Exact Algorithms for Mixed-Integer Multilevel Programming Problems

We examine multistage optimization problems, in which one or more decision makers solve a sequence of interdependent optimization problems. In each stage the corresponding decision maker determines values for a set of variables, which in turn parameterizes the subsequent problem by modifying its constraints and objective function. The optimization literature has covered multistage optimization problems in the form of bilevel programs, interdiction problems, robust optimization, and two-stage stochastic programming. One of the main differences among these research areas lies in the relationship between the decision makers. We analyze the case in which the decision makers are self-interested agents seeking to optimize their own objective function (bilevel programming), the case in which the decision makers are opponents working against each other, playing a zero-sum game (interdiction), and the case in which the decision makers are cooperative agents working towards a common goal (two-stage stochastic programming). Traditional exact approaches for solving multistage optimization problems often rely on strong duality either for the purpose of achieving single-level reformulations of the original multistage problems, or for the development of cutting-plane approaches similar to Benders\u27 decomposition. As a result, existing solution approaches usually assume that the last-stage problems are linear or convex, and fail to solve problems for which the last-stage is nonconvex (e.g., because of the presence of discrete variables). We contribute exact finite algorithms for bilevel mixed-integer programs, three-stage defender-attacker-defender problems, and two-stage stochastic programs. Moreover, we do not assume linearity or convexity for the last-stage problem and allow the existence of discrete variables. We demonstrate how our proposed algorithms significantly outperform existing state-of-the-art algorithms. Additionally, we solve for the first time a class of interdiction and fortification problems in which the third-stage problem is NP-hard, opening a venue for new research and applications in the field of (network) interdiction

The Rank Pricing Problem: models and branch-and-cut algorithms

International audienceOne of the main concerns in management and economic planning is to sell the right product to the right customer for the right price. Companies in retail and manufacturing employ pricing strategies to maximize their revenues. The Rank Pricing Problem considers a unit-demand model with unlimited supply and uniform budgets in which customers have a rank-buying behavior. Under these assumptions, the problem is first analyzed from the perspective of bilevel pricing models and formulated as a non linear bilevel program with multiple independent followers. We also present a direct non linear single level formulation bearing in mind the aim of the problem. Two different linearizations of the models are carried out and two families of valid inequalities are obtained which, embedded in the formulations by implementing a branch-and-cut algorithm, allow us to tighten the upper bound given by the linear relaxation of the models. We also study the polyhedral structure of the models, taking advantage of the fact that a subset of their constraints constitutes a special case of the Set Packing Problem, and characterize all the clique facets. Besides, we develop a preprocessing procedure to reduce the size of the instances. Finally, we show the efficiency of the formulations, the branch-and-cut algorithms and the preprocessing through extensive computational experiments

