Patrice Marcotte

• Université de Montréal

In this paper, we address a day-ahead pricing and load balancing problem, within an environment involving self-scheduled users whose utilities are optimized via a smart grid. The model is formulated as a mixed integer bilevel program for which we propose a single level reformulation. Next, we design two heuristic algorithms which, by relying on the...

This chapter is devoted to network design problems involving conflicting agents, referred to as the designer and the users, respectively. Such problems are best cast into the framework of bilevel programming, where the designer anticipates the reaction of rational users to its course of action, which fits many situations of interest. In this book c...

This paper is concerned with the design of efficient exact and heuristic algorithms for addressing a bilevel network pricing problem where demand is a nonlinear function of travel cost. The exact method is based on the piecewise linear approximation of the demand function, yielding mixed integer programming formulations, while heuristic procedures...

In this work, we address the problem of maximizing the revenue raised from tolls set on a multicommodity transportation network, taking into account that users are assigned to cheapest paths, and that demand is a linearly decreasing function of total path cost (initial cost of carrying the products plus toll). We propose for its numerical solution...

Firms selling perishable products use a variety of techniques to maximize revenue through the dynamic control of their inventories. One of the most powerful and simple approaches to address this issue consists of assigning threshold values (“bid prices”) to each resource, and to accept requests whenever their revenue exceeds the sum of the bid pric...

In revenue management, booking limits are commonly used to restrict access to classes of products, to subsequently make way for more profitable ones. Frequently, this inventory control policy assumes that products are nested in decreasing order of revenue, and that less profitable products are denied access first. In this article, we propose for th...

We consider an energy provider whose goal is to simultaneously set revenue-maximizing prices and meet a peak load constraint. In our bilevel setting, the provider acts as a leader (upper level) that takes into account a smart grid (lower level) that minimizes the sum of users' disutilities. The latter bases its decisions on the hourly prices set by...

This paper addresses a dynamic resource allocation problem which has its roots in airline revenue management, and where customers select the available product that ranks highest on a preset list of preferences. The problem is formulated as a flexible mathematical program that can easily embed technical and practical constraints, as well as accommod...

In this paper, we consider the problem of optimizing the portfolio of an aggregator that interacts with the energy grid via bilateral contracts. The purpose of the contracts is to achieve the pointwise procurement of energy to the grid. The challenge raised by the coordination of scattered resources and the securing of obligations over the planning...

In this paper, we address the numerical solution of a pricing problem where users are assigned according to a logit model onto the paths of a transportation network. Although this highly nonconvex problem admits a large number of local optima, we show that it is possible to devise strategies that allow us to find near-optimal solutions through a mi...

We derive worst-case bounds, with respect to the L p norm, on the error achieved by algorithms aimed at approximating a concave function of a single variable, through the evaluation of the function and its subgradient at a fixed number of points to be determined. We prove that, for p larger than 1, adaptive algorithms outperform passive ones. Next,...

Revenue management systems rely on customer data, and are thus affected by the absence of registered demand that arises when a product is no longer available. In the present work, we review the uncensoring (or unconstraining) techniques that have been proposed to deal with this issue, and develop a taxonomy based on their respective features. This...

This paper addresses a network pricing problem where users are assigned to the paths of a transportation network according to a mixed logit model, i.e., price sensitivity varies across the user population. For its solution, we propose algorithms based on combinatorial approximations, and show that the smoothing effect induced by both the discrete c...

Price optimization fits naturally the framework of bilevel programming, where a leader integrates within its decision process the reaction of rational customers. This paper addresses the situation where the users of a transportation network minimize a weighted sum of travel delay and out-of-pocket cost, each user having its own monetary valuation o...

This study analyses the use of neural networks to produce accurate forecasts of total bookings and cancellations before departure, of a major European rail operator. Effective forecasting models, can improve revenue performance of transportation companies significantly. The prediction model used in this research is an improved multi-layer perceptro...

In this paper, we propose an efficient Tabu Search procedure for solving the NP-hard network pricing problem. By exploiting the problem's features, the algorithm allows the near-optimal solution of problem instances that are out of reach of exact combinatorial methods.

