Optimal road maintenance investment in traffic networks with random demands

Abstract

The paper deals with a traffic network with random demands in which some of the roads need maintenance jobs. Due to budget constraints, a central authority has to choose which of them are to be maintained in order to decrease as much as possible the average total travel time spent by all the users, assuming that the network flows are distributed according to the Wardrop equilibrium principle. This optimal road maintenance problem is modeled as an integer nonlinear program, where the objective function evaluation is based on the solution of a stochastic variational inequality. We propose a regularization and approximation procedure for its computation and prove its convergence. Finally, the results of preliminary numerical experiments on some test networks are reported.

Appendix

In this section, we provide some details for the numerical approximation of the solution $\hat{u}$ of (8). First, we need a discretization of the space $X:= L^{p} (\mathbb {R}^d,\mathbb {P},\mathbb {R}^k).$ We introduce a sequence $\{ \pi _n \}_n$ of partitions of the support

of the probability measure $\mathbb {P}$ induced by the random elements R, S,  and D. For this, we set

$$\begin{aligned} \pi _n = (\pi _n ^R , \pi _n ^S, \pi _n ^D),\end{aligned}$$

where

These partitions give rise to an exhausting sequence $\{\Upsilon _n\}$ of subsets of $\Upsilon$, where each $\Upsilon _n$ is given by the finite disjoint union of the intervals:

$$\begin{aligned} I_{jkh}^n := [r_n^{j-1}, r_n^{j}[ \times [s_n^{k-1}, s_n^{k}[ \times I_h^n, \end{aligned}$$

where we use the multi-index $h = (h_1,\cdots ,h_m)$ and

$$\begin{aligned} I_h ^n := \prod _{i=1}^m \, [t_{n,i}^{h_i-1},t_{n,i}^{h_i}[. \end{aligned}$$

For each $n \in \mathbb {N},$ we consider the space of the $\mathbb {R}^l$-valued step functions ($l \in \mathbb {N}$) on $\Upsilon _n$, extended by 0 outside of $\Upsilon _n$:

$$\begin{aligned} X_n^l := \left\{ v_n:\ v_n (r,s,t)= \sum _j \sum _k \sum _h v^n_{jkh} 1_{I^n_{jkh}} (r,s,t),\ v^n_{jkh} \in \mathbb {R}^l \right\} , \end{aligned}$$

where $1_I$ denotes the $\{ 0,1\}$-valued characteristic function of a subset I.

To approximate an arbitrary function $w \in L^{p} (\mathbb {R}^d, \mathbb {P}, \mathbb {R}),$ we employ the mean value truncation operator $\mu _0 ^ n$ associated to the partition $\pi _n$ given by

$$\begin{aligned} \mu _0^n w := \sum _{j=1}^{N_n^{R}}\sum _{k=1}^{N_n^{S}} \sum _{h}(\mu _{jkh} ^n w)\,1_{I_{jkh}^n}\,, \end{aligned}$$

(22)

where

$$\begin{aligned} \mu _{jkh} ^n w := \left\{ \begin{array}{ll} \displaystyle \frac{1}{\mathbb {P}(I_{jkh})} \int _{I_{jkh}^n} w(y) \, d \mathbb {P} (y), &{} \text { if } \mathbb {P}(I^n_{jkh}) > 0,\\ 0, &{} \text{ otherwise. } \end{array} \right. \end{aligned}$$

Analogously, for a $L^{p}$ vector function $v=(v_1,\dots ,v_l)$, we define

$$\begin{aligned} \mu _0^n v := (\mu _0^n v_1, \dots , \mu _0^n v_l),\end{aligned}$$

for which one can prove that $\mu _0^n v$ converges to v, in $L^{p} (\mathbb {R}^d, \mathbb {P}, \mathbb {R}^l)$.

To construct approximations for

$$\begin{aligned} M_{\mathbb {P}} = \left\{ v \in L^{p} (\mathbb {R}^d, \mathbb {P},\mathbb {R}^k): G v (r,s,t) \le t\,, \;\mathbb {P}-\text {a.s.}\right\} , \end{aligned}$$

we introduce the orthogonal projector $q: (r,s,t) \in \mathbb {R}^d \mapsto t \in \mathbb {R}^m$ and define for each elementary cell $I_{jkh}^n$,

$$\begin{aligned} {\overline{q}}_{jkh}^n = ( \mu _{jkh}^{n} q) \in \mathbb {R}^m,\quad \;(\mu _0 ^ n q) = \sum _{jkh} {\overline{q}}_{jkh}^n \, 1_{I^n_{jkh}} \in X_n^m. \end{aligned}$$

This leads to the following sequence of convex and closed sets of the polyhedral type:

$$\begin{aligned} M_{\mathbb {P}}^n := \{v \in X_n^k :\ \ G v_{jkh}^n \le {\overline{q}}_{jkh}^n \,, \; \forall j,k,h \}. \end{aligned}$$

