existence theorem is enunciated.

1 Introduction

Many equilibrium problems arising from various field of science may be expressed, under general conditions, in a unified way   BuLu =0 Bu ≥ 0 (1)  u ∈ S where B and L are suitable operators defined in a suitable class S. The kind of equilibrium described by the structure (1) is, in general, different from the one obtained by minimization of a cost functional or of an energy inte- gral. Moreover, the structure (1) leads, in general, to a variational inequality on a convex, closed subset of S. This happens, for example, for unilateral problems in continuum mechanics (the celebrated Signorini problem), the obstacle problem, the spatial price problem, the financial problem, the Walras problem (see [8], [10], [11]). The equilibrium given by the structure (1) may be considered as an equilibrium from a “local point of view”. It is different, in general, from the one, which we call

“global”, obtained by minimizing the usual functionals, and represents a comple- mentary aspect which makes clear unknown features of equilibrium problems. The reason for which the past decades have witnessed an exceptional interest in the equilibrium problems of the type (1) rests on the fact that the Variational Inequality theory, which in general expresses the equilibrium conditions (1), pro- vides a powerful methodology, that in these last years has been improved by study- ing the connections with the Separation Theory, Gap Functions, the Lagrangean Theory and Duality and many computational procedures. The aim of this paper is to consider a continuum model of transportation network in presence of capacity constraints on the flow. The equilibrium conditions are expressed in terms of a Variational Inequality for which an existence theorem is enunciated. At first the author gives a definition of an equilibrium distribution flow in presence of capacity constraints; then establishes that 1) u is an equilibrium distribution if and only if u is a solution to a Variational Inequality, 2) this Variational Inequality admits solutions. In order to demonstrate the first point the author uses the duality theory, asso- ciating an equivalent maximization problem to a given minimization problem. For that the duality-Lagrangean-theory is applied in the infinite dimensional case.

2 The equilibrium conditions

2 Setting Ω for a simply connected bounded domain in R of generic point x = (x1,x2), with Lipschitz boundary ∂Ω, v(x)=(v1(x),v2(x)) for the unknown flow at each point x ∈ Ω, the components v1(x),v2(x) are the traffic density through a neighbourhood of x in the directions of the increasing axes x1 and x2. 2 2 ϕ =(ϕ1,ϕ2) ∈ L (∂Ω, R ) is the fixed flow on the boundary ∂Ω, c(x, u(x)) = (c1(x, u(x)),c2(x, u(x))) is the “personal cost” whose components c1(x, u(x)), c2(x, u(x)) represent the travel cost along the axes x1 and x2 respectively. The aim of this paper is to consider capacity constraints on the flow:

0 ≤ s(x) ≤ v(x) ≤ z(x) a. e. in Ω. (2)

The set of feasible ﬂows becomes ˜ 2 K = {v(x) ∈E(Ω, R ): s(x) ≤ v(x) ≤ z(x) a.e. in Ω,

div v(x)+t(x)=0,v1|∂Ω = ϕ1(x),v2|∂Ω = ϕ2(x)}.

The formulation of the equilibrium conditions is the following: ˜ Deﬁnition 1 u(x) ∈ K is an equilibrium distribution ﬂow if there exists a poten- tial µ ∈ H1(Ω) such that ∂µ if si(x)

∂µ if ui(x)=si(x),thenci(x, u(x)) ≥ ; (4) ∂xi ∂µ if ui(x)=zi(x),thenci(x, u(x)) ≤ . (5) ∂xi

