Congestion, equilibrium and learning: The minority game

The minority game is a simple congestion game in which the players' main goal is to choose among two options the one that is adopted by the smallest number of players. We characterize the set of Nash equilibria and the limiting behavior of several well-known learning processes in the minority game with an arbitrary odd number of players. Interestingly, different learning processes provide considerably different predictions

Towards a data-driven treatment of epilepsy: computational methods to overcome low-data regimes in clinical settings

Epilepsy is the most common neurological disorder, affecting around 1 % of the population. One third of patients with epilepsy are drug-resistant. If the epileptogenic zone can be localized precisely, curative resective surgery may be performed. However, only 40 to 70 % of patients remain seizure-free after surgery. Presurgical evaluation, which in part aims to localize the epileptogenic zone (EZ), is a complex multimodal process that requires subjective clinical decisions, often relying on a multidisciplinary team’s experience. Thus, the clinical pathway could benefit from data-driven methods for clinical decision support. In the last decade, deep learning has seen great advancements due to the improvement of graphics processing units (GPUs), the development of new algorithms and the large amounts of generated data that become available for training. However, using deep learning in clinical settings is challenging as large datasets are rare due to privacy concerns and expensive annotation processes. Methods to overcome the lack of data are especially important in the context of presurgical evaluation of epilepsy, as only a small proportion of patients with epilepsy end up undergoing surgery, which limits the availability of data to learn from. This thesis introduces computational methods that pave the way towards integrating data-driven methods into the clinical pathway for the treatment of epilepsy, overcoming the challenge presented by the relatively small datasets available. We used transfer learning from general-domain human action recognition to characterize epileptic seizures from video–telemetry data. We developed a software framework to predict the location of the epileptogenic zone given seizure semiologies, based on retrospective information from the literature. We trained deep learning models using self-supervised and semi-supervised learning to perform quantitative analysis of resective surgery by segmenting resection cavities on brain magnetic resonance images (MRIs). Throughout our work, we shared datasets and software tools that will accelerate research in medical image computing, particularly in the field of epilepsy

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