Given $a \in \mathbb R^3$, let $f(X) := a^T X a$. How to know whether the following minimization problem is convex or not?
$$ \min_{X \text{ is PSD}} \, f(X) $$
I know that if $f''(X) > 0$, it is convex. However, for my given function calculating $f''(X)$ gets too complicated and not straightforward. Please help.
Essentially, you have the semidefinite program (SDP)
$$\begin{array}{ll} \text{minimize} & \langle \mathrm a \mathrm a^\top, \mathrm X \rangle\\ \text{subject to} & \mathrm X \succeq \mathrm O_3\end{array}$$
Does this answer your question?