Since our objective is to approximate the random variables R and S,  we introduce

$$\begin{aligned}\rho _n =\sum _{j=1}^{N_n^R} r_n ^{j-1}\, 1_{[r_n^{j-1}, r_n ^j [} \in X_n \qquad \text {and} \qquad \sigma _n =\sum _{k=1}^{N_n^S} s_n ^{k-1}\, 1_{[s_n^{k-1}, s_n ^k [} \in X_n. \end{aligned}$$

Notice that

$$\begin{aligned} \sigma _n (r,s,t) \rightarrow \sigma (r,s,t)=s \quad \text {in}\ \ L^{\infty } (\mathbb {R}^d, \mathbb {P}) \ \ \text {and} \ \ \rho _n (r,s,t) \rightarrow \rho (r,s,t)=r \ \ \text {in}\ \ L^{p} (\mathbb {R}^d, \mathbb {P}). \end{aligned}$$

Combining the above ingredients, for $n \in \mathbb {N}$, we consider the following discretized variational inequality: Find $\hat{u} _n:=\hat{u}_n(y) \in M_{\mathbb {P}}^n$ such that for every $v_n \in M_{\mathbb {P}}^n$, we have

$$\begin{aligned} \int _{0}^{\infty } \int _{\underline{s}}^{\overline{s}}\int _{\mathbb {R}^d} [ \sigma _n(y) \, A(\hat{u}_n) + B(\hat{u}_n)]^{\top } [v_n - \hat{u}_n ] \, d\mathbb {P}(y) \ge \int _{0}^{\infty } \int _{\underline{s}}^{\overline{s}}\int _{\mathbb {R}^d} [ b + \rho _n(y) \, c]^{\top } [v_n - \hat{u}_n]\, d\mathbb {P}(y) \,. \end{aligned}$$

(23)

We also assume that the probability measures $P_R$, $P_S$, and $P_{D_i}$ have the probability densities $\varphi _R$, $\varphi _S$, and $\varphi _{D_i},$ $i=1, \dots , m$, respectively. Therefore, for $i=1,\dots , m,$ we have

$$\begin{aligned} dP_R (r)= \varphi _R (r)\, dr,\qquad dP_S (s)= \varphi _S (s)\, ds,\qquad dP_{D_i}(t_i) = \varphi _{D_i}(t_i)\,dt_i. \end{aligned}$$

It turns out that (23) can be split in a finite number of finite dimensional variational inequalities: For every $n \in \mathbb {N},$ and for every j, k, h,  find $\hat{u}^n_{jkh} \in M^n_{jkh}$ such that

$$\begin{aligned}{}[\tilde{T}_{k}^n (\hat{u}^n_{jkh})]^{\top } [v^n_{jkh}- \hat{u}^n_{jkh}] \ge [ \tilde{c}^{n}_{j}]^{\top } [ v^n_{jkh}- \hat{u}^n_{jkh}], \qquad \forall \ v^n_{jkh} \in M^n_{jkh}, \end{aligned}$$

(24)

where

$$\begin{aligned} M^n_{jkh} := \{v^n_{jkh} \in \mathbb {R}^k : G v^n_{jkh} \le {\overline{q}}_{jkh}^n \}, \quad \tilde{T}_{k}^n := s_n^{k-1} \, A + B, \quad \tilde{c}_{j}^n \,:= \, b + r_n^{j-1} \, c. \end{aligned}$$

Clearly, we have

$$\begin{aligned} \hat{u}_n = \sum _j \sum _k \sum _h \hat{u}^n_{jkh} \, 1_{I^n_{jkh}} \in X_n^k. \end{aligned}$$

We recall the following convergence result (see [9]).

Theorem 3

Assume that $T(\omega ,\cdot )$ is strongly monotone, uniformly with respect to $\omega \in \Omega$, that is

$$\begin{aligned} (T(\omega ,x)-T(\omega ,y))^{\top } ( x-y) \ge \alpha \Vert x-y\Vert ^{2}\quad \forall \ x,\,y,\ \text {a.e.}\ \omega \in \Omega , \end{aligned}$$

where $\alpha >0$ and that the growth condition (6) holds. Then the sequence $\{\hat{u}_n\},$ where $\hat{u}_n$ is the unique solution of (23), converges strongly in $L^{p} (\mathbb {R}^d, \mathbb {P},\mathbb {R}^k)$ to the unique solution $\hat{u}$ of (8).

In absence of strict monotonicity, the solution of (8) is not unique and we resort to a regularization technique as follows (see [10] for the details and proofs).

