Consider the following optimization problem $$ \min_{a_1,a_2,a_3} C \sqrt{\frac{a_1}{a_2 a_3}}, \quad C > 0$$ s.t. $$a_1+a_2+a_3=1, a_1>0, a_2>0, a_3 > 0$$
I found that $z^T H z \ge 0$, is it possible to find the optimal values of $a_1, a_2, a_3$? Is there some transformation that I can do?
Drop $C$ and the square root, they play no role in finding the minimiser (if it exists). If you write $a_1=1-a_2-a_3$ you just need to find the minimum of $g(a_2,a_3) = \frac{1-a_2-a_3}{a_2 a_3}$ with $a_2>0, a_3>0$ and $a_2+a_3 < 1$. Since these restrictions define an open set where the objective function is differentiable, the minimum must occur in a stationary point, which does not exist in that set... In the end the conclusion is that there is no minimum, there is an infimum. You can get arbitrarily close to zero, but you cannot reach zero in the set defined by those restrictions.