Convex Functions and Convex Optimization Problems

Definizione 10   A function $f : \mathbb{R}^n \rightarrow \mathbb{R}$ is convex if for $\forall \mathbf{x},\mathbf{y}$ belonging to the domain of $f$ and for $\lambda \in [0,1]$, it holds that

$$f \left( \lambda \mathbf{x} + (1-\lambda)\mathbf{y} \right) \leq \lambda f(\mathbf{x}) + (1-\lambda) f(\mathbf{y})$$ (3.63)

A function $f$ is defined as concave if $-f$ is convex.