We will regularize the above discrete variational inequality and show that its continuous analogue is recovered by the limiting process. For this, we choose a sequence $\{\varepsilon _n\}$ of regularization parameters and choose the regularization map to be the duality map $J:L^p(\mathbb {R}^d,\mathbb {P},\mathbb {R}^k)\rightarrow L^q(\mathbb {R}^d,\mathbb {P},\mathbb {R}^k).$ We assume that $\varepsilon _n>0$ for every $n\in \mathbb {N}$ and that $\varepsilon _n\downarrow 0$ as $n\rightarrow \infty .$ We consider the following regularized stochastic variational inequality: For $n \in \mathbb {N}$, find $w_n=w_n^{\varepsilon _n}(y) \in M_{\mathbb {P}}^n$ such that for every $v_n \in M_{\mathbb {P}}^n$, we have

$$\begin{aligned} {\begin{matrix} \int _{0}^{\infty } \int _{\underline{s}}^{\overline{s}} \int _{\mathbb {R}_{+}^m} [ \sigma _n(y) \, A(w_n (y)) + B(w_n (y))+\varepsilon _n J(w_n (y))]^{\top } (v_n (y)- w_n (y) ) \, d\mathbb {P}(y) \\ \ge \int _{0}^{\infty } \int _{\underline{s}}^{\overline{s}} \int _{\mathbb {R}_{+}^m} [ b + \rho _n(y) \, c]^{\top } (v_n (y) - w_n (y))\, d\mathbb {P}(y). \end{matrix}} \end{aligned}$$

(25)

As usual, the solution $w_n$ will be referred to as the regularized solution.

The following theorems highlight some of the features of the regularized solutions.

Theorem 4

The following statements hold.

• 1. For every $n\in \mathbb {N},$ the regularized stochastic variational inequality (10) has the unique solution $w_n.$

• 2. Any weak limit of the sequence $\{w_n\}$ of the regularized solutions is a solution of (8).

• 3. The sequence of the regularized solutions $\{w_n\}$ is bounded provided that the following coercivity condition holds: There exists a bounded sequence $\{\delta _n\}$ with $\delta _n\in M_{\mathbb {P}}^n$ such that

  $$\begin{aligned} \frac{\int _{0}^{\infty } \int _{\underline{s}}^{\overline{s}} \int _{\mathbb {R}_{+}^m}[\sigma _n(y) \, A(u_n (y))+B(u_n (y))]^{\top } (u_n(y)-\delta _n (y) ) \, d\mathbb {P}(y)}{\Vert u_n\Vert }\rightarrow \infty \quad \text {as}\ \Vert u_n\Vert \rightarrow \infty . \end{aligned}$$

  (26)

To obtain strong convergence we need to use the concept of fast Mosco convergence, as given by the following definition.

Definition 4

Let X be a normed space, let $\{K_n\}$ be a sequence of closed and convex subsets of X and let $K\subset X$ be closed and convex. Let $\varepsilon _n$ be a a sequence of positive real numbers such that $\varepsilon _n \rightarrow 0$. The sequence $\{K_n\}$ is said to converge to K in the fast Mosco sense, related to $\varepsilon _n$, if

• 1. For each $x\in K$, $\exists \{x_n\}\in K_n$ such that $\varepsilon _{n}^{-1} \Vert x_n -x \Vert \rightarrow 0$;

• 2. For $\{x_m\} \subset X$, if $\{x_m\}$ weakly converges to $x\in K$, then $\exists \{z_m\}\in K$ such that $\varepsilon _{m}^{-1} (z_m -x_m)$ weakly converges to 0.

Theorem 5

Assume that $M_{\mathbb {P}}^n$ converges to $M_{\mathbb {P}}$ in the fast Mosco sense related to $\varepsilon _n$. Moreover assume that $\varepsilon _{n}^{-1}\Vert \sigma _n -\sigma \Vert \rightarrow 0,\;\text{ and } \varepsilon _{n}^{-1}\Vert \rho _n -\rho \Vert \rightarrow 0$ as $n\rightarrow \infty$. Then the sequence of regularized solutions $\{w_n\}$ converges strongly to the minimal-norm solution of stochastic variational inequality (8) provided that $w_n$ is bounded.

We conclude this section by recalling the following general result that ensures the solvability of an infinite dimensional variational inequality like (7) or (8) (see [12] for a recent survey on existence results for variational inequalities).

Theorem 6

Let E be a reflexive Banach space and let $E^*$ denote its topological dual space. We denote the duality pairing between E and $E^*$ by $\langle \cdot ,\cdot \rangle _{E,E^{*}}$. Let K be a nonempty, closed, and convex subset of E, and $A: K \longrightarrow E^{*}$ be monotone and continuous on finite dimensional subspaces of K. Consider the variational inequality problem of finding $u\in K$ such that

$$\begin{aligned} \langle Au, v-u \rangle _{E,E^{*}}\ge 0,\qquad \forall \ v\in K. \end{aligned}$$

Then, a necessary and sufficient condition for the above problem to be solvable is the existence of $\delta >0$ such that at least a solution of the variational inequality:

$$\begin{aligned} \text {find } u_{\delta }\in K_{\delta } \text { such that } \langle Au_{\delta }, v-u_{\delta } \rangle _{E,E^{*}} \ge 0, \quad \forall \ v\in K_{\delta } \end{aligned}$$

satisfies $\Vert u_{\delta }\Vert <\delta ,$ where $K_{\delta }=\{v\in K:\ \Vert v\Vert \le \delta \}$.