#math

Convexity

Convexity is a very fundamental property: demarcation line in optimization between tractable and (possibly) intractable problems

Definition Convex Function

A function $f$ is convex over a domain $R$ , if for all $x, y \in R$ $$f(tx + (1-t)y) \leq tf(x) + (1-t)f(y), \quad \forall t \in [0;1]$$
If $f$ is differentiable, convexity is equivalent to the condition:

f (x) \geq f (y) + \nabla f (y) (x - y), \forall x, y \in R

if $f$ is twice differentiable, convexity on a convex domain $R$ is equivalent to the condition:

\nabla^{2} f (x) ⪰ 0, \forall x \in Int (R)