Convexity & the Hessian

A function is convex when its Hessian is PSD everywhere, guaranteeing that any local minimum is a global one — the key to trusting an optimizer's answer.

1. The idea

A twice-differentiable function $f : R^{n} \to R$ is convex if its Hessian matrix $H$ (the matrix of second partial derivatives) is positive semi-definite everywhere. Geometrically, the function curves upward or stays flat in every direction, never downward.

Convexity guarantees that any local minimum is automatically a global minimum, and that the set where $f (x) \leq c$ is a convex set. This is why convex optimization problems (like mean-variance portfolio problems) have unique, tractable solutions.

Checking convexity in practice means computing the Hessian and testing positive semi-definiteness (via eigenvalues or leading principal minors), the same tools used for covariance matrices.

2. Why it matters

Interviewers use convexity checks to confirm that a proposed optimization (portfolio variance minimization, likelihood maximization) actually has a well-behaved unique solution.

3. Core mathematics

Hessian

H_{ij} = \frac{\partial ^{2} f}{\partial x _{i} \partial x _{j}}

Convexity criterion

f convex ⟺ H ⪰ 0 \forall x

Strict convexity

H ≻ 0 \Rightarrow unique global minimum

4. How to recognize it

Think of this when…

'Show the optimization problem is convex' or 'is the minimum unique'
Checking a quadratic objective like $x^{T} A x$ for convexity via $A$
Questions about whether gradient descent will converge to a global optimum

5. Worked example

Worked example

Is $f (x, y) = x^{2} + 2 y^{2} + x y$ convex?

1
Compute the Hessian.
$H = (2114)$
2
Check leading principal minors.
$2 > 0, det (H) = 8 - 1 = 7 > 0$
3
Both positive, so $H$ is PD, and $f$ is strictly convex.
Answer: Yes, f is (strictly) convex; it has a unique global minimum.

6. Common mistakes

Confusing convexity of the function with convexity of the feasible set — both matter for the overall problem
Checking convexity at only one point instead of confirming the Hessian condition holds globally
Assuming a quadratic form is always convex without checking that its matrix is PSD

7. Quick check

A function is convex when its Hessian is:

The main benefit of convexity in optimization is:

8. Practice

2 QuantSolve questions use this concept.

Practice this concept

9. Related methods

Jensen's Inequality

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