Walid Krichene

A PDE-ODE Model for a Junction with Ramp Buffer

We consider the Lighthill--Whitham--Richards traffic flow model on a junction composed by one mainline, an onramp, and an offramp, which are connected by a node. The onramp dynamics is modeled using an ordinary differential equation describing the evolution of the queue length. The definition of the solution of the Riemann problem at the junction is based on an optimization problem and the use of a right-of-way parameter. The numerical approximation is carried out using a Godunov scheme, modified to take into account the effects of the onramp buffer. We present the result of some simulations and numerically check the convergence of the method.

Adjoint-Based Optimization on a Network of Discretized Scalar Conservation Laws with Applications to Coordinated Ramp Metering

The adjoint method provides a computationally efficient means of calculating the gradient for applications in constrained optimization. In this article, we consider a network of scalar conservation laws with general topology, whose behavior is modified by a set of control parameters in order to minimize a given objective function. After discretizing the corresponding partial differential equation models via the Godunov scheme, we detail the computation of the gradient of the discretized system with respect to the control parameters and show that the complexity of its computation scales linearly with the number of discrete state variables for networks of small vertex degree. The method is applied to the problem of coordinated ramp metering on freeway networks. Numerical simulations on the I15 freeway in California demonstrate an improvement in performance and running time compared with existing methods. In the context of model predictive control, the algorithm is shown to be robust to noise in the initial data and boundary conditions.

On the characterization and computation of Nash equilibria on parallel networks with horizontal queues

We study inefficiencies in parallel networks with horizontal queues due to the selfish behavior of players, by comparing social optima to Nash equilibria. The article expands studies on routing games which traditionally model congestion with latency functions that increase with the flow on a particular link. This type of latency function cannot capture congestion effects on horizontal queues. Latencies on horizontal queues increase as a function of density, and flow can decrease with increasing latencies. This class of latency functions arises in transportation networks. For static analysis of horizontal queues on parallel-link networks, we show that there may exist multiple Nash equilibria with different total costs, which contrasts with results for increasing latency functions. We present a novel algorithm, quadratic in the number of links, for computing the Nash equilibrium that minimizes total cost (best Nash equilibrium). The relative inefficiencies of best Nash equilibria are evaluated through analysis of the price of stability, and analytical results are presented for two-link networks. Price of stability is shown to be sensitive to changes in demand when links are near capacity, and congestion mitigation strategies are discussed, motivated by our results.

Online Learning of Nash Equilibria in Congestion Games

We study the repeated, nonatomic congestion game, in which multiple populations of players share resources and make, at each iteration, a decentralized decision on which resources to utilize. We investigate the following question: given a model of how individual players update their strategies, does the resulting dynamics of strategy profiles converge to the set of Nash equilibria of the one-shot game? We consider in particular a model in which players update their strategies using algorithms with sublinear discounted regret. We show that the resulting sequence of strategy profiles converges to the set of Nash equilibria in the sense of Cesaro means. However, convergence of the actual sequence is not guaranteed in general. We show that it can be guaranteed for a class of algorithms with a sublinear discounted regret and which satisfy an additional condition. We call such algorithms AREP (approximate replicator) algorithms, as they can be interpreted as a discrete-time approximation of the replicator equat...

Stackelberg Routing on Parallel Networks With Horizontal Queues

In order to address inefficiencies of Nash equilibria for congestion networks with horizontal queues, we study the Stackelberg routing game on parallel networks: assuming a coordinator has control over a fraction of the flow, and that the remaining players respond selfishly, what is an optimal Stackelberg strategy of the coordinator, i.e. a strategy that minimizes the cost of the induced equilibrium? We study Stackelberg routing for a new class of latency functions, which models congestion on horizontal queues. We introduce a candidate strategy, the non-compliant first strategy, and prove it to be optimal. Then we apply these results by modeling a transportation network in which a coordinator can choose the routes of a subset of the drivers, while the rest of the drivers choose their routes selfishly.

