Undergraduate Convexity

From Fourier and Motzkin to Kuhn and Tucker

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

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.
Key Features:

• Emphasis on viewing introductory convexity as a generalization of linear algebra in finding solutions to linear inequalities
• A key point is computation through concrete algorithms like the double description method. This enables students to carry out non-trivial computations alongside the introduction of the mathematical concepts
• Convexity is inherently a geometric subject. However, without computational techniques, the teaching of the subject turns easily into a reproduction of abstractions and definitions. The book addresses this issue at a basic level