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A $C^{2}$ function is convex exactly when its Hessian is positive semidefinite

Statement

A $C^{2}$ function on an open convex set is convex if and only if its Hessian is positive semidefinite at every point. Hessians and their quadratic forms have the conventions of The Hessian matrix and critical points of a scalar field, The Hessian of a $C^{2}$ scalar field is symmetric.

Facts & Assumptions

Given: An open convex $U \subseteq R^{n}$ , a $C^{2}$ function $f : U \to R$ , and the total chain rule The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ .

[L1]

A twice differentiable real function on an open interval is convex if and only if its second derivative is nonnegative throughout the interval (A twice-differentiable function on an open interval is convex if and only if its second derivative is nonnegative).

[F1]

A symmetric quadratic form is positive semidefinite when $q_{H} (h) \geq 0$ for every $h$ (Positive definite, negative definite, semidefinite, and indefinite quadratic forms).

Proof

technique · direct

1.1

For $x \in U$ and a direction $v$ , put $ϕ (t) = f (x + t v)$ on the open interval where the affine line lies in $U$ . Two applications of the chain rule give $ϕ^{''} (t) = ⟨ H_{f} (x + t v) v, v ⟩ .$

L1F1givenalgebra

2.1

For the forward implication, if $f$ is convex then every line restriction $ϕ$ is convex, so [L1] and step 1.1 give $⟨ H_{f} (x) v, v ⟩ \geq 0$ for every $v$ , which is [F1]. For the reverse implication, [F1] and step 1.1 make every line restriction have nonnegative second derivative; [L1] makes each restriction convex, yielding the two-point convexity inequality for $f$ .

step 1.1L1F1∎

Remarks

The convex-domain hypothesis cannot be dropped. On the open but nonconvex set $R ∖ {0}$ , the function $f (x) = x^{- 2}$ has $f^{''} (x) = 6 x^{- 4} > 0$ everywhere, but it is not a convex function on that domain in the sense of Convex and strictly convex functions on Euclidean convex sets. This is the boundary recorded in Boyd–Vandenberghe, Remark 3.1.

Depends on

Used by

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Sources

S. Boyd and L. Vandenberghe, Convex Optimization, §3.1.4 (standard reference, not scraped)

A C2C2 function is convex exactly when its Hessian is positive semidefinite