Product Description
Convex Analysis is the calculus of inequalities while Convex Optimization is its application. Analysis is inherently the domain of the mathematician while Optimization belongs to the engineer. In layman's terms, the mathematical science of Optimization is the study of how to make a good choice when confronted with conflicting requirements. The qualifier Convex means: when an optimal solution is found, then it is guaranteed to be a best solution; there is no better choice. Any Convex Optimization problem has geometric interpretation. Conversely, recent advances in geometry and in graph theory hold Convex Optimization within their proofs' core. This book is about Convex Optimization, convex geometry (with particular attention to distance geometry), and nonconvex, combinatorial, and geometrical problems that can be relaxed or transformed into convex problems. A virtual flood of new applications follows by epiphany that many problems, presumed nonconvex, can be so transformed. International Edition II ... Read more

Customer Reviews (3)

Epiphany after Epiphany
Convex Optimization & Euclidean Distance Geometry

I thought I'd use this book as a reference since the unusually large Index is a good place to locate the definitions.Dattorro starts from the basic premises and works through the algebra with many examples and many good illustrations.

I've found that Dattorro's perspective on each subject (optimization and distance geometry) is both algebraic and geometric.He bridges those unexpectedly well.His approach to rank minimization, for example, is how I would have thought of doing it, in terms of eigenvalues.It feels right to me.

Dattorro's notation is "progressive."A vector is represented by a single letter, say x, with no embellishment to distingush it from a real variable.That makes the presentation simple, but takes some getting used to as does his style of "missing articles" (e.g. the) and replacement everywhere of "i.e." with latin "id est."

The book is organized by convex optimzation first then distance geometry second,
three chapters devoted to each.The appendices support seven chapters total and take half the book!It's a big book.

Dattorro's treatment of distance geometry is the book's main strength.The main result is a new expression for the relationship between the semidefinite positive and Euclidean distance cones, and takes a long time to get there.Along the way, he goes back to 1935 and integrates the results of Schoenberg (before modern linear algebra), Cayley and Menger, Critchley, Gower, then augments that with some later results like Hayden, Wells, Liu, & Tarazaga, and then more contemporary results like Deza & Laurent, Wolkowicz, Saul and Weinberger to name only a few.Then, of course he shows how that all relates to optimization.I particularly liked the geographical map reconstruction examples where only distance ordering was known.

I recommend this book to anyone who wants both a good introduction to convex optimization and a reference to some latest techniques, a few of which Dattorro may have invented.There is a good review of semidefinite programming, and what he writes about distance geometry refreshes old math with new.

Masterly
"Dattorro has given a masterly account of the properties of convex squared-distance cones."

-Casper J. Albers, Frank Critchley, & John C. Gower, Group Average Representations in Euclidean Distance Cones, in Brito, Bertrand, Cucumel, and de Carvalho, editors, Selected Contributions in Data Analysis and Classification (Studies in Classification, Data Analysis, and Knowledge Organization), Springer, pp.445-454, 2007.

Get inspired for your research
I have read many books in optimization and in engineering, this is really one of the most exceptional books in my library. Dattorro is a talented writer. The book has plenty of illustrations that make the content of the book very easy to understand. The author is obviously influenced byBoyd Convex Optimization and this is actually a very good thing. In my opinion he has extended Boyd's book in the best possible way, by providing exciting applications and also a different perspective for understanding convexity, polytopes etc. The book is very useful for understanding Semidefinite {rogramming (SDP). To my knowledge this is the only book that gets so deeply in SDP (theory and mostly applications). The chapters on Euclidean Distance Geometry are unique in the sense that the only other books that I found in the literature on that topic date back to 1930. I think the biggest hit of the book is the algorithm for rank minimization.

Unfortunately the book hasn't gotten enough attention from the scientists and engineers probably because you can also get a free pdf copy for the author's website.Although I can get a free online version, I bought a copy becauseit is worth it (as it is for Boyd's book). I have recommended this book to may PhD students that got inspired for their research. ... Read more

Product Description
Presented in this monograph is the current state-of-the-art in the theory of convex structures. The notion of convexity covered here is considerably broader than the classic one; specifically, it is not restricted to the context of vector spaces. Classical concepts of order-convex sets (Birkhoff) and of geodesically convex sets (Menger) are directly inspired by intuition; they go back to the first half of this century. An axiomatic approach started to develop in the early Fifties. The author became attracted to it in the mid-Seventies, resulting in the present volume, in which graphs appear side-by-side with Banach spaces, classical geometry with matroids, and ordered sets with metric spaces. A wide variety of results has been included (ranging for instance from the area of partition calculus to that of continuous selection). The tools involved are borrowed from areas ranging from discrete mathematics to infinite-dimensional topology.

