Convexity characterized by positive semidefinite Hessian

A C^2 function on an open convex set is convex iff its Hessian is positive semidefinite everywhere

Convexity characterized by positive semidefinite Hessian

Theorem. Let $f:\Omega\to\mathbb{R}$ be twice continuously differentiable on a nonempty open convex set $\Omega\subset\mathbb{R}^n$ . Then $f$ is convex on $\Omega$ if and only if for every $x\in\Omega$ its Hessian matrix $\nabla^2 f(x)$ is positive semidefinite (in the sense of nonnegative operators ), i.e.,

\langle v,\nabla^2 f(x)\,v\rangle \ge 0 \quad \text{for all } v\in\mathbb{R}^n.

Context. This is the standard multivariable calculus criterion for convexity and underlies many second-order methods in optimization.

Proof sketch. Convexity of $f$ is equivalent to convexity of every restriction to a line: for fixed $x\in\Omega$ and direction $d\in\mathbb{R}^n$ , consider $\varphi(t)=f(x+td)$ on the maximal interval where $x+td\in\Omega$ . Apply the one-dimensional criterion f''≥0 ⇔ convex to $\varphi$ , noting that $\varphi''(t)=\langle d,\nabla^2 f(x+td)d\rangle$ .