When mathematicians say a function is “nonlinear” they often mean that it is not necessarily linear. In this sense “nonlinear” is not an assumption but rather the absence of an assumption. To make progress in studying a nonlinear problem, we have to make some assumption about how a function departs from linearity. We have to replace an assumption of linearity with a weaker assumption that still retains enough structure to allow us to prove theorems. Often that weaker assumption is convexity. In large-scale optimization, for example, convexity is just the right assumption in order to retain many of the benefits of the linear theory while greatly increasing its scope of application. The study of convex functions has become more popular as nonlinear problems have become more popular and researchers realize they need to assume a particular kind of nonlinearity.
Jonathan Borwein and Jon Vanderwerff have written a valuable resource entitled Convex Functions: Constructions, Characterizations and Counterexamples. As the authors note in the preface, the book is the result 15 years of collaboration; it is not a light read. However, neither is Convex Functions a dry reference. It is a textbook written to guide the reader through the material.
Convex Functions tells a story from beginning to end. It starts with examples of convex functions in order to motivate the reader. It then progresses further and further into the theory, introducing special cases before proceeding to more general theory. The book closes with a retrospective, revisiting the differences between convex functions over finite and infinite dimensional spaces. The authors introduce a small amount of redundancy to make the book easier to read.
One way to read Convex Functions is as a tour through a large amount of real and functional analysis, as seen from the perspective of convex functions. The book opens with simple geometric problems in Euclidean space and leads up to results on the classification of Banach spaces and the theory of monotone operators. As the authors suggest, the book could be used for a less advanced class by covering the early chapters and picking out the finite dimensional portions of the latter chapters.
John D. Cook is a research statistician at M. D. Anderson Cancer Center and a blogger.