Cooperation in traffic network problems via evolutionary split variational inequalities

In this paper, we construct an evolutionary (time-dependent) split variational inequality problem and show how to reformulate equilibria of the dynamic traffic network models of two cities as such problem. We also establish existence result for the proposed model. Primary numerical results of equilibria illustrate the validity and applicability of our results.


1.
Introduction. In the early of 1960s, the Italian mathematician Stampacchia [36] and Fichera [23,24] initiated a systematic study of variational inequality problems. Thereafter, Smith [35] and Dafermos [17] set up a traffic assignment problem in the terms of a finite dimensional variational inequality problem. Lawphongpanich and Hearn [26], and Panicucci et al. [31] studied the traffic assignment problems based on Wardrop user equilibrium principle via a variational inequality model. Recently, Chen and Huang [14] reformulated a traffic network equilibrium problem as a class of linearly constrained variational inequality problem and solved it by using the power penalty method. The industrial applications of variational inequality problems have been well documented in [34,38]. On the other hand, the evolutionary variational inequality problem was originated by mechanics and introduced by Lions and Stampacchia [27], and Brezis [7]. Firstly, Daniele et al. [19] constructed a time-dependent traffic network equilibrium problem in the terms of an evolutionary variational inequality problem. Several problems related to the economic world, such as spatial price equilibrium problem, internet problem with multiple classes of traffic, Nash equilibrium problem, pollution control problem, dynamic financial and oligopolistic market equilibrium problems have been studied via an evolutionary variational inequality problem in [3,5,6,13,18,30,33].
In continuation of above research works, the split inverse problems are being studied extensively due to their applicability in image reconstruction, cancer treatment planning and many more. A split inverse problem concerns a model in which two vector spaces X and Y , and a bounded linear operator A : X → Y are given. In addition, two inverse problems are involved. The first one, denoted by IP 1 , is formulated in the space X and the second one, denoted by IP 2 , is formulated in the space Y . Given these data, the split inverse problem is formulated as follows: find a point x * ∈ X that solves IP 1 (1) and such that the point y * = Ax * ∈ Y solves IP 2 .
Censor and Elfving [10] introduced the first example of split inverse problem, known as split convex feasibility problem in which the two problems IP 1 and IP 2 are convex feasibility problems each. Censor et al. [11] used this problem for solving an inverse problem in radiation therapy treatment planning. Many results in this area were developed in the recent decades, for example split common fixed-point problem by Moudafi [29] and split common null point problem by Byrne et al. [8].
The most recent and general split problem reformulation was presented by Censor et al. [12], called as split variational inequality problem. They also constructed iterative algorithms that solve such problems and discussed their some new special cases in Euclidean space as well. These prominent roles of the split inverse problems in different areas of science, medical and real world show the necessity of their some new formulations and applications. To contribute in this direction, we define an evolutionary split variational inequality problem. In order to show its applicability in the economic world, we formulate equilibrium flow of the dynamic traffic network models of any two cities in terms of the introduced evolutionary split variational inequality problem. Further, we define a split Wardrop condition (one of the another novelty of this paper) and establish its equivalent relation with the formulated equilibrium flow of the dynamic traffic network model. Additionally, we motivate our formulated definition of equilibrium flow by an application to the dynamic traffic network models of two cities for pizza deliveries. Moreover, we also establish the existence and uniqueness for the formulated equilibria. At last, we provide two methods for solving the introduced evolutionary split variational inequality problem, which are helpful to calculate the equilibrium flow for the dynamic traffic network models of any two cities. The first method is motivated by the projected dynamical system theory, in which we define a split projected dynamical system on the feasible sets (one of the another novelty of this paper) and establish an equivalent relation between its critical point and solution of the introduced evolutionary split variational inequality problem. The second method is inspired by an iterative algorithm method, introduced by Censor et al. [12]. There are two key significance of our formulated dynamic traffic network model and the derived results. The first one is in the area of traffic network analysis and second one is in the area of split inverse problems. In essence, our formulated dynamic traffic network model in the terms of an evolutionary split variational inequality problem can deal with the two different dynamic traffic network models simultaneously in the same time intervals and our derived results can be adopted to study the dynamic traffic network model, defined by Daniele et al. [19] and explored by Cojocaru et al. [15], Barbagallo [4], and Aussel and Cotrina [2]. The detailed arguments regarding these are given in the Subsections 2.2 and 4.1. Moreover, the outcomes of the present paper give a new approach towards the real world applications of the split variational inequality problem, defined by Censor et al. [12]. The paper is organized as follows: preliminaries, formulation of the equilibrium flow for the dynamic traffic network models of two cities in the terms of an evolutionary split variational inequality problem, an equivalence of the formulated equilibrium flow with the split Wardrop condition, and a motivated example are given in Section 2. The existence and uniqueness results for the formulated equilibria are established in Section 3. The numerical illustrations and procedures for solving the introduced evolutionary split variational inequality problem are given in Section 4. Eventually, Section 5 concludes our paper.
