Bilevel linear programs: generalized models for the lower-level reaction set and related problems

Bilevel programming forms a class of optimization problems that model hierarchical relation between two independent decision-makers, namely, the leader and the follower, in a collaborative or conflicting setting. Decisions in this hierarchical structure are made sequentially where the leader decides first and then the follower responds by solving an optimization problem, which is parameterized by the leader's decisions. The follower's reaction, in return, affects the leader's decision, usually through shaping the leader's objective function. Thus, the leader should take into account the follower's response in the decision-making process. A key assumption in bilevel optimization is that both participants, the leader and the follower, solve their problems optimally. However, this assumption does not hold in many important application areas because: (i) there is no known efficient method to solve the lower-level formulation to optimality; (ii) the follower either is not sufficiently sophisticated or does not have the required computational resources to find an optimal solution to the lower-level problem in a timely manner; or (iii) the follower might be willing to give up a portion of his/her optimal objective function value in order to inflict more damage to the leader. This dissertation mainly focuses on developing approaches to model such situations in which the follower does not necessarily return an optimal solution of the lower-level problem as a response to the leader's action. That is, we assume that the follower's reaction set may include both exact and inexact solutions of the lower-level problem. Therefore, we study a generalized class of the follower's reaction sets. This is arguably the case in many application areas in practice, thus our approach contributes to closing the gap between the theory and practice in the bilevel optimization area. In addition, we develop a method to solve bilevel problems through single-level reformulations under the assumption that the lower-level problem is a linear program. The most common technique for such transformations is to replace the lower-level linear optimization problem by its KKT optimality conditions. We propose an alternative technique for a broad class of bilevel linear integer problems, based on the strong duality property of linear programs and compare its performance against the current methods. Finally, we explore bilevel models in an application setting of the pediatric vaccine pricing problem

A Framework for Generalized Benders' Decomposition and Its Application to Multilevel Optimization

We describe a framework for reformulating and solving optimization problems that generalizes the well-known framework originally introduced by Benders. We discuss details of the application of the procedures to several classes of optimization problems that fall under the umbrella of multilevel/multistage mixed integer linear optimization problems. The application of this abstract framework to this broad class of problems provides new insights and a broader interpretation of the core ideas, especially as they relate to duality and the value functions of optimization problems that arise in this context