Co-evolutionary Hybrid Bi-level Optimization

Multi-level optimization stems from the need to tackle complex problems involving multiple decision makers. Two-level optimization, referred as ``Bi-level optimization'', occurs when two decision makers only control part of the decision variables but impact each other (e.g., objective value, feasibility). Bi-level problems are sequential by nature and can be represented as nested optimization problems in which one problem (the ``upper-level'') is constrained by another one (the ``lower-level''). The nested structure is a real obstacle that can be highly time consuming when the lower-level is $\mathcal{NP}-hard$. Consequently, classical nested optimization should be avoided. Some surrogate-based approaches have been proposed to approximate the lower-level objective value function (or variables) to reduce the number of times the lower-level is globally optimized. Unfortunately, such a methodology is not applicable for large-scale and combinatorial bi-level problems. After a deep study of theoretical properties and a survey of the existing applications being bi-level by nature, problems which can benefit from a bi-level reformulation are investigated. A first contribution of this work has been to propose a novel bi-level clustering approach. Extending the well-know ``uncapacitated k-median problem'', it has been shown that clustering can be easily modeled as a two-level optimization problem using decomposition techniques. The resulting two-level problem is then turned into a bi-level problem offering the possibility to combine distance metrics in a hierarchical manner. The novel bi-level clustering problem has a very interesting property that enable us to tackle it with classical nested approaches. Indeed, its lower-level problem can be solved in polynomial time. In cooperation with the Luxembourg Centre for Systems Biomedicine (LCSB), this new clustering model has been applied on real datasets such as disease maps (e.g. Parkinson, Alzheimer). Using a novel hybrid and parallel genetic algorithm as optimization approach, the results obtained after a campaign of experiments have the ability to produce new knowledge compared to classical clustering techniques combining distance metrics in a classical manner. The previous bi-level clustering model has the advantage that the lower-level can be solved in polynomial time although the global problem is by definition $\mathcal{NP}$-hard. Therefore, next investigations have been undertaken to tackle more general bi-level problems in which the lower-level problem does not present any specific advantageous properties. Since the lower-level problem can be very expensive to solve, the focus has been turned to surrogate-based approaches and hyper-parameter optimization techniques with the aim of approximating the lower-level problem and reduce the number of global lower-level optimizations. Adapting the well-know bayesian optimization algorithm to solve general bi-level problems, the expensive lower-level optimizations have been dramatically reduced while obtaining very accurate solutions. The resulting solutions and the number of spared lower-level optimizations have been compared to the bi-level evolutionary algorithm based on quadratic approximations (BLEAQ) results after a campaign of experiments on official bi-level benchmarks. Although both approaches are very accurate, the bi-level bayesian version required less lower-level objective function calls. Surrogate-based approaches are restricted to small-scale and continuous bi-level problems although many real applications are combinatorial by nature. As for continuous problems, a study has been performed to apply some machine learning strategies. Instead of approximating the lower-level solution value, new approximation algorithms for the discrete/combinatorial case have been designed. Using the principle employed in GP hyper-heuristics, heuristics are trained in order to tackle efficiently the $\mathcal{NP}-hard$ lower-level of bi-level problems. This automatic generation of heuristics permits to break the nested structure into two separated phases: \emph{training lower-level heuristics} and \emph{solving the upper-level problem with the new heuristics}. At this occasion, a second modeling contribution has been introduced through a novel large-scale and mixed-integer bi-level problem dealing with pricing in the cloud, i.e., the Bi-level Cloud Pricing Optimization Problem (BCPOP). After a series of experiments that consisted in training heuristics on various lower-level instances of the BCPOP and using them to tackle the bi-level problem itself, the obtained results are compared to the ``cooperative coevolutionary algorithm for bi-level optimization'' (COBRA). Although training heuristics enables to \emph{break the nested structure}, a two phase optimization is still required. Therefore, the emphasis has been put on training heuristics while optimizing the upper-level problem using competitive co-evolution. Instead of adopting the classical decomposition scheme as done by COBRA which suffers from the strong epistatic links between lower-level and upper-level variables, co-evolving the solution and the mean to get to it can cope with these epistatic link issues. The ``CARBON'' algorithm developed in this thesis is a competitive and hybrid co-evolutionary algorithm designed for this purpose. In order to validate the potential of CARBON, numerical experiments have been designed and results have been compared to state-of-the-art algorithms. These results demonstrate that ``CARBON'' makes possible to address nested optimization efficiently

Synthesis, Interdiction, and Protection of Layered Networks

This research developed the foundation, theory, and framework for a set of analysis techniques to assist decision makers in analyzing questions regarding the synthesis, interdiction, and protection of infrastructure networks. This includes extension of traditional network interdiction to directly model nodal interdiction; new techniques to identify potential targets in social networks based on extensions of shortest path network interdiction; extension of traditional network interdiction to include layered network formulations; and develops models/techniques to design robust layered networks while considering trade-offs with cost. These approaches identify the maximum protection/disruption possible across layered networks with limited resources, find the most robust layered network design possible given the budget limitations while ensuring that the demands are met, include traditional social network analysis, and incorporate new techniques to model the interdiction of nodes and edges throughout the formulations. In addition, the importance and effects of multiple optimal solutions for these (and similar) models is investigated. All the models developed are demonstrated on notional examples and were tested on a range of sample problem sets

Le problème du prix de rang : une approche d'optimisation linéaire en nombres entiers mixtes