Revenue management (RM) is the process of understanding and anticipating customer behavior in order to maximize revenue raised from the sale of perishable resources available in limited quantities. While RM systems have been in operation for quite some time, they cannot take into account the full dynamic and stochastic nature of the problem, hence...

Motivated by an application in highway pricing, we consider the problem that consists in setting profit-maximizing tolls on a clique subset of a multicommodity transportation network. We formulate the problem as a linear mixed integer program and propose strong valid inequalities, some of which define facets of the two-commodity polyhedron. The num...

This work focuses on an improved exact algorithm for addressing an NP-hard network pricing problem. The method involves an efficient and partial generation of candidate solutions, a recursive scheme for generating improved upper bounds, and a column generation procedure for solving the network-structured subproblems. Its efficiency is assessed agai...

This study addresses bilevel linear multi-objective problem issues i.e. the special case of bilevel linear pro-gramming problems where each decision maker has several objective functions conflicting with each other. We introduce an artificial multi-objective linear programming problem of which resolution can permit to generate the whole feasible se...

Consider the problem that consists in maximizing the revenue generated by tolls set on a subset of arcs of a transportation network, and where origin-destination flows are assigned to shortest paths with respect to the sum of tolls and initial costs. In this work, we address the instance where toll arcs must be connected, as occurs on highways. Our...

In this study, we establish a parallel between two classes of pricing problems that have attracted the attention of researchers in marketing, theoretical computer science and operations research, each community addressing issues from its own vantage point. More precisely, we contrast the problems of pricing a network or a product line, in order to...

A noncooperative game is formulated on a transportation network with congestion. The players are associated with origin-destination pairs; each player supplies from an origin a destination, where demand is variable. A Nash-Cournot equilibrium is defined and conditions for existence and uniqueness are provided. The asymptotic behaviour of the equili...

In this paper, we investigate toll setting as a policy tool to regulate the use of roads for dangerous goods shipments. We propose a mathematical formulation as well as a solution method for the hazardous materials toll problem. Based on a comparative analysis of proposed mathematical models, we show that toll policies can be more effective than th...

In this work we provide a simple proof of the existence of optimal tolls for multiclass network equilibrium problems where the value-of-time parameter varies continuously throughout the population. The main argument, based on a finite-dimensional reformulation of the problem, also allows us to determine in a simple fashion revenue minimizing link t...

Preventive healthcare aims at reducing the likelihood and severity of potentially life-threatening illnesses by protection and early detection. The level of participation in preventive healthcare programs is a critical determinant in terms of their effectiveness and efficiency. This article presents a methodology for designing a network of preventi...

This paper is concerned with the characterization of optimal strategies for a service firm acting in an oligopolistic environment. The decision problem is formulated as a leader-follower game played on a transportation network, where the leader firm selects a revenue-maximizing price schedule that takes explicitly into account the rational behavior...

In this work, we establish a parallel between two classes of pricing problems that have attracted the attention of researchers in economics, theoretical computer science and operations research, each community addressing issues from its own vantage point. More precisely, we contract the problems of pricing a network or a product line, in order to a...

To optimize revenue, service firms must integrate within their pricing policies the rational reaction of customers to their price schedules. In the airline or telecommunication industry, this process is all the more complex due to interactions resulting from the structure of the supply network. In this paper, we consider a streamlined version of th...

We consider the problem of maximizing the revenue raised from tolls set on the links of a multi- commodity transportation network. Since an optimal policy must balance toll levels against utilization rates, this problem lends itself naturally to a bilevel formulation, where network users assign themselves to shortest paths with respect to a disutil...

Keywords See also References

This chapter presents the main theoretical and algorithmical results pertaining to the traffic equilibrium problem (TEP), along the way improving theoretical results that were established some 20 years ago, and describes the relationship between the Nash and Wardrop concepts. The subject of traffic equilibrium is the description, through analytical...

This paper is devoted to bilevel optimization, a branch of mathematical program- ming of both practical and theoretical interest. Starting with a simple example, we proceed towards a general formulation. We then present fields of application, focus on solution ap- proaches, and make the connection with MPECs (Mathematical Programs with Equilibrium...

