Iterated Chvátal--Gomory cuts and the geometry of numbers

Chvátal--Gomory cutting planes (CG-cuts for short) are a fundamental tool in integer programming. Given any single CG-cut, one can derive an entire family of CG-cuts, by “iterating' its multiplier vector modulo one. This leads naturally to two questions: first, which iterates correspond to the strongest cuts, and, second, can we find such strong cuts efficiently? We answer the first question empirically, by showing that one specific approach for selecting the iterate tends to perform much better than several others. The approach essentially consists of solving a nonlinear optimization problem over a special lattice associated with the CG-cut. We then provide a partial answer to the second question, by presenting a polynomial-time algorithm that yields an iterate that is strong in a certain well-defined sense. The algorithm is based on results from the algorithmic geometry of numbers

Generating general-purpose cutting planes for mixed-integer programs

Franz WesselmannPaderborn, Univ., Diss., 201

On the complexity of branching proofs

We consider the task of proving integer infeasibility of a bounded convex K in Rn using a general branching proof system. In a general branching proof, one constructs a branching tree by adding an integer disjunction ax ≤ b or ax ≥ b + 1, a ∈ Zn, b ∈ Z, at each node, such that the leaves of the tree correspond to empty sets (i.e., K together with the inequalities picked up from the root to leaf is empty). Recently, Beame et al (ITCS 2018), asked whether the bit size of the coefficients in a branching proof, which they named stabbing planes (SP) refutations, for the case of polytopes derived from SAT formulas, can be assumed to be polynomial in n. We resolve this question in the affirmative, by showing that any branching proof can be recompiled so that the normals of the disjunctions have coefficients of size at most (nR)O(n2), where R ∈ N is the radius of an `1 ball containing K, while increasing the number of nodes in the branching tree by at most a factor O(n). Our recompilation techniques works by first replacing each disjunction using an iterated Diophantine approximation, introduced by Frank and Tardos (Combinatorica 1986), and proceeds by “fixing up” the leaves of the tree using judiciously added Chvátal-Gomory (CG) cuts. As our second contribution, we show that Tseitin formulas, an important class of infeasible SAT instances, have quasi-polynomial sized cutting plane (CP) refutations. This disproves a conjecture that Tseitin formulas are (exponentially) hard for CP. Our upper bound follows by recompiling the quasi-polynomial sized SP refutations for Tseitin formulas due to Beame et al, which have a special enumerative form, into a CP proof of the same length using a serialization technique of Cook et al (Discrete Appl. Math. 1987). As our final contribution, we give a simple family of polytopes in [0, 1]n requiring exponential sized branching proofs

Integrality and cutting planes in semidefinite programming approaches for combinatorial optimization

Many real-life decision problems are discrete in nature. To solve such problems as mathematical optimization problems, integrality constraints are commonly incorporated in the model to reflect the choice of finitely many alternatives. At the same time, it is known that semidefinite programming is very suitable for obtaining strong relaxations of combinatorial optimization problems. In this dissertation, we study the interplay between semidefinite programming and integrality, where a special focus is put on the use of cutting-plane methods. Although the notions of integrality and cutting planes are well-studied in linear programming, integer semidefinite programs (ISDPs) are considered only recently. We show that manycombinatorial optimization problems can be modeled as ISDPs. Several theoretical concepts, such as the Chvátal-Gomory closure, total dual integrality and integer Lagrangian duality, are studied for the case of integer semidefinite programming. On the practical side, we introduce an improved branch-and-cut approach for ISDPs and a cutting-plane augmented Lagrangian method for solving semidefinite programs with a large number of cutting planes. Throughout the thesis, we apply our results to a wide range of combinatorial optimization problems, among which the quadratic cycle cover problem, the quadratic traveling salesman problem and the graph partition problem. Our approaches lead to novel, strong and efficient solution strategies for these problems, with the potential to be extended to other problem classes

Reinforcement Learning: New Algorithms and An Application for Integer Programming

Reinforcement learning (RL) is a generic paradigm for the modeling and optimization of sequential decision making. In the recent decade, progress in RL research has brought about breakthroughs in several applications, ranging from playing video games, mastering board games, to controlling simulated robots. To bring the potential benefits of RL to other domains, two elements are critical: (1) Efficient and general-purpose RL algorithms; (2) Formulations of the original applications into RL problems. These two points are the focus of this thesis. We start by developing more efficient RL algorithms. In Chapter 2, we propose Taylor Expansion Policy Optimization, a model-free algorithmic framework that unifies a number of important prior work as special cases. This unifying framework also allows us to develop a natural algorithmic extension to prior work, with empirical performance gains. In Chapter 3, we propose Monte-Carlo Tree Search as Regularized Policy Optimization, a model-based framework that draws close connections between policy optimization and Monte-Carlo tree search. Building on this insight, we propose Policy Optimization Zero (POZero), a novel algorithm which leverages the strengths of regularized policy search to achieve significant performance gains over MuZero. To showcase how RL can be applied to other domains where the original applications could benefit from learning systems, we study the acceleration of integer programming (IP) solvers with RL. Due to the ubiquity of IP solvers in industrial applications, such research holds the promise of significant real life impacts and practical values. In Chapter 4, we focus on a particular formulation of Reinforcement Learning for Integer Programming: Learning to Cut. By combining cutting plane methods with selection rules learned by RL, we observe that the RL-augmented cutting plane solver achieves significant performance gains over traditional heuristics. This serves as a proof-of-concept of how RL can be combined with general IP solvers, and how learning augmented optimization systems might achieve significant acceleration in general

