Based on undergraduate teaching to students in computer science, economics and mathematics at Aarhus University, this is an elementary introduction to convex sets and convex functions with emphasis on concrete computations and examples.
Starting from linear inequalities and Fourier–Motzkin elimination, the theory is developed by introducing polyhedra, the double description method and the simplex algorithm, closed convex subsets, convex functions of one and several variables ending with a chapter on convex optimization with the Karush–Kuhn–Tucker conditions, duality and an interior point algorithm.
Study Guide here
Sample Chapter(s)
Chapter 1: Fourier–Motzkin Elimination (627 KB)
Contents:
- Fourier–Motzkin Elimination
- Affine Subspaces
- Convex Subsets
- Polyhedra
- Computations with Polyhedra
- Closed Convex Subsets and Separating Hyperplanes
- Convex Functions
- Differentiable Functions of Several Variables
- Convex Functions of Several Variables
- Convex Optimization
- Appendices:
- Analysis
- Linear (In)dependence and the Rank of a Matrix
Readership: Undergraduates focusing on convexity and optimization.
“Overall, the author has managed to keep a sound balance between the different approaches to convexity in geometry, analysis, and applied mathematics. The entire presentation is utmost lucid, didactically well-composed, thematically versatile and essentially self-contained. The large number of instructive examples and illustrating figures will certainly help the unexperienced reader grasp the abstract concepts, methods and results, all of which are treated in a mathematically rigorous way. Also, the emphasis on computational, especially algorithmic methods is a particular feature of this fine undergraduate textbook, which will be a great source for students and instructors like-wise … the book under review is an excellent, rather unique primer on convexity in several branches of mathematics.”
Zentralblatt MATH
“Undergraduate Convexity would make an excellent textbook. An instructor might choose to have students present some of the examples while he or she provides commentary, perhaps alternating coaching and lecturing. A course taught from this book could be a good transition into more abstract mathematics, exposing students to general theory then giving them the familiar comfort of more computational exercises. One could also use the book as a warm-up to a more advanced course in optimization.”
MAA Review
“The book is didactically written in a pleasant and lively style, with careful motivation of the considered notions, illuminating examples and pictures, and relevant historical remarks. This is a remarkable book, a readable and attractive introduction to the multi-faceted domain of convexity and its applications.”
Nicolae Popovici
Stud. Univ. Babes-Bolyai Math
“Compared to most modern undergraduate math textbooks, this book is unusually thin and portable. It also contains a wealth of material, presented in a concise and delightful way, accompanied by figures, historical references, pointers to further reading, pictures of great mathematicians and snapshots of pages of their groundbreaking papers. There are numerous exercises, both of computational and theoretical nature. If you want to teach an undergraduate convexity course, this looks like an excellent choice for the textbook.”
MathSciNet
