The optimization function is defined as
$\frac{1}{2}.x^T.A.x$ where $A=\begin{pmatrix} 1 & 0.5 \\ 0.5 & 1 \end{pmatrix}$
How to check if this is a convex or not? I know about the second derivative test and it gives $A$ which should be greater than $0$ in order to be convex but what are we really checking here? how is $A$ compared against $0$?
You have to check wheather $A$ (the hessian of the objetctive function $\frac{1}{2}x^T A x$) is positive semidefinit or not. Here, $A$ is diagonally dominant and symmetric, which implies directly, that A is positive semidefinit, which implies, that your objective function $\frac{1}{2}x^T A x$ is convex.