2. Problem formulations and motivation example.

General setting of evolutionary split variational inequality problem.
For the City X: The traffic network is made of the set of nodes N (airports, railway stations, crossings, etc.), the set of directed links L between the nodes, origindestination pair W and the set of routes V . Each route r ∈ V connects exactly one origin-destination pair. The set of all r ∈ V which link a given w ∈ W is denoted by V (w). Let f (t) ∈ R V be the time-dependent flow trajectory, where t varies in the fixed time interval [0, T ] and f r (t), r ∈ V represents the flow trajectory over time t in the route r. Every feasible flow has to satisfy the following time-dependent capacity constraints λ(t) ≤ f (t) ≤ µ(t), a.e. in [0, T ], and the traffic conservation law/demand requirements where the bounds λ(t) ≤ µ(t) and the demand ρ(t) ≥ 0 are given, the pair link incidence matrix φ = φ r,w is 1 if route r links the pair w and 0 otherwise, and "a.e." stands for almost everywhere and same will be followed through out the paper. Due to technical reasons, we take the functional setting for the flow trajectories as a reflexive Banach space L p ([0, T ], R V ), p > 1 with the dual space L q ([0, T ], R V ), 1 p + 1 q = 1. We consider λ(t) and µ(t) belong to L p ([0, T ], R V ) and ρ(t) belongs to L p ([0, T ], R W ). We also consider the following which implies a nonempty set of feasible flows The canonical bilinear form on where ., . denotes the Euclidean inner product.
For the City Y: The traffic network is made of the set of nodes N , the set of directed links L between the nodes, the origin-destination pair W and the set of routes V . Each route r ∈ V connects exactly one origin-destination pair. The set of all r ∈ V which link a given w ∈ W is denoted by V (w). Let g(t) ∈ R V be a time-dependent flow trajectory, where t varies in the fixed time interval [0, T ] and g r (t), r ∈ V represents the flow trajectory over time t in the route r. Every feasible flow has to satisfy the following time-dependent capacity constraints λ(t) ≤ g(t) ≤ µ(t), a.e. in [0, T ], and the traffic conservation law/demand requirements where the bounds λ(t) ≤ µ(t) and the demand ρ(t) ≥ 0 are given, and the pair link incidence matrix φ = φ r,w is 1 if route r links the pair w and 0 otherwise. Due to technical reasons, we take the functional setting for the flow trajectories as a reflexive Banach space We consider λ(t) and µ(t) belong to L p ([0, T ], R V ) and ρ(t) belongs to L p ([0, T ], R W ). We also consider the following which implies a nonempty set of feasible flows The canonical bilinear form on where ., . denotes the Euclidean inner product.
Remark 1. Evidently, the feasible sets K and K are convex, closed and bounded, consequently both are weakly compact sets.