Traffic prediction and bilevel network design

Cette thèse porte sur la modélisation du trafic dans les réseaux routiers et comment celle-ci est intégrée dans des modèles d'optimisation. Ces deux sujets ont évolué de manière plutôt disjointe: le trafic est prédit par des modèles mathématiques de plus en plus complexes, mais ce progrès n'a pas été incorporé dans les modèles de design de réseau dans lesquels les usagers de la route jouent un rôle crucial. Le but de cet ouvrage est d'intégrer des modèles d'utilités aléatoires calibrés avec de vraies données dans certains modèles biniveaux d'optimisation et ce, par une décomposition de Benders efficace. Cette décomposition particulière s'avère être généralisable par rapport à une grande classe de problèmes communs dans la litérature et permet d'en résoudre des exemples de grande taille. Le premier article présente une méthodologie générale pour utiliser des données GPS d'une flotte de véhicules afin d'estimer les paramètres d'un modèle de demande dit recursive logit. Les traces GPS sont d'abord associées aux liens d'un réseau à l'aide d'un algorithme tenant compte de plusieurs facteurs. Les chemins formés par ces suites de liens et leurs caractéristiques sont utilisés afin d'estimer les paramètres d'un modèle de choix. Ces paramètres représentent la perception qu'ont les usagers de chacune de ces caractéristiques par rapport au choix de leur chemin. Les données utilisées dans cet article proviennent des véhicules appartenant à plusieurs compagnies de transport opérant principalement dans la région de Montréal. Le deuxième article aborde l'intégration d'un modèle de choix de chemin avec utilités aléatoires dans une nouvelle formulation biniveau pour le problème de capture de flot de trafic. Le modèle proposé permet de représenter différents comportements des usagers par rapport à leur choix de chemin en définissant les utilités d'arcs appropriées. Ces utilités sont stochastiques ce qui contribue d'autant plus à capturer un comportement réaliste des usagers. Le modèle biniveau est rendu linéaire à travers l'ajout d'un terme lagrangien basé sur la dualité forte et ceci mène à une décomposition de Benders particulièrement efficace. Les expériences numériques sont principalement menés sur un réseau représentant la ville de Winnipeg ce qui démontre la possibilité de résoudre des problèmes de taille relativement grande. Le troisième article démontre que l'approche du second article peut s'appliquer à une forme particulière de modèles biniveaux qui comprennent plusieurs problèmes différents. La décomposition est d'abord présentée dans un cadre général, puis dans un contexte où le second niveau du modèle biniveau est un problème de plus courts chemins. Afin d'établir que ce contexte inclut plusieurs applications, deux applications distinctes sont adaptées à la forme requise: le transport de matières dangeureuses et la capture de flot de trafic déterministe. Une troisième application, la conception et l'établissement de prix de réseau simultanés, est aussi présentée de manière similaire à l'Annexe B de cette thèse.The subject of this thesis is the modeling of traffic in road networks and its integration in optimization models. In the literature, these two topics have to a large extent evolved independently: traffic is predicted more accurately by increasingly complex mathematical models, but this progress has not been incorporated in network design models where road users play a crucial role. The goal of this work is to integrate random utility models calibrated with real data into bilevel optimization models through an efficient Benders decomposition. This particular decomposition generalizes to a wide class of problems commonly found in the literature and can be used to solved large-scale instances. The first article presents a general methodology to use GPS data gathered from a fleet of vehicles to estimate the parameters of a recursive logit demand model. The GPS traces are first matched to the arcs of a network through an algorithm taking into account various factors. The paths resulting from these sequences of arcs, along with their characteristics, are used to estimate parameters of a choice model. The parameters represent users' perception of each of these characteristics in regards to their path choice behaviour. The data used in this article comes from trucks used by a number of transportation companies operating mainly in the Montreal region. The second article addresses the integration of a random utility maximization model in a new bilevel formulation for the general flow capture problem. The proposed model allows for a representation of different user behaviors in regards to their path choice by defining appropriate arc utilities. These arc utilities are stochastic which further contributes in capturing real user behavior. This bilevel model is linearized through the inclusion of a Lagrangian term based on strong duality which paves the way for a particularly efficient Benders decomposition. The numerical experiments are mostly conducted on a network representing the city of Winnipeg which demonstrates the ability to solve problems of a relatively large size. The third article illustrates how the approach used in the second article can be generalized to a particular form of bilevel models which encompasses many different problems. The decomposition is first presented in a general setting and subsequently in a context where the lower level of the bilevel model is a shortest path problem. In order to demonstrate that this form is general, two distinct applications are adapted to fit the required form: hazmat transportation network design and general flow capture. A third application, joint network design and pricing, is also similarly explored in Appendix B of this thesis

Learning Active Constraints to Efficiently Solve Linear Bilevel Problems

Bilevel programming can be used to formulate many engineering and economics problems. However, common reformulations of bilevel problems to mixed-integer linear programs (through the use of Karush-Kuhn-Tucker conditions) make solving such problems hard, which impedes their implementation in real-life. In this paper, we significantly improve solution speed and tractability by introducing decision trees to learn the active constraints of the lower-level problem, while avoiding to introduce binaries and big-M constants. The application of machine learning reduces the online solving time, and becomes particularly beneficial when the same problem has to be solved multiple times. We apply our approach to power systems problems, and especially to the strategic bidding of generators in electricity markets, where generators solve the same problem many times for varying load demand or renewable production. Three methods are developed and applied to the problem of a strategic generator, with a DCOPF in the lower-level. We show that for networks of varying sizes, the computational burden is significantly reduced, while we also manage to find solutions for strategic bidding problems that were previously intractable.Comment: 11 pages, 5 figure

Bilevel Optimization Approaches to Decide the Feasibility of Bookings in the European Gas Market

The European gas market is organized as a so-called entry-exit system with the main goal to decouple transport and trading. To this end, gas traders and the transmission system operator (TSO) sign so-called booking contracts that grant capacity rights to traders to inject or withdraw gas at certain nodes up to this capacity. On a day-ahead basis, traders then nominate the actual amount of gas within the previously booked capacities. By signing a booking contract, the TSO guarantees that all nominations within the booking bounds can be transported through the network. This results in a highly challenging mathematical problem. Using potential-based flows to model stationary gas physics, feasible bookings on passive networks, i.e., networks without controllable elements, have been characterized in the recent literature. In this paper, we consider networks with linearly modeled active elements such as compressors and control valves that do not lie on cycles of the network. Since these active elements allow the TSO to control the gas flow, the single-level approaches from the literature are no longer applicable. We thus present a bilevel approach to decide the feasibility of bookings in networks with active elements. Besides the classical Karush-Kuhn-Tucker reformulation, we obtain three problem-specific optimal-value-function reformulations, which also lead to novel characterizations of feasible bookings in active networks. We compare the performance of our methods by a case study based on data from the GasLib

