Appendix A: Convex Sets. Convex Functions

This appendix introduces basic definitions, graphic insights, and useful properties related to convex sets and convex and concave functions, used throughout the book in the characterization of convex network programs. Strict convexity and concavity are also introduced, given their importance in guaranteeing a unique optimal solution. Strongly convex and concave functions are presented since gradient algorithms have better convergence properties with them. A set of rules is provided to determine, in a simple form, the convexity and concavity of composite functions. This will permit us to study the curvature of multiple functions in network problems throughout the book, without the need of a laborious application of the definition, or the determination of the eigenvalues of the hessian matrix.