From now on, for notational simplicity we write the time-dependent flow trajectories without mentioning t, for instance, f is written in the place of time-dependent flow trajectory f (t). Further, for each f ∈ K and g ∈ K, the cost trajectories are represented by F : K → L q ([0, T ], R V ) and G : K → L q ([0, T ], R V ), respectively. Also, we consider A : is a bounded linear operator. Now, the evolutionary split variational inequality problem is formulated as the following: and such that The solution set of (ESVIP) is given as where X * and Y * are the solution sets of classical variational inequality problems (3) and (4), respectively.
By virtue of the definition of equilibrium flow for the dynamic traffic network model defined by Daniele et al. [19], we interpret the following definition for the dynamic traffic network models of two cities X and Y in the terms of introduced (ESVIP).

Definition 2.1. H ∈ K is an equilibrium flow if and only if H ∈ Z.
Several authors have studied the equilibrium flow of traffic network problems in the terms of Wardrop condition. Basically in traffic analysis, the meaning of Wardrop equilibrium is that the road users choose minimum cost paths. Daniele et al. [19] formulated the traffic network equilibrium problem as a classical variational inequality problem, which is equivalent to a Wardrop equilibrium condition. Raciti [32] dealt with the vector form of Wardrop equilibrium condition. In the similar manner, we define the following split Wardrop condition.
(SWC) For arbitrary f ∈ K the split Wardrop condition is defined as and such that the point g = Af ∈ K satisfies Remark 2. It is evident that the expressions (5) and (6) represent the generalized Wardrop conditions for the classical variational inequality problems (3) and (4), respectively, which are discussed in [19] and [4].
The following theorem provides the equivalent form of equilibrium flow of the formulated dynamic traffic network model in the terms of (ESVIP) by means of the split Wardrop condition. Proof. Firstly, we suppose that f ∈ K satisfies the (SWC). For w ∈ W and w ∈ W , we define the following sets, It follows from (SWC) that and such that the point g = Af ∈ K satisfies Inequality (7) yields that there exist real numbers l and l ∈ R such that Assume that h ∈ K and k ∈ K are the arbitrary flows. Then, we have for a.e. in [0, T ] Thus, r / ∈ P follows that f r = µ r and (h r − f r ) ≤ 0, consequently we obtain In the inequality (8), the terms equal to zero because of the another form of traffic conservation law/demand requirements Since h ∈ K and k ∈ K are arbitrary, (8) yields Thus, f is the equilibrium flow. Conversely, we consider that f is an equilibrium flow and it does not satisfy the (SWC). It follows that there exist w ∈ W , u, s ∈ V (w) and w ∈ W , v, s ∈ V (w) together with a set J ⊂ [0, T ] having positive measure, and we have the following cases: 1.
Similarly, we can also define the time-dependent flow trajectory and k = g out side of J.
Then, evidently h ∈ K such that h = f out side of J and k ∈ K such that k = g out side of J. Now, we have and similarly the point g = Af ∈ K satisfies which implies that f is not an equilibrium flow. By using the same techniques, we can easily find that f is not an equilibrium flow for the cases 2. and 3.. Further, due to the presence of g = Af / ∈ K in the cases 4., 5., 6. and 7., it is obvious that f is not an equilibrium flow. Eventually, we have the contradiction, which completes the proof.
Remark 3. Due to the decomposed form of (SWC), it is more responsive to the user than its equivalent form given by Definition 2.1. We can say that (SWC) is a user-oriented equilibrium.

