constraints

We present a practical algorithm for computing the minimum volume n-dimensional ellipsoid that must contain m given points a 1 , . . . , am . This convex constrained problem arises in a variety of applied computational settings, particularly in data mining and robust statistics. Its structure makes it particularly amenable to solution by interior-point methods, and it has been the subject of much theoretical complexity analysis. Here we focus on computation. We present a combined interior-point and active-set method for solving this problem. Our computational results demonstrate that our method solves very large problem instances (m = 30, 000 and n = 30) to a high degree of accuracy in under 30 seconds on a personal computer.

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

A wide variety of nonlinear convex optimization problems can be cast as problems involving linear matrix inequalities (LMIs), and hence e ciently solved using recently developed interior-point methods. In this paper, we will consider two classes of optimization problems with LMI constraints: The semide nite programming problem, i.e., the problem of minimizing a linear function subject to a linear matrix inequality. Semide nite 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. The problem of maximizing the determinant of a positive de nite matrix subject to linear matrix inequalities. This problem has applications in computational geometry, experiment design, information and communication theory, and other elds. We review some of these applications, including some interesting applications that are less well known and arise in statistics, optimal experiment design and VLSI. 1 Optimization problems involving LMI constraints We consider convex optimization problems with linear matrix inequality (LMI) constraints, i.e., constraints of the form F (x) =F0+x1F1+ +xmFm 0; (1) where the matrices Fi = F T i 2 R n n are given, and the inequality F (x) 0 means F (x) is positive semide nite. The LMI (1) is a convex constraint in the variable x 2 R m. Conversely, a wide variety of nonlinear convex constraints can be expressed as LMIs (see the recent