Convex Analysis and Global Optimization

This is a very pleasant text on global optimization problems, concentrating on those problems that have some kind of convexity property. It describes itself as a Ph.D. level text, but in fact it is easy to read and develops all the necessary background, so it could be used by any sufficiently-motivated student. It only deals with global optimization (finding global maxima or minima) and deals with techniques specifically for doing this, rather than the more familiar methods of finding local extrema and then showing that one is the global extremum.

The organizing principle of the book is “dc” (difference of convex): it studies how to optimize dc functions (those that are the difference of two convex functions) on a dc set (one that is a set-difference of two convex sets). Optimizing convex functions defined on convex sets is already a familiar problem in optimization. Linear programming is a well-known example, because the objective function is linear (and therefore convex) and the feasible region is the intersection of half-planes (and therefore convex). It turns out that most of the important properties of convex functions and regions can be generalized to dc functions and dc regions. It also turns out that most optimization problems that are important in practice can be expressed in terms of dc functions and dc regions, so this is a very powerful approach.

The book has an introductory section of about 120 pages that develops all the necessary properties from scratch. The remaining 400 pages of the book apply these to various classes of optimization problems. Each chapter has a good selection of problems, both proof problems and specific problems to optimize.

My only complaint with the book is that the index is weak; a lot of terms do not have their definition indexed, and there are lots of acronyms and abbreviations and these are not indexed either.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is allenstenger.com. His mathematical interests are number theory and classical analysis.