A motivational example:
The dynamic traffic network model of two cities for pizza deliveries. In Figure 1, the transportation network patterns of two cities X and Y are shown. We consider, a pizza company has the branches at P 1 and P 2 in City X and at P 1 and P 4 in City Y . In City X, the pizza delivery boys of the branches P 1 and P 2 have to deliver the pizzas at P 3 and P 5 , respectively. In City Y , the pizza delivery boys of the branches P 1 and P 4 have to deliver the pizzas at P 2 and P 3 , respectively. Thus, for City X, the network consists six nodes and eight links, we assume the origin destination pairs are w 1 = (P 1 , P 3 ) and w 2 = (P 2 , P 5 ), which are respectively connected by the following paths: V 1 = (P 1 , P 2 ) ∪ (P 2 , P 3 ) V 2 = (P 1 , P 6 ) ∪ (P 6 , P 5 ) ∪ (P 5 , P 2 ) ∪ (P 2 , P 3 ), The set of feasible flows is a.e. in [0, T ]} and cost function on the path is F : for City Y , the network consists five nodes and seven links, we assume the origin destination pairs are w 1 = (P 1 , P 2 ) and w 2 = (P 4 , P 3 ), which are respectively connected by the following paths: The set of feasible flows is We can easily seen that with the help of above introduced model, we can find the equilibrium flows of the traffic of both cities X and Y simultaneously in the same time interval [0, T ]. 3. Existence of equilibria. Several existence results for the different kinds of variational inequality problems have been established in the literature, for instance see [1,9]. In these existing results, two standard techniques are used. The first one is by using KKM-Fan Theorem with monotonicity assumption and second is Brouwer's fixed-point theorem without monotonicity assumption. By keeping the view of proofs of [37], we will prove the existence of equilibria of the dynamic traffic network model defined in the previous section but by using the concept of graph of a operator A, which is defined as follows, We also consider that It is easy to prove that M is the convex set. The bounded linear operator A implies that it is also continuous. Then by the closed graph theorem, we get that the graph M of A is closed with the product topology. Thus, M is a nonempty, closed and convex subset of K × K. Remark 1 yields that K × K is the weakly compact set. Hence, M is the weakly compact set. Moreover, we also need the following definitions and lemma, which are motivated by [9,22], for proving our existence result.
Definition 3.2. The cost functions F and G are called sequentially continuous at the point a ∈ K and b ∈ K, respectively, if and only if F (x n ) → F (a) for all sequences x n ∈ K, x n → a and G(y n ) → G(b) for all sequences y n ∈ K, y n → b, where the symbol " → " stands for convergence.
The following corollary gives the uniqueness of the solution of (ESVIP). Corollary 1. If the cost functions F and G are strict monotone on K and K, respectively, then (ESVIP) has a unique solution.
Proof. Let x 1 ∈ K be a solution of (ESVIP). We have Again, let x 2 ∈ K be a solution of (ESVIP) and x 1 = x 2 , we get The inequality (11) can be rewritten as Strict monotonicities of the functions F and G together with the fact y 1 = y 2 , we obtain By adding the corresponding inequalities of (13) and (14), we get F (x 2 ), x 1 − x 2 X < 0 and G(y 2 ), y 1 − y 2 Y < 0, which contradicts the inequality (12), i.e., x 2 is not the solution of (ESVIP). Therefore, (ESVIP) has the unique solution.
4. Numerical illustration of the dynamic traffic network model of two cities. In this section, we shall demonstrate two methods to solve the introduced (ESVIP). The first one is by projected dynamical system (PDS) theory which is motivated by [15] and the second one is by an iterative method, introduced by Censor et al. [12]. Moreover, we shall also discuss that which method is more amenable to the user. The projected dynamical system was first introduced by Dupuis and Nagurney [21]. They also defined its relation with a classical variational inequality problem. However, this relation was first noted in Dupuis and Ishii [20]. The historical and applicative point of views of the projected dynamical systems are well defined by Cojocaru et al. [15] and Giuffré et al. [25]. Inspired by these works, we define the split projected dynamical system on the feasible sets K and K for p = 2 and p = 2, respectively, as following: proj K (.) and proj K (.) are the projections of the given vectors on the sets K and K, respectively, which are defined after this paragraph. Further, to avoid confusion between the times t and τ , we represent elements of the sets L 2 ([0, T ], R V ) and L 2 ([0, T ], R V ) at fixed moments t ∈ [0, T ] as x(.) and y(.), respectively. Indeed, in the formulation of (SPDS), the time τ is different from the time t. For all t ∈ [0, T ] a solution of (ESVIP) represents a static state of the underlying system and the static states define one or more equilibrium curves with the variation of t over the interval [0, T ]. On the other hand, τ defines the dynamics of the system over the interval [0, ∞) until it reaches one of the equilibria on the curves. It is clear that the solutions of (SPDS) lie in the class of absolutely continuous functions with respect to τ , from [0, ∞) to K. In order to describe the procedure to solve the (ESVIP), we need to demonstrate the following definitions which are inspired by [16,28].

