Mathematics of Optimization: How to do Things Faster

Contents: Preface. Part 1. Classical Algorithms. 1. Efficient Multiplication, I. 2. Efficient Multiplication, II. Part 2. Introduction to Linear Programming. 3. Introduction to Linear Programming. 4. The Canonical Linear Programming Problem. 5. Symmetries and Dualities. 6. Basic Feasible and Basic Optimal Solutions. 7. The Simplex Method. Part 3. Advanced Linear Programming. 8. Integer Programming. 9. Integer Optimization. 10. Multi-Objective and Quadratic Programming. 11. The Traveling Salesman Problem. 12. Introduction to Stochastic Linear Programming. Part 4 . Fixed Point Theorems. 13. Introduction to Fixed Point Theorems. 14. Contraction Maps. 15. Sperner's Lemma. 16. Brouwer's Fixed Point Theorem. Part 5. Advanced Topics. 17. Gale-Shapley Algorithm. 18. Interpolating Functions. 19. The Four Color Problem. 20. The Kepler Conjecture. Bibliography. Index. Optimization Theory is an active area of research with numerous applications; many of the books are designed for engineering classes, and thus have an emphasis on problems from such fields. Covering much of the same material, there is less emphasis on coding and detailed applications as the intended audience is more mathematical. There are still several important problems discussed (especially scheduling problems), but there is more emphasis on theory and less on the nuts and bolts of coding. A constant theme of the text is the why and the how in the subject. Why are we able to do a calculation efficiently? How should we look at a problem? Extensive effort is made to motivate the mathematics and isolate how one can apply ideas/perspectives to a variety of problems. As many of the key algorithms in the subject require too much time or detail to analyze in a first course (such as the run-time of the Simplex Algorithm), there are numerous comparisons to simpler algorithms which students have either seen or can quickly learn (such as the Euclidean algorithm) to motivate the type of results on run-time savings.…