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constraints

this document. 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, CONTENTS 3 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. Chapter 1 Conic programming 1.1 Introduction

A wide variety of nonlinear convex optimization problems can be cast as problems involving linear matrix inequalities (LMIs), and hence efficiently solved using recently developed interior-point methods. In this paper, we will consider two classes of optimization problems with LMI constraints: ffl The semidefinite programming problem, i.e., the problem of minimizing a linear function subject to a linear matrix inequality. Semidefinite programming is an important numerical tool for analysis and synthesis in systems and control theory. It has also been recognized in combinatorial optimization as a valuable technique for obtaining bounds on the solution of NP-hard problems.

We consider convex optimization problems with linear matrix inequality (LMI) constraints, i.e., constraints of the form F (x) =F0+x1F1+ +xmFm 0; (1.1) where the matrices Fi = F T

The paper considers a general approach for classifying objects using mathematical programming algorithms. The approach is based on optimizing a utility function, which is quadratic in indicator parameters and is linear in control parameters (which need to be identified). Qualitative characteristics of the utility function, such as monotonicity in some variables, are included using additional constraints. The methodology was tested with a "credit cards scoring" problem. Credit scoring is a way of separating specific subgroups in a population of objects (such as applications for credit), which have significantly different credit risk characteristics. A new feature of our approach is incorporating expert judgments in the model. For instance, the following preference was included with an additional constraint: “give more preference to customers with higher incomes. ” Numerical experiments showed that including constraints based on expert judgments improves the performance of the algorithm. Introduction. In this paper, we consider a general approach for classifying objects and explain it with credit cards scoring problem. Classification can be defined by a classification function assigning to each object some categorical value called the class number. However, this classification