Remark 5.
For each x(.) ∈ L p ([0, T ], R V ), proj K (x(.)) satisfies the following property Definition 4.3. The polar set K • associated to K is defined as Definition 4.4. The tangent cone to the set K at a point x(.) ∈ K is defined as where cl stands for closure.
Definition 4.5. The normal cone of K at a point x(.) is defined as and it can be written as The following theorem enables us to find a relation between the solution of (ESVIP) and the critical point of (SPDS). Above inequalities imply −F (x * (.)) ∈ N K (x * (.)) and − G(y * (.)) ∈ N K (y * (.)).
By using same values for all the given arguments F , G, t i , K ti and K ti as defined in the previous method for solving the (ESVIP) by (SPDS) theory and keeping the fact of uniqueness of the solution of (ESVIP) in the mind, we get the same solutions for static split variational inequality problem (17) as given in Table 1 and shown by Figure 2 . We can clearly see that this method is computationally expansive than (SPDS) theory method because of the several iterative steps. Therefore, one can speak the previous method is user-oriented.

4.1.
A comparative study. In this section, we will compare our formulated dynamic traffic network model with the given and investigated dynamic traffic network models of [19,15,4,2]. We can see that the introduced evolutionary split variational inequality problem (ESVIP) is combination of two evolutionary variational inequality problems given by the inequalities (3) and (4). Firstly, Daniele et al. [19] defined the equilibrium flow of a dynamic traffic network model in the terms of an evolutionary variational inequality problem of the form as given by the inequalities (3) and (4). Thereafter, the existence of this equilibrium flow was investigated and improved by Cojocaru et al. [15], Barbagallo [4], and Aussel and Cotrina [2]. Clearly, if x * ∈ K is a solution of (ESVIP) then it is also the solution of evolutionary variational inequality problem (3) on the feasible set K and y * = Ax * ∈ K is the solution of evolutionary variational inequality problem (4) on the feasible set K. Therefore, x * is the equilibrium flow of the dynamic traffic network model for City X and y * is the equilibrium flow of the dynamic traffic network model for City Y (according to the definition of equilibrium flow of the dynamic traffic network model, defined by Daniele et al. [19]). After solving the evolutionary variational inequality problems (3) and (4) by using the same arguments as given in the above section of numerical illustration, we get the unique solutions set of evolutionary variational inequality problem (3) as given in Table 1 and traffic network pattern of the City X can be shown by Figure 2. In continuation, we also get the unique solutions set of evolutionary variational inequality problem (4) as given in Table 2 and the traffic network pattern of the City Y is shown in Figure 3. Conclusively, we get the equilibrium flows of the dynamic traffic network models, studied in literature with the help of our formulated dynamic traffic network model in the terms of evolutionary split variational inequality problem and the derived results.   Figure 3. Traffic network pattern of City Y

5.
Conclusion. The present paper is a contribution in the study of traffic network and split inverse problems. We formulated the equilibrium flows of the dynamic traffic network models for any two cities in the terms of an evolutionary split variational inequality problem and established an equivalent relationship between the equilibria of the proposed model and split Wardrop condition. We also dealt with the existence and uniqueness of the formulated equilibria. In order to calculate the equilibrium flow of the dynamic traffic network models for two cities, we proposed two methods and computed the solutions of the evolutionary split variational inequality problem.