Exercise · ZW-CONVEXITY-AND-THE-HESSIAN-006 · ★★☆ Intermediate

Compute the Hessian of \(f(x,y)=x^2+3xy\) and determine whether it is positive semidefinite.

Choose one option.

Show answer

C. \(\begin{pmatrix}2&3\\3&0\end{pmatrix}\), not positive semidefinite

Explanation

The Hessian has entry 0 in position (2,2) and 3 off the diagonal, so a suitable vector gives a negative quadratic form. The option with diagonal 2 and 0 off it drops the xy mixing term.

Study Continuous optimization in ZWMath →

Review the concept: Convexity and the Hessian

Criterion characterizing convexity of a twice differentiable function through positive semidefiniteness of its Hessian matrix \(\nabla^2 f(x)\) at every point of the domain.

\[f \text{ convexa} \iff \nabla^2 f(x) \succeq 0 \ \forall x \in \operatorname{dom} f\]

More practice problems: convexity and the hessian

ZW-CONVEXITY-AND-THE-HESSIAN-001 What condition on the Hessian guarantees that f is convex?
ZW-CONVEXITY-AND-THE-HESSIAN-002 If ∇^2 f(x) is negative semidefinite everywhere, what kind of function is f?
ZW-CONVEXITY-AND-THE-HESSIAN-003 For f(x)=x^2, what is its Hessian ∇^2 f(x)?
ZW-CONVEXITY-AND-THE-HESSIAN-004 What does it mean for a symmetric matrix to be positive semidefinite?
ZW-CONVEXITY-AND-THE-HESSIAN-005 What kind of set must the domain of f be to apply the Hessian criterion?
ZW-CONVEXITY-AND-THE-HESSIAN-007 For which values of a is f(x,y)=x^2+axy+y^2 convex on ℝ^2?
ZW-CONVEXITY-AND-THE-HESSIAN-008 Let f(x,y)=e^x+y^2. Is f convex on ℝ^2?
ZW-CONVEXITY-AND-THE-HESSIAN-009 Let f(x,y)=x^3+y^2. What can be concluded about the convexity of f on \x≥ 0\?