I have the following optimization problem in $q \in \Bbb R^d$
$$\min_{q \geq 0} \quad q^T M q - w^T q $$
where matrix $M \in \Bbb R^{d \times d}$ is symmetric and positive semidefinite (PSD), $M_{ij} \ge 0$, $w \in \Bbb R^d$.
Since this is a convex function, using Lagrange multipliers and KKT, I get
$$ 2Mq-w-\beta=0\\ q_i\ge0\\ \beta_i\ge0\\ q_i\beta_i=0\\ $$
Now I don't know how to continue. Could anyone give me some advice? Generally, what methods can be applied when equations derived from KKT condition are too difficult to solve?