Solving Bilevel Knapsack Problem using Graph Neural Networks

The Bilevel Optimization Problem is a hierarchical optimization problem with two agents, a leader and a follower. The leader make their own decisions first, and the followers make the best choices accordingly. The leader knows the information of the followers, and the goal of the problem is to find the optimal solution by considering the reactions of the followers from the leader's point of view. For the Bilevel Optimization Problem, there are no general and efficient algorithms or commercial solvers to get an optimal solution, and it is very difficult to get a good solution even for a simple problem. In this paper, we propose a deep learning approach using Graph Neural Networks to solve the bilevel knapsack problem. We train the model to predict the leader's solution and use it to transform the hierarchical optimization problem into a single-level optimization problem to get the solution. Our model found the feasible solution that was about 500 times faster than the exact algorithm with $1.7\%$ optimal gap. Also, our model performed well on problems of different size from the size it was trained on.Comment: 27 pages, 2 figure

Quand l'optimisation à deux niveaux rencontre les réseaux de gaz : Faisabilité des réservations sur le marché européen entrée-sortie du gaz

Transport and trade of gas are decoupled after the liberalization of the European gasmarkets, which are now organized as so-called entry-exit systems. At the core of thismarket system are bookings and nominations, two special capacity-right contractsthat grant traders access to the gas network. The latter is operated by a separateentity, known as the transmission system operator (TSO), who is in charge of thetransport of gas from entry to exit nodes. In the mid to long term, traders signa booking contract with the TSO to obtain injection and withdrawal capacities atentry and exit nodes, respectively. On a day-ahead basis, they then nominate withinthese booked capacities a balanced load flow of the planned amounts of gas to beinjected into and withdrawn from the network the next day. The key property isthat by signing a booking contract, the TSO is obliged to guarantee transportabilityfor all balanced load flows in compliance with the booked capacities. To assess thefeasibility of a booking, it is therefore necessary to check the feasibility of infinitelymany nominations. As a result, deciding if a booking is feasible is a challengingmathematical problem, which we investigate in this dissertation.Our results range from passive networks, consisting of pipes only, to activenetworks, containing controllable elements to influence gas flows. Since the study ofthe latter naturally leads to a bilevel framework, we first consider some more generalproperties of bilevel optimization. For the case of linear bilevel optimization, weconsider the hardness of validating the correctness of big-M s often used in solvingthese problems via a single-level reformulation. We also derive a family of validinequalities to be used in a bilevel-tailored branch-and-cut algorithm as a big-M -freealternative.We then turn to the study of feasible bookings. First, we present our results onpassive networks, for which bilevel approaches are not required. A characterization offeasible bookings on passive networks is derived in terms of a finite set of nominations.While computing these nominations is a difficult task in general, we present polynomialcomplexity results for the special cases of tree-shaped or single-cycle passive networks.Finally, we consider networks with linearly modeled active elements. After obtaininga bilevel optimization model that allows us to determine the feasibility of a bookingin this case, we derive various single-level reformulations to solve the problem. Inaddition, we obtain novel characterizations of feasible bookings on active networks,which generalize our characterization in the passive case. The performance of thesevarious approaches is compared in a case study on two networks from the literature,one of which is a simplified version of the Greek gas network.Transport et commerce de gaz sont découplés depuis la libéralisation des marchéseuropéens du gaz, qui sont désormais organisés en systèmes dit d’entrée-sortie. Aucœur de ce système de marché se trouvent les réservations et les nominations, deuxcontrats spéciaux de droit à la capacité qui permettent aux négociants d’accéder auréseau de gaz. Ce dernier est exploité par une entité distincte, appelée gestionnairede réseau de transport (GRT), qui est chargée du transport du gaz entre les nœudsd’entrée et de sortie. À moyen et long terme, les négociants signent un contrat deréservation avec le GRT pour obtenir des capacités d’injection et d’extraction auxnœuds d’entrée et de sortie, respectivement. Au jour le jour, ils désignent ensuite,dans les limites des capacités réservées, un flux de charge équilibrée des quantités degaz prévues à injecter et à extraire le lendemain. La propriété essentielle est qu’ensignant un contrat de réservation, le GRT est obligé de garantir la transportabilitéde tous les flux de charge équilibrée respectant les capacités réservées. Pour évaluerla faisabilité d’une réservation, il est donc nécessaire de vérifier la faisabilité d’uneinfinité de nominations. Par conséquent, décider si une réservation est réalisable estun problème mathématique difficile, que nous étudions dans cette thèse.Nos résultats vont des réseaux passifs, constitués uniquement de pipelines, auxréseaux actifs, contenant des éléments contrôlables pour influencer les flux de gaz.Comme l’étude de ces derniers conduit naturellement à un cadre biniveau, nousconsidérons d’abord certaines propriétés plus générales de l’optimisation biniveau.Pour le cas de l’optimisation biniveau linéaire, nous étudions la difficulté de validerl’exactitude des constantes de type big-M souvent utilisées dans la résolution de cesproblèmes via une reformulation à un seul niveau. Nous déduisons également unefamille d’inégalités valides à utiliser dans un algorithme de branch-and-cut adapté aubiniveau comme alternative à l’approche utilisant des big-M s.Nous nous tournons ensuite vers l’étude des réservations réalisables. D’abord,nous présentons nos résultats sur les réseaux passifs, pour lesquels les approchesbiniveaux ne sont pas nécessaires. Une caractérisation des réservations réalisablessur les réseaux passifs est déduite en termes d’un ensemble fini de nominations. Bienque le calcul de ces nominations soit une tâche difficile en général, nous présentonsdes algorithmes polynomiaux pour les cas particuliers des réseaux passifs en formed’arbre ou contenant un cycle unique. Enfin, nous considérons les réseaux avec deséléments actifs modélisés à l’aide de contraintes linéaires. Après avoir obtenu unmodèle biniveau, permettant de déterminer la faisabilité d’une réservation dans cecas, nous dérivons diverses reformulations à un seul niveau pour résoudre le problème.En outre, nous obtenons de nouvelles caractérisations des réservations réalisablessur les réseaux actifs, qui généralisent notre caractérisation dans le cas passif. Laperformance de ces différentes approches est comparée dans une étude de cas réaliséesur deux réseaux de la littérature, dont l’un est une version simplifiée du réseau degaz grec

