Second-order condition

Second-order condition

Convexity

If , second derivative will be matrix

Theorem

Let be twice-differentiable.
Then is convex if and only if for all .
(i.e. is positive semi-Definite).

Proof

Let satisfy the second-order condition.
Then

Taylor series for multivariate

Take between and s.t.

(we used positive-semidefinite here)
Since the first-order condition holds, must be convex

To prove the converse, consider a new function
(for some )
Use Taylor series for at and take the limit as .