$$\begin{array}{ll} \underset{W}{\text{minimize}} & - \displaystyle\sum_{t=1}^T \frac{W^\top A a_t}{\| W^\top A \|_2 \| a_t \|_2}\\ \text{subject to} & \sum_{t=1}^T w_t =1\\ & w_t \geq 0 \text { for } t \in T\end{array}$$
where $W = [w_1, w_2,\dots, w_T]^\top \in \Bbb R^{T \times 1}$, $A = [a_1, a_2,..., a_T] \in \Bbb R^{T \times n}$, $a_t \in \Bbb R^{n\times 1}$.
It is intuitive that the problem is convex, and I have tied to prove that it is convex by using the definition of convex function. However, I have failed.
Can I have your help? Is the problem convex? If it is, is there any hint of the proof?