Robust optimization methods for chance constrained, simulation-based, and bilevel problems

The objective of robust optimization is to find solutions that are immune to the uncertainty of the parameters in a mathematical optimization problem. It requires that the constraints of a given problem should be satisfied for all realizations of the uncertain parameters in a so-called uncertainty set. The robust version of a mathematical optimization problem is generally referred to as the robust counterpart problem. Robust optimization is popular because of the computational tractability of the robust counterpart for many classes of uncertainty sets, and its applicability in wide range of topics in practice. In this thesis, we propose robust optimization methodologies for different classes of optimization problems. In Chapter 2, we give a practical guide on robust optimization. In Chapter 3, we propose a new way to construct uncertainty sets for robust optimization using the available historical data information. Chapter 4 proposes a robust optimization approach for simulation-based optimization problems. Finally, Chapter 5 proposes approximations of a specific class of robust and stochastic bilevel optimization problems by using modern robust optimization techniques

Data-driven Inverse Optimization with Imperfect Information

In data-driven inverse optimization an observer aims to learn the preferences of an agent who solves a parametric optimization problem depending on an exogenous signal. Thus, the observer seeks the agent's objective function that best explains a historical sequence of signals and corresponding optimal actions. We focus here on situations where the observer has imperfect information, that is, where the agent's true objective function is not contained in the search space of candidate objectives, where the agent suffers from bounded rationality or implementation errors, or where the observed signal-response pairs are corrupted by measurement noise. We formalize this inverse optimization problem as a distributionally robust program minimizing the worst-case risk that the {\em predicted} decision ({\em i.e.}, the decision implied by a particular candidate objective) differs from the agent's {\em actual} response to a random signal. We show that our framework offers rigorous out-of-sample guarantees for different loss functions used to measure prediction errors and that the emerging inverse optimization problems can be exactly reformulated as (or safely approximated by) tractable convex programs when a new suboptimality loss function is used. We show through extensive numerical tests that the proposed distributionally robust approach to inverse optimization attains often better out-of-sample performance than the state-of-the-art approaches