Undergraduate convexity

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 algorithm1. Fourier-Motzkin elimination. 1.1. Linear inequalities. 1.2. Linear optimization using elimination. 1.3. Polyhedra. 1.4. Exercises. 2. Affine subspaces. 2.1. Definition and basics. 2.2. The affine hull. 2.3. Affine subspaces and subspaces. 2.4. Affine independence and the dimension of a subset. 2.5. Exercises. 3. Convex subsets. 3.1. Basics. 3.2. The convex hull. 3.3. Faces of convex subsets. 3.4. Convex cones. 3.5. Carathéodory's theorem. 3.6. The convex hull, simplicial subsets and Bland's rule. 3.7. Exercises. 4. Polyhedra. 4.1. Faces of polyhedra. 4.2. Extreme points and linear optimization. 4.3. Weyl's theorem. 4.4. Farkas's lemma. 4.5. Three applications of Farkas's lemma. 4.6. Minkowski's theorem. 4.7. Parametrization of polyhedra. 4.8. Doubly stochastic matrices: the Birkhoff polytope. 4.9. Exercises. 5. Computations with polyhedra. 5.1. Extreme rays and minimal generators in convex cones. 5.2. Minimal generators of a polyhedral cone. 5.3. The double description method. 5.4. Linear programming and the simplex algorithm. 5.5. Exercises. 6. Closed convex subsets and separating hyperplanes. 6.1. Closed convex subsets. 6.2. Supporting hyperplanes. 6.3. Separation by hyperplanes. 6.4. Exercises. 7. Convex functions. 7.1. Basics. 7.2. Jensen's inequality. 7.3. Minima of convex functions. 7.4. Convex functions of one variable. 7.5. Differentiable functions of one variable. 7.6. Taylor polynomials. 7.7. Differentiable convex functions. 7.8. Exercises. 8. Differentiable functions of several variables. 8.1. Differentiability. 8.2. The chain rule. 8.3. Lagrange multipliers. 8.4. The arithmetic-geometric inequality revisited. 8.5. Exercises. 9. Convex functions of several variables. 9.1. Subgradients. 9.2. Convexity and the Hessian. 9.3. Positive definite and positive semidefinite matrices. 9.4. Principal minors and definite matrices. 9.5. The positive semidefinite cone. 9.6. Reduction of symmetric matrices. 9.7. The spectral theorem. 9.8. Quadratic forms. 9.9. Exercises. 10. Convex optimization. 10.1. A geometric optimality criterion. 10.2. The Karush-Kuhn-Tucker conditions. 10.3. An example. 10.4. The Langrangian, saddle points, duality and game theory. 10.5. An interior point method. 10.6. Maximizing convex functions over polytopes. 10.7. Exercises