Chapter 3: Convex functions

Now that we know from the direct method of the calculus of variations when a functional \(F : X \to \overline{\mathbb{R}} := \mathbb{R} \cup \{\infty\}\) admits a minimizer \(\bar{x} \in X\), our next goal is to characterize such minimizers using optimality conditions, i.e., without comparing its function value to that at every other point. If \(F\) is differentiable at \(\bar{x}\), the classical optimality condition is, by Fermat’s principle, \(F'(\bar{x}) = 0,\) and we can use calculus rules to evaluate this derivative in order to make this condition as explicit as possible. We wish to extend this as far as possible to nonsmooth \(F\), i.e., not classically (Fréchet or Gateaux) differentiable. Clearly, not being differentiable is not much to work with, so we have to assume other properties instead. One possibility is to replace the local property of differentiability with the global property of convexity. As we will see in this and the following chapters, this property will allow us to recover a satisfying calculus for a class of relevant nonsmooth functionals. We begin in this chapter by deriving several fundamental properties of convex functions relevant for optimization, while the corresponding Fermat principle and calculus rules are the topic of Chapter 4.