Given a unit vector $d$, I would like to find a weight vector $x$, which has non-negative elements and sums up to one, such that the $Ax+b$ direction aligns with $d$ as much as possible. That is, $$\max_x \langle\frac{Ax+b}{\|Ax+b\|},d \rangle \quad \text{s.t} \quad x\geq0, \sum_{i}x_i=1$$
where $\langle,\rangle$ is the inner product.
I think this type of problem should appear a lot, but I fail to find literature on it. Is there a way to solve this efficiently? Or is there a similar optimization formulation that can be solved efficiently, such as having a close-formed solution or a linear programming problem?