Multi-robot mission optimisation : an online approach for optimised, long range inspection and sampling missions

Mission execution optimisation is an essential aspect for the real world deployment of robotic systems. Execution optimisation can affect the outcome of a mission by allowing longer missions to be executed or by minimising the execution time of a mission. This work proposes methods for optimising inspection and sensing missions undertaken by a team of robots operating under communication and budget constraints. Regarding the inspection missions, it proposes the use of an information sharing architecture that is tolerant of communication errors combined with multirobot task allocation approaches that are inspired by the optimisation literature. Regarding the optimisation of sensing missions under budget constraints novel heuristic approaches are proposed that allow optimisation to be performed online. These methods are then combined to allow the online optimisation of long-range sensing missions performed by a team of robots communicating through a noisy channel and having budget constraints. All the proposed approaches have been evaluated using simulations and real-world robots. The gathered results are discussed in detail and show the benefits and the constraints of the proposed approaches, along with suggestions for further future directions

Semidefinite Programming. methods and algorithms for energy management

La présente thèse a pour objet d explorer les potentialités d une méthode prometteuse de l optimisation conique, la programmation semi-définie positive (SDP), pour les problèmes de management d énergie, à savoir relatifs à la satisfaction des équilibres offre-demande électrique et gazier.Nos travaux se déclinent selon deux axes. Tout d abord nous nous intéressons à l utilisation de la SDP pour produire des relaxations de problèmes combinatoires et quadratiques. Si une relaxation SDP dite standard peut être élaborée très simplement, il est généralement souhaitable de la renforcer par des coupes, pouvant être déterminées par l'étude de la structure du problème ou à l'aide de méthodes plus systématiques. Nous mettons en œuvre ces deux approches sur différentes modélisations du problème de planification des arrêts nucléaires, réputé pour sa difficulté combinatoire. Nous terminons sur ce sujet par une expérimentation de la hiérarchie de Lasserre, donnant lieu à une suite de SDP dont la valeur optimale tend vers la solution du problème initial.Le second axe de la thèse porte sur l'application de la SDP à la prise en compte de l'incertitude. Nous mettons en œuvre une approche originale dénommée optimisation distributionnellement robuste , pouvant être vue comme un compromis entre optimisation stochastique et optimisation robuste et menant à des approximations sous forme de SDP. Nous nous appliquons à estimer l'apport de cette approche sur un problème d'équilibre offre-demande avec incertitude. Puis, nous présentons une relaxation SDP pour les problèmes MISOCP. Cette relaxation se révèle être de très bonne qualité, tout en ne nécessitant qu un temps de calcul raisonnable. La SDP se confirme donc être une méthode d optimisation prometteuse qui offre de nombreuses opportunités d'innovation en management d énergie.The present thesis aims at exploring the potentialities of a powerful optimization technique, namely Semidefinite Programming, for addressing some difficult problems of energy management. We pursue two main objectives. The first one consists of using SDP to provide tight relaxations of combinatorial and quadratic problems. A first relaxation, called standard can be derived in a generic way but it is generally desirable to reinforce them, by means of tailor-made tools or in a systematic fashion. These two approaches are implemented on different models of the Nuclear Outages Scheduling Problem, a famous combinatorial problem. We conclude this topic by experimenting the Lasserre's hierarchy on this problem, leading to a sequence of semidefinite relaxations whose optimal values tends to the optimal value of the initial problem.The second objective deals with the use of SDP for the treatment of uncertainty. We investigate an original approach called distributionnally robust optimization , that can be seen as a compromise between stochastic and robust optimization and admits approximations under the form of a SDP. We compare the benefits of this method w.r.t classical approaches on a demand/supply equilibrium problem. Finally, we propose a scheme for deriving SDP relaxations of MISOCP and we report promising computational results indicating that the semidefinite relaxation improves significantly the continuous relaxation, while requiring a reasonable computational effort.SDP therefore proves to be a promising optimization method that offers great opportunities for innovation in energy management.PARIS11-SCD-Bib. électronique (914719901) / SudocSudocFranceF

Traveling Salesman Problem

This book is a collection of current research in the application of evolutionary algorithms and other optimal algorithms to solving the TSP problem. It brings together researchers with applications in Artificial Immune Systems, Genetic Algorithms, Neural Networks and Differential Evolution Algorithm. Hybrid systems, like Fuzzy Maps, Chaotic Maps and Parallelized TSP are also presented. Most importantly, this book presents both theoretical as well as practical applications of TSP, which will be a vital tool for researchers and graduate entry students in the field of applied Mathematics, Computing Science and Engineering

