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.

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