It is true that the following: ˜ Theorem 1 u ∈ K is an equilibrium distribution if and only if ˜ c(x, u(x))(v(x) − u(x)) dx ≥ 0 ∀v(x) ∈ K (6) Ω and the Variational Inequality (6) admits solutions. (The proof is in [7]) Theorem 1 ensures the existence of a solution fulfilling the equilibrium conditions (3)–(5). As it is well known, these conditions provide an equilibrium flow that follows the so–called “user’s optimization” approach. This equilibrium flow is different from the equilibrium flow obtained minimizing a cost functional.The ˜ convex set K satisfies the constraints qualification conditions introduced in [1]. ˜ It is important to note that the “quasi relative interior of K non empty”, (see also [9]), replaces the standard Slater condition for the infinite dimensional case. In order to prove Theorem 1 we need the following Lemmas: ψ(v)= c(x, u(x))(v(x) − u(x))dx Lemma 1 Setting Ω , the problem min ψ(v)(= ψ(u)=0) (7) ˜ v∈K is equivalent to the problem ∂v ∂v min sup ψ(v)+ µ(x) 1 + 2 + t(x) dx+ 2 (µ,λ ,Λ ,λ ,Λ )∈C∗ Ω ∂x1 ∂x2 v∈Eϕ(Ω,R ) 1 1 2 2

− λ1(x)(v1(x) − s1(x)) dx − Λ1(x)(z1(x) − v1(x)) dx+ Ω Ω − λ2(x)(v2(x) − s2(x)) dx − Λ2(x)(z2(x) − v2(x)) dx , Ω Ω (8) where ∗ 2 C = {(µ, λ1, Λ1,λ2, Λ2):µ, λi, Λi ∈ L (Ω),µ,λi, Λi ≥ 0, i =, 1, 2 a.e. in Ω} and 2 2 Eϕ(Ω, R )={v ∈E(Ω, R ):v1|∂Ω = ϕ1,v2|∂Ω = ϕ2}.

Let us consider the dual problem: ∂v ∂v max inf ψ(v)+ µ(x) 1 + 2 + t(x) dx+ ∗ (µ,λ1,Λ1,λ2,Λ2)∈C v∈Eϕ Ω ∂x1 ∂x2

− λ1(x)(v1(x) − s1(x)) dx − Λ1(x)(z1(x) − v1(x)) dx+ (9) Ω Ω − λ2(x)(v2(x) − s2(x)) dx − Λ2(x)(z2(x) − v2(x)) dx , Ω Ω and the problem: max Λ (10) Λ∈∆ where ∂v ∂v ∆={Λ ∈ R : ψ(v)+ µ(x) 1 + 2 + t(x) dx+ Ω ∂x1 ∂x2

− λ1(x)(v1(x) − s1(x)) dx − Λ1(x)(z1(x) − v1(x)) dx+ Ω Ω

− λ2(x)(v2(x) − s2(x)) dx − Λ2(x)(z2(x) − v2(x)) dx ≥ Λ Ω Ω

∗ ∀v ∈Eϕ, ∀(µ, λ1, Λ1,λ2, Λ2) ∈C }.

(The proof is in [7]) ∗ Lemma 2 (µ, λ1, Λ1, λ2, Λ2) ∈C is a maximal solution to the dual problem (9) if and only if ∂v ∂v Λ= max inf ψ(v)+ µ(x) 1 + 2 + t(x) dx+ ∗ (µ,λ1,Λ1,λ2,Λ2)∈C v∈Eϕ Ω ∂x1 ∂x2

− λ1(x)(v1(x) − s1(x)) dx − Λ1(x)(z1(x) − v1(x)) dx+ Ω Ω − λ2(x)(v2(x) − s2(x)) dx − Λ2(x)(z2(x) − v2(x)) dx Ω Ω isasolutionto(10). (The proof is in [7]) Lemma 3 If the primal problem (8) (or (7)) is solvable, then the dual problem (9) is also solvable and the extremal values of the two problems are equal. (The proof is in [7])

∗ Lemma 4 A point (u, µ, λ1, Λ1, λ2, Λ2) ∈Eϕ×C is a saddle point of the Lagran- gean function L, that is

L(u, µ, λ1, Λ1,λ2, Λ2) ≤L(u, µ, λ1, Λ1, λ2, Λ2) ≤L(v, µ, λ1, Λ1, λ2, Λ2)

∗ ∀(µ, λ1, Λ1,λ2, Λ2) ∈C , ∀v ∈Eϕ, if and only if u is a solution of the primal problem, (µ, λ1, Λ1, λ2, Λ2) is a solution to the dual problem (9) and the extremal values of the two problems are equal. (The proof is in [7]) The author in [4] gives a numerical example of a solution. The computational procedure is based on the adaptation of the subgradient method.

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