Five recent book reviews
This book provides an excellent overview of the theory, methods and applications of continuous optimization problems. The book deals with one-dimensional optimization (chapter 1), unconstrained optimization (chapter 2), optimization under equality constraints (chapter 3), optimization under inequality constraints (chapter 4), linear programming (chapter 6), convex optimization (chapter 7), mixed smooth-convex optimization (chapter 10), dynamic programming in discrete time (chapter 11), and dynamic optimization in continuous time (chapter 12). The following four-step procedure, conveniently adapted to each of the above families of continuous optimization problems, is proposed for computing an optimal solution of those problems that are analytically solvable (i.e. by a formula):
1. Establish the existence of global solutions.
2. Write down the first order necessary conditions.
3. Investigate these conditions.
4. Write down the conclusions.
The main tools in step 1 are the classical Weierstrass theorem (applied to some bounded non-empty sublevel set) and its extension to coercive functions. Step 2 is based upon the so-called Fermat-Lagrange principle (i.e. conditions of multiplier type) for finite-dimensional optimization problems and the Pontryagin principle for dynamic optimization problems. The particular versions of the Fermat-Lagrange principle for the different classes of mathematical programming problems considered in this book (all of them involving differentiable and/or convex functions, i.e. functions admitting linear approximations) are obtained via the tangent space theorem (closely related to the implicit function theorem) for smooth problems and by means of the supporting hyperplane theorem (related to the separation theorem) in the case of convex problems. Typically, step 3 provides a list of candidates and step 1 allows the identification of optimal solutions among them if the given problem is solvable. The second order conditions in chapter 5 give some insight although they do not play a crucial role in this approach because they can seldom be checked in practice. A similar point of view is adopted regarding the constraint qualifications.
Concerning those optimization problems that cannot be solved analytically but which admit a convex reformulation, the authors propose to solve them by means of interior point methods (such as the self-concordant barrier method) or cutting plane methods (such as the ellipsoid method). These and other complementary methods (e.g. line search methods and linear optimization methods) are described in chapters 6 and 7.
Each chapter contains a rich list of selected applications (many of them rare in the standard textbooks) that are presented as either motivating examples or as complementary exercises. Classical optimization problems (most of them relating to geometric objects), physical laws and economic models usually admit analytic solutions whereas real life (or pragmatic) applications may require the use of numerical methods. The solutions are given in appendix H. Chapters 8 and 9 are devoted to economic and mathematical applications, many of them surprising.
This description of the content of the book suggests that it is just one more of the many good textbooks that have been written on continuous optimization. Nevertheless, this book presents the following novel features:
1. The presentation of the material is as friendly, simple and suggestive as possible, including many illustrative examples and historical notes about results, methods and applications. These notes illustrate the important contributions of Russian mathematicians to optimization and related topics that have been ignored up to now in the occidental academic world.
2. The book is totally self-contained, although some familiarity with basic calculus and algebra could help the reader. The first explanation of each topic is intuitive, showing the geometric meaning of concepts (including those of basic calculus), results and numerical methods by means of suitable pictures. For instance, all the main ideas about one-dimensional optimization in the book are sketched in chapter 1 and this section could be read by college students. Subsection 1.5.1, entitled ‘All methods of continuous optimization in a nutshell’, is an excellent introduction to this topic. In order to understand the basic proofs and to be able to obtain analytic solutions, the reader must recall some elements of linear algebra, real analysis and continuity (Weierstrass theorem) that can be found in appendices A, B and C respectively.
3. The basic part of the book is written in a personal way, emphasizing the underlying ideas that are usually hidden behind technical details. For instance, in the authors’ opinion, the secret power of the Lagrange multiplier method consists in reversing the order of the two tasks of elimination and differentiation and not in the use of multipliers (as many experts think). In the same vein, personal anecdotes are also given, e.g. mention is made of some open problems (most of them extending classical optimization problems) that they posed in their lectures and were brilliantly solved by their students.
4. The book provides new simple proofs of important results on optimization theory (such as the Pontryagin principle in appendix G) and also results on other mathematical fields that can be derived from optimization theory (such as the fundamental theorem of algebra in chapter 2). Most formal proofs are confined to the last two appendices, which are written in a fully analytic style.
This book can be used for different courses on continuous optimization, from introductory to advanced level, for any field for which optimization is relevant: mathematics, engineering, economics, physics, etc. The introduction of each chapter describes a “royal road” containing the essential tools for problem solving. Examples of possible courses based on materials contained in this textbook are:
Basic optimization course: chapters 1, 2, 3, 4, and appendix D (Crash course on problem solving).
Intermediate optimization course: chapters 5, 6, 7, 10 and appendices E and F.
Advanced-course on the applications of optimization: chapters 8 and 9.
Advanced-course on dynamic optimization: chapters 11, 12 and Appendix G.
The website of the first author (people.few.eur.nl/brinkhuis) contains references to implementations of optimization algorithms and a list of corrections for the few shortcomings that are unavoidable in a first edition, despite its careful production. In my opinion, this stimulating textbook will be for the teaching of optimization what Spivak's "Calculus" was for the teaching of that subject (and even real analysis) in the 70’s.
