An Introduction to Convexity

Contents 1 The basic concepts 1 1.1 Is convexity useful? 1 1.2 Nonnegative vectors 4 1.3 Linear programming 5 1.4 Convex sets, cones and polyhedra 6 1.5 Linear algebra and affine sets 11 1.6 Exercises 14 2 Convex hulls and Carathéodory's theorem 17 2.1 Convex and nonnegative combinations 17 2.2 The convex hull 19 2.3 Affine independence and dimension 22 2.4 Convex sets and topology 24 2.5 Carathéodory's theorem and some consequences 29 2.6 Exercises 32 3 Projection and separation 36 3.1 The projection operator 36 3.2 Separation of convex sets 39 3.3 Exercises 45 4 Representation of convex sets 47 4.1 Faces of convex sets 47 4.2 The recession cone 50 4.3 Inner representation and Minkowski's theorem 53 4.4 Polytopes and polyhedra 55 4.5 Exercises 63 5 Convex functions 67 5.1 Convex functions of one variable 67 5.2 Convex functions of several variables 75 5.3 Continuity and differentiability 81 5.4 Exercises 88 6 Nonlinear and convex optimization 91 6.1 Local and global minimum in convex optimization 91 6.2 Optimality conditions for convex optimization 93 6.3 Feasible direction methods 95 6.4 Nonlinear optimization and Lagrange multipliers 97 6.5 Nonlinear optimization: inequality constraints 101 6.6 An augmented Lagrangian method 106 6.7 Exercises 107