Algoritmos evolutivos para alguns problemas em telecomunicações

Orientadores: Flavio Keidi Miyazawa, Mauricio Guilherme de Carvalho ResendeTese (doutorado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: Nos últimos anos, as redes de telecomunicação tem experienciado um grande aumento no fluxo de dados. Desde a utilização massiva de vídeo sob demanda até o incontável número de dispositivos móveis trocando texto e vídeo, o tráfego alcançou uma escala capaz de superar a capacidade das redes atuais. Portanto, as companhias de telecomunicação ao redor do mundo tem sido forçadas a aumentar a capacidade de suas redes para servir esta crescente demanda. Como o custo de instalar uma infraestrutura de rede é geralmente muito grande, o projeto de redes usa fortemente ferramentas de otimização para manter os custos tão baixos quanto possível. Nesta tese, nós analisamos vários aspectos do projeto e implementação de redes de telecomunicação. Primeiramente, nós apresentamos um novo problema de projeto de redes usado para servir demandas sem fio de dispositivos móveis e rotear tal tráfego para a rede principal. Tais redes de acesso são baseadas em tecnologias sem fio modernos como Wi-Fi, LTE e HSPA. Este problema consideramos várias restrições reais e é difícil de ser resolvido. Nós estudamos casos reais nas vizinhanças de uma grande cidade nos Estados Unidos. Em seguida, nós apresentamos uma variação do problema de localização de hubs usado para modelar as redes principais (backbones ou laços centrais). Este problema também pode ser utilizado para modelar redes de transporte de cargas e passageiros. Nós também estudamos o problema de clusterização correlacionada com sobreposições usado para modelar o comportamento dos usuários quando utilizam seus equipamentos móveis. Neste problema, nós podemos rotular um objeto usando múltiplos rótulos e analisar a conexão entre eles. Este problema é adequado para análise de mobilidade de equipamentos que pode ser usada para estimar o tráfego em uma dada região. E finalmente, nós analisamos o licenciamento de espectro sobre uma perspectiva governamental. Nestes casos, uma agência do governo deseja vender licenças para companhias de telecomunicação para que operem em uma dada faixa de espectro. Este processo usualmente é conduzido usando leilões combinatoriais. Para todos problemas, nós propomos algoritmos genéticos de chaves aleatórias viciadas e modelos de programação linear inteira mista para resolvê-los (exceto para o problema de clusterização correlacionada com sobreposição, devido sua natureza não-linear). Os algoritmos que propusemos foram capazes de superar algoritmos do estado da arte para todos problemasAbstract: Cutting and packing problems are common problems that occur in many industry and business process. Their optimized resolution leads to great profits in several sectors. A common problem, that occur in textil and paper industries, is to cut a strip of some material to obtain several small items, using the minimum length of material. This problem, known by Two Dimensional Strip Packing Problem (2SP), is a hard combinatorial optimization problem. In this work, we present an exact algorithm to 2SP, restricted to two staged cuts (known by Two Dimensional Level Strip Packing, 2LSP). The algorithm uses the branch-and-price technique, and heuristics based on approximation algorithms to obtain upper bounds. The algorithm obtained optimal or almost optimal for small and moderate sized instancesAbstract: In last twenty years, telecommunication networks have experienced a huge increase in data utilization. From massive on-demand video to uncountable mobile devices exchanging text and video, traffic reached scales that overcame the network capacities. Therefore, telecommunication companies around the world have been forced to increase their capacity to serve this increasing demand. As the cost to deploy network infrastructure is usually very large, the design of a network heavily uses optimization tools to keep costs as low as possible. In this thesis, we analyze several aspects of the design and deployment of communication networks. First, we present a new network design problem used to serve wireless demands from mobile devices and route the traffic to the core network. Such access networks are based on modern wireless access technologies such as Wi-Fi, LTE, and HSPA. This problem has several real world constraints and it is hard to solve. We study real cases of the vicinity of a large city in the United States. Following, we present a variation of the hub-location problem used to model these core networks. Such problem is also suitable to model transportation networks. We also study the overlapping correlation clustering problem used to model the user's behavior when using their mobile devices. In such problem, one can label an object with multiple labels and analyzes the connections between them. Although this problem is very generic, it is suitable to analyze device mobility which can be used to estimate traffic in geographical regions. Finally, we analyze spectrum licensing from a governmental perspective. In these cases, a governmental agency wants to sell rights for telecommunication companies to operate over a given spectrum range. This process usually is conducted using combinatorial auctions. For all problems we propose biased random-key genetic algorithms and mixed integer linear programming models (except in the case of the overlapping correlation clustering problem due its non-linear nature). Our algorithms were able to overcome the state of the art algorithms for all problemsDoutoradoCiência da ComputaçãoDoutor em Ciência da Computaçã