The subtitle of the book is “Where Engineering and Mathematics meets”; indeed, one of the authors is an engineer and the other is a mathematician. The book illustrates how physical models can be created from mathematical ones. To be more precise, the book is devoted to various relations between geometry and mechanics. Various topics, presented in 13 chapters, include construction of linkages producing exact and approximate movements along straight lines and other curves, balancing bodies and dissecting and recomposing planar figures. The topics treated in the book are usually first briefly explained from the (correct) geometrical point of view and then applied to physical constructions. The book is very nicely printed and contains many nice figures and photographs of physical models, as well as an extensive bibliography. It can be recommended as a formal or recreational lecture both for mathematicians and engineers.
Even though there are many excellent monographs on Banach spaces, the topic of ordered Banach spaces is not very often included. And even if it is, the authors usually deal only with the more particular case of Banach lattices. The aim of this book is to fill (at least partially) this gap and to provide an up-to-date study of the structure of convex cones contained in Banach spaces and some of the related operators. The book originates from three different sources. The first is conical measures (introduced by G. Choquet and used for developing integral representation theory). The second is the well-established theory of Banach lattices and the third is the Krivine theorem on finite representability of finite dimensional ℓp spaces in Banach lattices. By blending them together, the author obtains an original and instructive contribution to classical functional analysis.
The book starts by presenting various forms of the Hahn-Banach theorem and proving theorems of M. G. Krein, V. L. Klee and T. Ando on cones in Banach spaces. The second chapter is devoted to a presentation of B. Maurey’s theorem on factorization of operators through Lp spaces. An important ingredient here and later on is the Rosenthal lemma. To study convex cones, chapter 3 introduces conical measures and their basic properties. The next part contains the first main results of the book on p-summing operators and their factorization. The author then proceeds with an investigation of representability of finite ℓp spaces in normal cones of Banach spaces. He also shows a relationship between type and cotype of a Banach space with the index introduced in the previous chapter. The author then studies positive operators starting in C(K) spaces and he continues the investigation of p-summing operators on convex cones. The Pietsch inequality and a composition theorem are proved. The last chapter describes the situation of tensor products and positive maps. Five appendices and several open problems complete the book making the presentation rather self-contained.
This book consists of two parts. The first part gives a general introduction to the modern theory of automorphic forms with applications to spectral questions. In particular, it deals with the spectrum of differential forms on congruence hyperbolic manifolds. It contains, for example, the Selberg type theorem on the first eigenvalue of the Laplace operator acting on differential forms, using representation theoretic methods and techniques of proof. The second part of the book has a more differential geometric flavour. The main motivation of this chapter comes from Arthur conjectures, which imply strong restrictions on the spectrum of arithmetic manifolds and conjectural properties of the geometry of hyperbolic manifolds (proved in a weak form in some particular cases).
This is a remarkably comprehensive treatise on modern, as well as classical, measure theory and integration. Volume 1 covers constructions and extensions of measures, the (abstract) Lebesgue integral, Lp-spaces, signed measures, product measures (including infinite products), change of variables and connections between the integral and derivative (covering theorems, the maximal function, functions of bounded variation and absolutely continuous functions). Of course the Radon-Nikodym theorem, convolution and basic facts on the Fourier transform are included. Also, less traditional topics are discussed: uniform integrability, strong convergence of measures and a concise introduction to the Henstock-Kurzweil integral. The core material of volume 1 (500 pages in total) is divided into five chapters and the exposition is presented on about 170 pages.
What makes both volumes exceptional, interesting and extremely valuable are the sections “Supplement and exercises” attached to each chapter. These sections provide important additional material. Let us mention just a few subjects: set-theoretic problems in measure theory, invariant extensions of Lebesgue measure, Whitney’s decomposition, Steiner’s symmetrization, Hausdorff measures, the Brunn-Minkowski inequality, mixed volumes, weak compactness in L1, Hellinger’s integral, additive set functions, density of point sets, differentiation of measures, BMO, the area and coarea formulas, surface measures and the Calderon-Zygmund decomposition. Some of the exercises are marked as problems accessible for individual work of students while others extend the basic exposition and include plenty of material with hints and references. The concluding part “Bibliographical and Historical Comments” offers a rich, detailed and competent picture of the development and the present state of measure and integration theory.
Volume 2 is organised analogously with five chapters, each accompanied by “Supplement and exercises”. The selection of material reflects the research orientation of the author and is written for analysts as well as for probabilists. The arrangement is not necessarily designed for linear reading; individual chapters are high quality detailed surveys on important parts of modern measure theory: Borel, Baire and Souslin sets, topological measure theory, weak convergence of measures, transformation of measures and isomorphisms and conditional measures and conditional expectations. There is again a wealth of material, which cannot be described here in detail. However, what should be mentioned is the collection of 2038 references also representing large contributions from the Russian mathematical school. This is an excellent and impressive monograph, which I can strongly recommended to researchers in analysis and probability, to university teachers as well as to students. I am convinced that this two volume treatise cannot be missing from university libraries and the shelves of mathematicians interested in measure and integration.