CHAPTER 4.2 - Convex Functions

Abstract

This chapter discusses convex functions. Any convex function can be taken to be lower semicontinuous, with some possible redefinitions on the boundary. The convex programming problem, which is to minimize a convex function f over a constraint set K, can be replaced, according to the Rockafellar–Fenchel theorem with the dual problem of maximizing the sum of the conjugate of f and the indicator function of the set K. A function can have a subdifferential at points where it cannot have a derivative in the sense of either Gateaux or Frechet. A convex function of a single real variable has numerous differentiability properties by virtue of its convexity. The derivative f '(xo) convex function is a continuous linear transformation only if the function f is continuous at xo. A Banach space is an Asplund space only if every separable closed subspace has a separable dual.