In this paper, we provide a heuristic procedure, that performs well from a global optimality point of view, for an important and difficult class of bilevel programs. The algorithm relies on an interior point approach that can be interpreted as a combination of smoothing and implicit programming techniques. Although the algorithm cannot guarantee gl...

Motivated by the study of parametric convex programs, we consider approximation of concave functions by piecewise affine functions. Using dynamic programming, we derive a procedure for selecting the knots at which an oracle provides the function value and one supergradient. The procedure is adaptive in that the choice of a knot is dependent on the...

Without Abstract

A version of the toll setting problem consists in determining profit maximizing tolls on a subset of arcs of a transportation network, given that users travel on shortest paths. This yields a bilevel program for which we propose efficient algorithms based on path generation.

Consider the problem of maximizing the revenue generated by tolls set on a subset of arcs of a transportation network, and where origin-destination flows are assigned to shortest paths with respect to the sum of tolls and initial costs. This work is concerned with two new combinatorial formulations of this problem and provides a framework for deriv...

Whereas pricing models have in the past often been approached from a purely academic standpoint, optimal pricing is nowadays considered as a central financial and operational tool in several industries. The optimal pricing problem, involving two decision makers acting non cooperatively and in a sequential way, can be adequately modelled as a bileve...

Bilevel programming is a branch of optimization where a subset of variables is constrained to lie in the optimal set of an auxiliary mathematical prograri. This chapter presents an overview of two specific classes cf bilevel programs, and in particular their relationship to well-known combinatorial problems.

We consider the problem of maximizing the revenue raised from tolls set on the arcs of a transportation network, under the constraint that users are assigned to toll-compatible shortest paths. We first prove that this problem is strongly NP-hard. We then provide a polynomial time algorithm with a worst-case precision guarantee of ${1/2}\log_2 m_T+1...

Bilevel programming problems are hierarchical optimization problems where an objective function is to be minimized over the graph of the solution set mapping of a second parametric optimization problem. It is the aim of the paper to give a survey for this living research area indicating main recent approaches to solve such problems and to de- scrib...

We consider the approximation of nonlinear bilevel mathematical programs by solvable programs of the same type, i.e., bilevel programs involving linear approximations of the upper-level objective and all constraint-defining functions, as well as a quadratic approximation of the lower-level objective. We describe the main features of the algorithm a...

Convexity has recently received a lot of attention in the machine learning community, and the lack of convexity has been seen as a major disadvantage of many learning algorithms, such as multi-layer artificial neural networks. We show that training multi-layer neural networks in which the number of hidden units is learned can be viewed as a convex...

We consider the problem of jointly determining installed capacity levels and associated tariffs on the arcs of a multicommodity transportation network. We model this situation as a joint pricing and network capacity setting problem. Capacities are available at discrete, non uniform levels. This problem is first formulated as a mixed integer bilevel...

In this paper, we propose a model of dynamic traffic assignment where strategic choices are an integral part of user behaviour. The model is based on a discrete-time description of flow variation through a road network involving arcs with rigid capacities. In such network, a driver's strategy consists in a rule that assigns to each node of the netw...

he multiclass network equilibrium problem is expressed in general as a nonmonotone, asymmetric, varia- tional inequality problem. We show that in spite of the nonmonotonicity of the cost operator, the problem may actually satisfy a weaker property, induced by the hierarchical nature of the travel cost interactions. This property allows a natural de...

In a transit network involving vehicles with rigid capacities, we advocate the use of strategies for describing consumer behavior. At each boarding node, a user sorts the transit lines in decreasing order of pref- erence, and boards the first vehicle in this list whose residual capacity is nonzero. Since a user's position in the queue varies from d...

We show that the travelling salesman problem is polynomially reducible to a bilevel toll optimization program. Based on natural bilevel programming techniques, we recover the lifted Miller–Tucker–Zemlin constraints. Next, we derive an O(n2) multi-commodity extension whose LP relaxation is comparable to the exponential formulation of Dantzig, Fulker...

