Given vectors $a, b \in \mathbb{R}^d$, consider the following optimization problem in $x,y\in \mathbb{R}^d$.
$$ \underset{x, y \in \mathbb{R}^d}{\text{maximize}} \quad x^\top y + a^\top x+b^\top y \quad\quad \text{subject to} \quad \|x\|_2\le 1, \|y\|_2 \le 1 $$
This problem is not a DCP because the expression $x^\top y$ not convex (suitable change of variables turns it into $\|x\|^2 - \|y\|^2$). However, if I still want to optimize it, what can I do?
I tried to use iterative methods: fix $x$, solve $y$, then fix $y$ solve $x$. But it seems this method cannot guarantee to converge. I'm totally stuck. Is it in the world that this problem have a solution that garantee to converge? Or even analytic solutions? Any advice would be helpful.