Maximize correlation subject to nonconvex correlation constraints

Question

Let $r, z$ and each of $u_i$ be a length $n$ vector. I’d like to maximize the correlation between $z$ and $r$ (when that correlation is positive) while keeping $z$ “away” from $u_i$’s. Formally,

\begin{align} \max_z &\quad \text{corr}^+(z,r) \\ \text{s.t.} &\quad \text{corr}(z,u_i)\leq a_i,\ i = 1, \dots, k \end{align}

where $\text{corr}(z,r) \stackrel{\text{def}}{=} \frac{z^T r}{\sqrt{z^Tz}\sqrt{r^Tr}}$, and $\text{corr}^+(z,r) \stackrel{\text{def}}{=} \max(0, \text{corr}(z,r))$, and $0 \leq a_i \leq 1$ for all $i$

The trouble is that the constraints are non-convex. Any leads? Thanks!

Answer 1

You could add the non-convex constraint $z^Tz = 1$. That would make the objective function and other constraints linear. So this would be a Linear Programming problem, but for a single non-convex equality constraint. Use either a global non-convex optimization solver (if the problem can be solved fast enough) or local non-convex optimization solver to solve it.