This work pleads for the use of the concept of strategies, and their network-theoretic representation as hyperpaths, for modeling network assignment problems. While this concept describes adequately the behavior of users in transit systems, we show that it can apply as well to networks where arc capacities are rigid. This opens up a whole new field...

The airline revenue management problem can be decomposed into four distinct but related sub-problems that are usually treated separately: demand forecasting, overbooking, capacity allocation and pricing. In recent decades, much interest has been devoted to overbooking and capacity allocation issues and, today, most major airlines rely on computeris...

In this paper, we survey applications and algorithms pertaining to an important class of price setting problems formulated as bilevel programs.

We present an iterative algorithm for solving variational inequalities under the weakest monotonicity condition proposed so far. The method relies on a new cutting plane and on analytic centers.

Generalized bilevel programs were proposed to be solved by a trust region approach. This approach mixed continuous and discrete optimization, trust regions and linesearches were in stark contrast with more traditional descent methods whose convergence properties were weak. The core of the algorithm consisted in solving at each iteration the affine...

We consider the problem of determining a set of optimal tolls on the arcs of a multicommodity transportation network. The problem is formulated as a bilevel mathematical program where the upper level consists in a firm that raises revenues from tolls set on arcs of the network, while the lower level is represented by a group of users travelling on...

We consider a bilevel programming formulation of a freight tariff setting problem where the leader consists in one among a group of competing carriers and the follower is a shipper. At the upper level, the leader's revenue corresponds to the total tariffs levied, while the shipper minimizes its transportation cost, given the tariff schedule set by...

Let VIP(F; C) denote the variational inequality problem associated with the mapping F and the closed convex set C. In this paper we introduce weak conditions on the mapping F that allow the development of a convergent cutting-plane framework for solving VIP(F; C). In the process we introduce, in a natural way, new and useful notions of generalized...

In this work we give sufficient conditions for the finite convergence of descent algorithms for solving variational inequalities involving generalized monotone mappings.

The optimal setting of taxes or subsidies on goods and services can be naturally modelled as a bilinear bilevel program. We analyze this class of hierarchical problems both from the theoretical and algorithmical points of view, devoting special attention to the problem of setting profit-maximizing tolls on a transportation network.

The continuous dynamic network loading problem (CDNLP) consists in determining, on a congested network, time-dependent arc volumes, together with arc and path travel times, given the time-varying path flow departue rates over a finite time horizon. This problem constitutes an intrinsic part of the dynamic traffic assignment problem. In this paper,...

This paper is concerned with the existence of solutions to a dynamic network equilibrium problem modelled as an infinite dimensional variational inequality. Our results are based on properties of operators that map path flow departure rates to consistent time-dependent path flows and other link performance functions. The existence result requires t...

The descent auxiliary problem method allows one to find the solution of minimization problems by solving a sequence of auxiliary problems which incorporate a linesearch strategy. We derive the basic algorithm and study its convergence properties within the framework of infinite dimensional pseudoconvex minimization. We also introduce a partial desc...

Recently, Fukushima proposed a differentiable optimization framework for solving strictly monotone and continuously differentiable variational inequalities. The main result of this paper is to show that Fukushima's results can be extended to monotone (not necessarily strictly monotone) and Lipschitz continuous (not necessarily continuously differen...

We present a framework for descent algorithms that solve the monotone variational inequality problem VIPv which consists in finding a solutionv *∈Ω v satisfyings(v *)T(v−v *)⩾0, for allv∈Ω v. This unified framework includes, as special cases, some well known iterative methods and equivalent optimization formulations. A descent method is developed f...

. It is well-known (see Pang and Chan [7]) that Newton's method, applied to strongly monotone variational inequalities, is locally and quadratically convergent. In this paper we show that Newton's method yields a descent direction for a nonconvex, nondifferentiable merit function, even in the abscence of strong monotonicity. This result is then use...

Let X and Y be two compact spaces endowed with respective measures μ and ν satisfying the condition µ(X) = v(Y). Let c be a continuous function on the product space X x Y. The mass transfer problem consists in determining a measure ξ on X x Y whose marginals coincide with μ and ν, and such that the total cost ∫ ∫ c(x,y)dξ(x,y) be minimized. We firs...