This doctorate is entirely devoted to an in-depth study of the Rank Pricing Problem (RPP) and two generalizations. The RPP is a combinatorial optimization problem which aims at setting the prices of a series of products of a company to maximize its revenue. This problem is specified by a set of unit-demand customers, that is, customers interested in a subset of the products offered by the company which intend to buy at most one of them. To do so, they count on a fixed budget, and they rank the products of their interest from the “best” to the “worst”. Once the prices are established by the company, they will purchase their highest-ranked product among the ones they can afford. In the RPP, it is assumed an unlimited supply of products, which is consistent with a company having enough copies of a product to satisfy the demand, or with a setting where the products can be produced quickly at negligible cost (e.g., digital goods). This dissertation consists of four chapters. The first chapter introduces the RPP problem and the mathematical concepts present in the work, whereas each of the next three chapters tackles the resolution of each of the problems of study: the RPP and two generalizations. Thus, Chapter 3 is dedicated to the Rank Pricing Problem with Ties (RPPT), an extension of the RPP where we consider that customers can express indifference among products in their preference list. And the last chapter of the thesis is devoted to a generalization of the problem that we have named the Capacitated Rank Pricing Problem (CRPP) with envy. For this generalization, we have considered reservation prices of customers for the different products that reflect their willingness to pay, instead of a single budget per customer. However, the main difference is that, in the CRPP, the company has a limited supply of products and might not be able to satisfy all the customers’ requests. This is a realistic assumption that we can find in many companies.The aim of this thesis is the proposal of mixed-integer linear formulations for the three problems of study, and their theoretical and/or computational comparison. The methodology used is based on the introduction of decision variables and adequate restrictions to model the problems. Another objective consists in strengthening the formulations by means of valid inequalities that reduce the feasible region of the relaxed problem and allow us to obtain better linear relaxation bounds. Finally, a third goal is to derive resolution algorithms for each of these models and compare them computationally, using commercial solvers.Cette thèse est consacrée à une étude approfondie du Rank Pricing Problem (RPP) et de deux généralisations. Le RPP est un problème d'optimisation combinatoire qui vise à fixer le prix des produits d'une entreprise afin de maximiser son profit. Elle concerne les clients à la demande, c'est-à-dire les clients qui sont intéressés par plusieurs produits de l'entreprise, mais qui n'ont l'intention d'en acheter qu'un. Les clients disposent d'un budget fixe et classent les produits qui les intéressent du "meilleur" au "pire". Lorsque l'entreprise fixe les prix, chaque client achètera son produit préféré parmi ceux qu'il peut se permettre. Dans le RPP, nous supposons que les produits sont offerts en quantité illimitée, ce qui convient si l'on considère que l'entreprise a suffisamment de produits pour satisfaire la demande, ou lorsque les produits peuvent être fabriqués rapidement avec un coût négligeable (par exemple, les biens numériques).Cette thèse se compose de quatre chapitres. Le premier est un chapitre d'introduction au problème et aux concepts mathématiques présents dans la thèse, tandis que les trois chapitres suivants se concentrent sur chacun des problèmes étudiés : le RPP et deux généralisations. Ainsi, le troisième chapitre est consacré à l'étude du Rank Pricing Problem with Ties (RPPT). Dans cette extension du problème, nous supposons que les clients peuvent exprimer leur indifférence entre les produits qui les intéressent au moyen de liens dans leur liste de préférences. Enfin, le dernier chapitre de la thèse comprend l'étude du Capacitated Rank Pricing Problem (CRPP) avec envie. Dans cette extension, nous avons supposé des prix de réserve pour les clients qui reflètent ce qu'ils sont prêts à payer pour chaque produit, plutôt qu'un budget unique par consommateur. Cependant, la principale différence est que dans le cas du CRPP, l'entreprise dispose d'un nombre limité de produits et peut ne pas être en mesure de satisfaire la demande de tous les clients. L'objectif de la thèse est d'obtenir des formulations linéaires en nombres entiers mixtes pour les trois problèmes étudiés, et de les comparer sur le plan théorique et/ou computationnel. À cette fin, la méthodologie utilisée est basée sur la proposition de variables de décision et de contraintes appropriées qui modélisent le problème. L'objectif suivant est l'amélioration de ces formulations au moyen d'inégalités valides qui réduisent l’ensemble admissible de la relaxation du problème et permettent d'obtenir une meilleure borne de la relaxation linéaire. Et enfin, un troisième objectif est la proposition d'algorithmes de résolution pour chacun de ces modèles, et leur comparaison ultérieure au moyen d'études computationnelles et de résolution au moyen de logiciels d'optimisation commerciaux

Deep Reinforcement Learning for Distribution Network Operation and Electricity Market

The conventional distribution network and electricity market operation have become challenging under complicated network operating conditions, due to emerging distributed electricity generations, coupled energy networks, and new market behaviours. These challenges include increasing dynamics and stochastics, and vast problem dimensions such as control points, measurements, and multiple objectives, etc. Previously the optimization models were often formulated as conventional programming problems and then solved mathematically, which could now become highly time-consuming or sometimes infeasible. On the other hand, with the recent advancement of artificial intelligence technologies, deep reinforcement learning (DRL) algorithms have demonstrated their excellent performances in various control and optimization fields. This indicates a potential alternative to address these challenges. In this thesis, DRL-based solutions for distribution network operation and electricity market have been investigated and proposed. Firstly, a DRL-based methodology is proposed for Volt/Var Control (VVC) optimization in a large distribution network, to effectively control bus voltages and reduce network power losses. Further, this thesis proposes a multi-agent (MA)DRL-based methodology under a complex regional coordinated VVC framework, and it can address spatial and temporal uncertainties. The DRL algorithm is also improved to adapt to the applications. Then, an integrated energy and heating systems (IEHS) optimization problem is solved by a MADRL-based methodology, where conventionally this could only be solved by simplifications or iterations. Beyond the applications in distribution network operation, a new electricity market service pricing method based on a DRL algorithm is also proposed. This DRL-based method has demonstrated good performance in this virtual storage rental service pricing problem, whereas this bi-level problem could hardly be solved directly due to a non-convex and non-continuous lower-level problem. These proposed methods have demonstrated advantageous performances under comprehensive case studies, and numerical simulation results have validated the effectiveness and high efficiency under different sophisticated operation conditions, solution robustness against temporal and spatial uncertainties, and optimality under large problem dimensions