Convergence of mirror descent dynamics in the routing game

We consider a routing game played on a graph, in which different populations of drivers (or packet routers) iteratively make routing decisions and seek to minimize their delays. The Nash equilibria of the game are known to be the minimizers of a convex potential function, over the product of simplexes which represent the strategy spaces of the populations. We consider a class of population dynamics which only uses local loss information, and which can be interpreted as a mirror descent on the convex potential. We show that for vanishing, non-summable learning rates, mirror descent dynamics are guaranteed to converge to the set of Nash equilibria, and derive convergence rates as a function of the learning rate sequences of each population, and illustrate these results on numerical examples.

On learning how players learn: estimation of learning dynamics in the routing game

The routing game models congestion in transportation networks, communication networks, and other cyber physical systems in which agents compete for shared resources. We consider an online learning model of player dynamics: at each iteration, every player chooses a route (or a probability distribution over routes, which corresponds to a flow allocation over the physical network), then the joint decision of all players determines the costs of each path, which are then revealed to the players. We pose the following estimation problem: given a sequence of player decisions and the corresponding costs, we would like to estimate the learning model parameters. We consider in particular entropic mirror descent dynamics, reduce the problem to estimating the learning rates of each player. We demonstrate this method using data collected from a routing game experiment, played by human participants: We develop a web application to implement the routing game. When players log in, they are assigned an origin and destination on the graph. They can choose, at each iteration, a distribution over their available routes, and each player seeks to minimize her own cost. We collect a data set using this interface, then apply the proposed method to estimate the learning model parameters. We observe in particular that after an exploration phase, the joint decision of the players remains within a small distance of the Nash equilibrium. We also use the estimated model parameters to predict the flow distribution over routes, and compare these predictions to the actual distribution. Finally, we discuss some of the qualitative implications of the experiments, and give directions for future research.

On Stackelberg routing on parallel networks with horizontal queues

This paper presents a game theoretic framework for studying Stackelberg routing games on parallel networks with horizontal queues, such as transportation networks. First, we introduce a new class of latency functions that models congestion due to the formation of physical queues. For this new class, some results from the classical congestion games literature (in which latency is assumed to be a non-decreasing function of the flow) do not hold. In particular, we find that there may exist multiple Nash equilibria that have different total costs. We provide a simple polynomial-time algorithm for computing the best Nash equilibrium, i.e., the one which achieves minimal total cost. Then we study the Stackelberg routing game: assuming a central authority has control over a fraction of the flow on the network (compliant flow), and that the remaining flow (non-compliant) responds selfishly, what is the best way to route the compliant flow in order to minimize the total cost? We propose a simple Stackelberg strategy, the Non-Compliant First (NCF) strategy, that can be computed in polynomial time. We show that it is optimal for this new class of latency on parallel networks. This work is applied to modeling and simulating congestion relief on transportation networks, in which a coordinator (traffic management agency) can choose to route a fraction of compliant drivers, while the rest of the drivers choose their routes selfishly.

Efficient Bregman projections onto the simplex

We consider the problem of projecting a vector onto the simplex Δ = {x ∈ ℝ+d : Σi=1d xi = 1}, using a Bregman projection. This is a common problem in first-order methods for convex optimization and online-learning algorithms, such as mirror descent. We derive the KKT conditions of the projection problem, and show that for Bregman divergences induced by ω-potentials, one can efficiently compute the solution using a bisection method. More precisely, an ω-approximate projection can be obtained in O(d log 1/ω). We also consider a class of exponential potentials for which the exact solution can be computed efficiently, and give a O(d log d) deterministic algorithm and O(d) randomized algorithm to compute the projection. In particular, we show that one can generalize the KL divergence to a Bregman divergence which is bounded on the simplex (unlike the KL divergence), strongly convex with respect to the ℓ1 norm, and for which one can still solve the projection in expected linear time.

A heterogeneous routing game

Most literature on routing games make the assumption that drivers or vehicles are of the same type and, hence, experience the same latency or cost when traveling along the edges of the network. In contrast, in this article, we propose a heterogeneous routing game in which each driver or vehicle belongs to a certain type. The type determines the cost of traveling along an edge as a function of the flow of all types of drivers or vehicles over that edge. We examine the existence of a Nash equilibrium in this heterogeneous routing game. We study the conditions for which the problem of finding a Nash equilibrium can be posed as a convex optimization problem and is therefore numerically tractable. Numerical simulations are presented to validate the results.