Although addressed primarily to the researcher, parts of this monograph can be used as a basis for a well-balanced, one-semester graduate course. ... Read more

Product Description
This research-level book presents up-to-date information concerning recent developments in convex functions and partial orderings and some applications in mathematics, statistics, and reliability theory. The book will serve researchers in mathematical and statistical theory and theoretical and applied reliabilists.

Key Features
* Presents classical and newly published results on convex functions and related inequalities
* Explains partial ordering based on arrangement and their applications in mathematics, probability, statsitics, and reliability
* Demonstrates the connection of partial ordering with other well-known orderings such as majorization and Schur functions
* Will generate further research and applications ... Read more

Product Description
Triangulations appear everywhere, from volume computations and meshing to algebra and topology. This book studies the subdivisions and triangulations of polyhedral regions and point sets and presents the first comprehensive treatment of the theory of secondary polytopes and related topics.A central theme of the book is the use of the rich structure of the space of triangulations to solve computational problems (e.g., counting the number of triangulations or finding optimal triangulations with respect to various criteria), and to establish connections to applications in algebra, computer science, combinatorics, and optimization.With many examples and exercises, and with nearly five hundred illustrations, the book gently guides readers through the properties of the spaces of triangulations of "structured" (e.g., cubes, cyclic polytopes, lattice polytopes) and "pathological" (e.g., disconnected spaces of triangulations) situations using only elementary principles. ... Read more

Product Description

Mirrors and Reflections is a systematic and elementary treatment of finite groups generated by reflections. The approach is based on fundamental geometric considerations in Coxeter complexes, and emphasizes the intuitive geometric aspects of the theory of reflection groups.

Key features: • Many important concepts in the proofs are illustrated in simple drawings, which give easy access to the theory • A large number of exercises at various levels of difficulty • Some Euclidean geometry is included along with the theory of convex polyhedra • Few prerequisites are necessary beyond linear algebra and the basic notions of group theory.

The exposition is directed at advanced undergraduates and first-year graduate students.

Product Description
This book is suitable as a graduate text and reference work in the areas of convex functions and bodies, global geometric problems, and nonlinear elliptic boundary value problems with special emphasis on Monge-Ampere equations. The theory of convex functions and bodies is presented first so that it can be used to study the other areas. In fact, the author makes a point of emphasizing the interrelationship of all the areas mentioned above. This enables the reader to obtain a working knowledge of the material.
Specific topics of the book include the Minkowski problem, mixed volumes of convex bodies, the Brunn-Minkowski inequalities, geometric maximum principles, the normal mapping of convex hypersurfaces, the R-curvature of convex functions. ... Read more

Product Description

This undergraduate textbook is written for a junior/senior level course on linear optimization. Unlike other texts, the treatment follows the "modified Moore method" approach in which examples and proof opportunities are worked into the text in order to encourage students to develop some of the content through their own experiments and arguments while they are reading the text. Additionally, the focus is on the mathematics underlying the ideas of optimizing linear functions under linear constraints and the algorithms used to solve them. In particular, the author uses the Simplex Algorithm to motivate these concepts. The text progresses at a gentle and inviting pace. The presentation is driven by examples and illustrations. Ample exercises are provided at the end of each chapter for mastering the material.

The instructor's version of the text contains solutions embedded within the text, rather than in an appendix. It also has extra material and suggestions for the teacher’s benefit. Junior/senior level undergraduate students will benefit from the book. Future secondary school mathematics teachers will also find this book useful.

Product Description
This book focuses on three disciplines of applied mathematics:control theory, location science and computational geometry. Theauthors show how methods and tools from convex geometry in a widersense can help solve various problems from these disciplines. Moreprecisely they consider mainly the tent method (as an application of ageneralized separation theory of convex cones) in nonclassicalvariational calculus, various median problems in Euclidean and otherMinkowski spaces (including a detailed discussion of theFermat--Torricelli problem) and different types of partitioningsof topologically complicated polygonal domains into a minimum numberof convex pieces. Figures are used extensively throughout the book andthere is also a large collection of exercises. Audience: Graduate students, teachers and researchers. ... Read more

Product Description
High Quality Content by WIKIPEDIA articles! A polygon mesh or unstructured grid is a collection of vertices, edges and faces that defines the shape of a polyhedral object in 3D computer graphics and solid modeling. The faces usually consist of triangles, quadrilaterals or other simple convex polygons, since this simplifies rendering, but may also be composed of more general concave polygons, or polygons with holes.The study of polygon meshes is a large sub-field of computer graphics and geometric modeling. Different representations of polygon meshes are used for different applications and goals. The variety of operations performed on meshes may include Boolean logic, smoothing, simplification, and many others. ... Read more