convex function

Definition

A function $f$ is convex iff for any $\mathbf{x},\mathbf{y},\lambda \in [0,1]$:

(see Jensen's inequality)

convexity in one dimension

A twice-differentiable function $f : \mathbb{R} → R$ is:

• convex if and only if $f''(x) ≥ 0$ for all $x$.
• β-smooth if $f''(x) ≤ β$.
• α-strongly convex if $f''(x) ≥ α$.

Definition (function convexity, gradient)

A function $f$ is convex if and only if for any $\mathbf{x},\mathbf{y}$:

equivalently,

See also:

• convex set

References:

1. https://www.cs.cornell.edu/courses/cs6783/2018fa/lec16-supplement.pdf