Distances to lattice points in rational polyhedra

Let a ∈ Z n >0 , n ≥ 2 , gcd(a) := gcd(a1 , . . . , an ) = 1, b ∈ Z≥0 and denote by k · k∞ the ℓ∞-norm. Consider the knapsack polytope P(a, b) = { x ∈ R n ≥0 : a T x = b and assume that P(a, b) ∩ Z n 6= ; holds. The main result of this thesis states that for any vertex x ∗ of the knapsack polytope P(a, b) there exists a feasible integer point z ∗ ∈ P(a, b) such that, denoting by s the size of the support of z ∗ , i.e. the number of nonzero components in z ∗ and upon assuming s > 0 , the inequality kx ∗ − z ∗ k∞ 2 s−1 s < kak∞ holds. This inequality may be viewed as a transference result which allows strengthening the best known distance (proximity) bounds if integer points are not sparse and, vice versa, strengthening the best known sparsity bounds if feasible integer points are sufficiently far from a vertex of the knapsack polytope. In particular, this bound provides an exponential in s improvement on the previously best known distance bounds in the knapsack scenario. Further, when considering general integer linear programs, we show that a resembling inequality holds for vertices of Gomory’s corner polyhedra [49, 96]. In addition, we provide several refinements of the known distance and support bounds under additional assumption

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

Energy Minimization for Multiple Object Tracking

Multiple target tracking aims at reconstructing trajectories of several moving targets in a dynamic scene, and is of significant relevance for a large number of applications. For example, predicting a pedestrian’s action may be employed to warn an inattentive driver and reduce road accidents; understanding a dynamic environment will facilitate autonomous robot navigation; and analyzing crowded scenes can prevent fatalities in mass panics. The task of multiple target tracking is challenging for various reasons: First of all, visual data is often ambiguous. For example, the objects to be tracked can remain undetected due to low contrast and occlusion. At the same time, background clutter can cause spurious measurements that distract the tracking algorithm. A second challenge arises when multiple measurements appear close to one another. Resolving correspondence ambiguities leads to a combinatorial problem that quickly becomes more complex with every time step. Moreover, a realistic model of multi-target tracking should take physical constraints into account. This is not only important at the level of individual targets but also regarding interactions between them, which adds to the complexity of the problem. In this work the challenges described above are addressed by means of energy minimization. Given a set of object detections, an energy function describing the problem at hand is minimized with the goal of finding a plausible solution for a batch of consecutive frames. Such offline tracking-by-detection approaches have substantially advanced the performance of multi-target tracking. Building on these ideas, this dissertation introduces three novel techniques for multi-target tracking that extend the state of the art as follows: The first approach formulates the energy in discrete space, building on the work of Berclaz et al. (2009). All possible target locations are reduced to a regular lattice and tracking is posed as an integer linear program (ILP), enabling (near) global optimality. Unlike prior work, however, the proposed formulation includes a dynamic model and additional constraints that enable performing non-maxima suppression (NMS) at the level of trajectories. These contributions improve the performance both qualitatively and quantitatively with respect to annotated ground truth. The second technical contribution is a continuous energy function for multiple target tracking that overcomes the limitations imposed by spatial discretization. The continuous formulation is able to capture important aspects of the problem, such as target localization or motion estimation, more accurately. More precisely, the data term as well as all phenomena including mutual exclusion and occlusion, appearance, dynamics and target persistence are modeled by continuous differentiable functions. The resulting non-convex optimization problem is minimized locally by standard conjugate gradient descent in combination with custom discontinuous jumps. The more accurate representation of the problem leads to a powerful and robust multi-target tracking approach, which shows encouraging results on particularly challenging video sequences. Both previous methods concentrate on reconstructing trajectories, while disregarding the target-to-measurement assignment problem. To unify both data association and trajectory estimation into a single optimization framework, a discrete-continuous energy is presented in Part III of this dissertation. Leveraging recent advances in discrete optimization (Delong et al., 2012), it is possible to formulate multi-target tracking as a model-fitting approach, where discrete assignments and continuous trajectory representations are combined into a single objective function. To enable efficient optimization, the energy is minimized locally by alternating between the discrete and the continuous set of variables. The final contribution of this dissertation is an extensive discussion on performance evaluation and comparison of tracking algorithms, which points out important practical issues that ought not be ignored

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

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

Geometric aspects of linear programming : shadow paths, central paths, and a cutting plane method

Most everyday algorithms are well-understood; predictions made theoretically about them closely match what we observe in practice. This is not the case for all algorithms, and some algorithms are still poorly understood on a theoretical level. This is the case for many algorithms used for solving optimization problems from operations reserach. Solving such optimization problems is essential in many industries and is done every day. One important example of such optimization problems are Linear Programming problems. There are a couple of different algorithms that are popular in practice, among which is one which has been in use for almost 80 years. Nonetheless, our theoretical understanding of these algorithms is limited. This thesis makes progress towards a better understanding of these key algorithms for lineair programming, among which are the simplex method, interior point methods, and cutting plane methods