We consider a bilevel model where the leader wants to maximize revenues from a taxation scheme, while the follower rationally reacts to those tax levels. We focus our attention on the special case of a toll-setting problem defined on a multicommodity transportation network. We show that the general problem is NP-complete, while particular instances...

We consider an analytic center algorithm for solving generalized monotone variational inequalities in R n , which adapts a recent result due to Goffin, Luo and Ye [5] to the numerical resolution of continuous pseudomonotone variational inequalities. Keywords: Variational inequalities -- Interior point methods -- Cutting planes -- Generalized monoto...

. Let ¯ and be two probability measures on the real line and c be a lower semicontinuous function on the plane. The mass transfer problem consists in determining a measure ¸ with respective marginals ¯ and that minimizes the functional R c d¸. In this paper we show that, whenever the measure c is strictly superadditive, the solution corresponding t...

Consider a network facility location problem where congestion arises at facilities, and is represented by delay functions that approximate the queueing process. We strive to minimize the sum of customers' transportation and waiting times, and facilities' fixed and variable costs. The problem is solved using a column generation technique within a Br...

Bilinear bilevel programs occur naturally in the context of the optimal setting of taxes or subsidies on goods and services. We analyze this class of hierarchical problems both from the theoretical and algorithmical points of view. We devote special attention to the problem of setting profit-maximizing tolls on a transportation network.

This work is concerned with the analysis of convergence properties of feasible descent methods for solving monotone variational inequalities in Banach spaces.

Traditional models of traffic assignment usually assume that flow is distributed along paths of the underlying network. In this presentation, we argue that several models can be enhanced by considering that the physical path travelled by a given user depends on a sequence of events (random or not) that occur at the nodes of the network, and whose o...

Each chapter in Equilibrium and Advanced Transportation Modelling develops a topic from basic concepts to the state-of-the-art, and beyond. All chapters relate to aspects of network equilibrium. Chapter One advocates the use of simulation models for the representation of traffic flow movements at the microscopic level. Chapter Two presents travel d...

This work is concerned with the analysis of convergence properties of feasible descent methods for solving monotone variational inequalities in Banach spaces. Keywords: Banach spaces, variational inequalities, Frank-Wolfe algorithm, global convergence, projection. 1 Introduction Let X be a reflexive Banach space, X the topological dual space of con...

This paper is concerned with a class of dynamic and stochastic problems known as real-time decision problems. The objective is to provide responses of a required quality in a continuously evolving environment, within a prescribed time frame, using limited resources and information that is often incomplete or uncertain. Furthermore, the outcome of a...

We consider an analytic center algorithm for solving generalized monotone variational inequalities in , which adapts a recent result due to Goffin et al. (1993) to the numerical resolution of continuous pseudomonotone variational inequalities.

This paper stresses the role of co-coercivity and related notions in the convergence of iterative schemes for solving monotone, but not necessarily strongly or even strictly monotone variational inequalities. The analysis will be conducted within the conceptual framework of the "auxiliary problem principle". Keywords. Mathematical programming. Vari...

. In this work we consider a bicriterion extension of equilibrium problems formulated as variational inequalities, and propose for its solution a generalization of the Frank-Wolfe method. Under suitable monotonicity assumptions on the cost function and a reasonable regularity assumption, we prove that the Frank-Wolfe iterates converge linearly to t...

We consider a hierarchical system where a leader incorporates into its strategy the reaction of the follower to its decision. The follower's reaction is quite generally represented as the solution set to a monotone variational inequality. For the solution of this nonconvex mathematical program a penalty approach is proposed, based on the formulatio...

The linear Bilevel Programming Problem (BLP) is an instance of a linear hierarchical decision process where the lower level constraint set is dependent on decisions taken at the upper level. In this paper we propose to solve this NP-hard problem using an adaptive search method related to the Tabu Search metaheuristic. Numerical results on large sca...

We consider a bicriterion traffic assignment model where the network users have different values of time. We present an infinite-dimensional formulation of this model that lends itself to an extremely simple and efficient algorithm that takes into account the network structure of the problem.