PATTERN SEPARATION VIA ELLIPSOIDS AND CONIC PROGRAMMING Machine learning is a scientific discipline whose purpose is to design computer procedures that are able to perform classification tasks. For example, given a certain number of medical characteristics about a patient (e.g. age, weight, blood pressure, etc.), we would like to infer automatically whether he or she is healthy or not. A special case of machine learning problem is the separation problem, which asks to find a way to classify patterns that are known to belong to different well-defined classes. This is equivalent to finding a procedure that is able to recognize to which class each pattern belongs. The obvious utility of such a procedure is its use on unknown patterns, in order to determine to which one of the classes they are most likely to belong From a completely different point of view, mathematical programming is a branch of optimization which seeks to minimize or maximize a function of several variables under a set of constraints. Linear programming is a special case for which a very efficient algorithm is known since a long time, the simplex method. Another completely different kind of efficient method has been developed much later to solve the linear programming problem : the so-called interior-point methods. A certain type of interior-point methods has been very recently generalized to a broader class of problems called the SQL conic programs. The objective of this work is to solve the pattern separation problem using SQL conic programming. In addition to that statement, the main idea of this thesis is to use ellipsoids to perform the pattern separation. The first chapter is about mathematical programming. We will start by describing how and why researchers were led to study special types of mathematical programs, namely convex programs and conic programs. We will also provide a detailed discussion about conic duality and give a classification of conic programs. We will then describe what are self-scaled cones and why they are so useful in conic programming. Finally, we will give an overview of what can be modelled using a SQL conic program, keeping in mind our pattern separation problem. Since most of the material in the chapter is standard, many of the proofs are omitted. The second chapter will concentrate on pattern separation. After a short description of the problem, we will successively describe four different separation methods using SQL conic programming. For each method, various properties are investigated. Each algorithm has in fact been successively designed with the objective of eliminating the drawbacks of the previous one, while keeping its good properties. We conclude this chapter with a small section describing the state of the art in pattern separation with ellipsoids. The third chapter reports some computational experiments with our four methods, and provides a comparison with other separation procedures. Finally, we conclude this work by providing a short summary, highlighting the author's personal contribution and giving some interesting perspectives for further research.