Minimum-volume ellipsoids. Theory and algorithms (Q2825979)

This very nice book is dedicated to various aspects, both theoretical and algorithmic, regarding minimum-volume ellipsoids. As special as this subject might look like at a first glance, it encompasses different aspects and ideas from fields like (matrix) optimization, convex analysis, geometry and linear algebra and has some important applications, and this almost 150-page volume provides a thorough initiation to it. The book is divided into six chapters, each of them ending with a short section on notes and references, containing also a preface and lists of figures, algorithms, Matlab codes, terms and references as well as some background material on matrix analysis, convex analysis, optimization and other related issues. The first, introductory, chapter provides a motivation of what comes next, with emphasis on history, contributors, connections to other research areas and applications. And, as the author puts it, ``in the rest of this book you will learn more than you ever (thought you) wanted to learn about these geometric objects [i.e. ellipsoids] and this convex function [i.e. the logdet function]''. The second chapter presents theoretical investigations on minimum-volume ellipsoids, such as duality and optimality conditions, relaxations of the centered restriction and quality of the approximation by a minimum-volume ellipsoid. The next one is focused on algorithms for the centered minimum-volume enclosing ellipsoid problem, for which investigations on local and global convergence and complexity, and also connections to other fields, like graph theory, are delivered. In the following two chapters a natural extension of the minimum-volume enclosing ellipsoid problem is presented, namely the minimum-area ellipsoidal cylinder problem, for which both theoretical (duality, optimality conditions) and numerical (convergence and complexity results for algorithms) are provided, together with related issues such as collision detection and rank deficiency. Last but not least in the last chapter several related problems and algorithms are discussed, namely conditional minimum-volume ellipsoids, approximations by parallelotopes and maximum-volume ellipsoids inscribed in a polyhedron.