Math professor publishes new book to help optimize algorithms
This year, Springer has published the new book, "Convex Analysis and Beyond, Volume I: Basic Theory," by Professor Boris Mordukhovich, of Wayne State University's Department of Mathematics and his co-author, Professor Nguyen Mau Nam of Portland State University. Professor Nam was also a doctoral student of Professor Mordukhovich, graduating from WSU in 2007. Their book is part of the Springer Series in Operations Research and Financial Engineering.
The famous simplex method of George Dantzig is an algorithm for finding extrema of linear functions on convex polyhedra in R^n. Meanwhile, even more famously (featured in freshman calculus classes!), one can find extrema of smooth functions on compact subsets of the real line by evaluating at endpoints and at zeroes of the derivative. One naturally would like to be able to take the best of both of these methods, finding extrema (i.e., solving optimization problems) of not-necessarily-linear functions on arbitrary convex polyhedra in many dimensions, including in infinite-dimensional spaces, not only closed intervals in the real line. This motivates the subject of convex analysis, which offers tools for solving nonlinear optimization problems on convex polyhedra and other finite- and infinite-dimensional domains in which one must carefully consider singular behavior on the boundary.
The newly-published first volume of "Convex Analysis and Beyond" is a textbook account of the theoretical foundations of the subject, with a particular emphasis on methods employing set-valued mappings, a particular area of specialization of the authors. "Convex Analysis and Beyond" is intended to have a second volume in the future, featuring applications of the ideas in the newly-published first volume. The published first volume does not assume its reader already has much exposure to analysis and covers the required basic ideas from topology and algebra as well as analysis. This makes the book suitable for students as well as for mathematicians in other subjects who are curious about convex analysis.
The ideas are introduced clearly, and supplemented with useful exercises and also explanations of the history of the subject. Much of what the book covers are what would be standard ideas from undergraduate and graduate analysis courses if one were working with smooth functions on an open domain, but the ideas become more subtle, requiring a careful development (given in the book) of much less widely-known foundations, because of the weaker hypotheses made on the functions and their domains, and the corresponding wider level of generality. The fifth chapter covers particularly rewarding conceptual connections between Gateaux differentiability and Frechet differentiability.