This monumental treatise is not for the fainthearted or the uninitiated. The prospective reader might well begin by perusing chapter 5, a 97-page summary of relevant mathematical background, but should not give up prematurely. The author expounds seminal ideas that serve to unify the theoretical analysis of a wide range of optimization problems, ideas that have algorithmic implications. The preface provides an exegesis. Given the abstract approach and the dearth of examples, fully appreciating what the author has accomplished requires substantial prior preparation in mathematics and optimization theory plus a prodigious investment of effort. The prospective audience, consequently, may be smaller than intended or merited.

Chapter 1 (166 pages) and chapter 2 (201 pages) survey finite-dimensional optimization problems: unconstrained in the first, and constrained and min-max in the second. The approach and selection of topics are designed to establish the paradigms to be elaborated on later. Chapter 3 (114 pages) extends ideas developed in chapter 2 to semi-infinite optimization problems, where the minimizer is a finite-dimensional vector but an infinite number of functions specify the desired minimizer. Chapter 4 (164 pages) brings a large class of optimal control problems under the aegis of the formalism.

There are a fair number of obvious typographical errors, perhaps not surprising for a book of this size and complexity, but they bring to mind the possibility that the book contains less obvious and more pernicious errors.