Undergraduate Convexity: From Fourier and Motzkin to Kuhn and Tucker

Content: 1. 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.