I'm not asking for a solution, I would just need to know what type of optimization is the following problem? Find $\mathbf{m}$ that minimizes:
$$ \sum_{k=1}^{N}\left|\frac{1}{\sin^{m}\left(\frac{k}{N}\pi\right)} \prod_{i=1}^{m} \left|\sin\left(\frac{k\ m_i}{N}\pi\right)\right| - h_{k}\right| $$
where $ 0 \leq h_k \leq 1$ is known, $\mathbf{m} = [m_1, ... , m_m]$ , $N = \sum_{i=1}^{m} m_i - m +1$ and $m_i$ is integer, $i = 1, 2, ..., m$.
In other words $\mathbf{m}$ is a vector of $m$ integers and there are $N$ "points" for which the "distance" needs to be minimized. The distance has been written with absolute values, but it could be also expressed as a quadratic function. Where I am completely lost is if there is some theory about this kind of optimizations on integers and what is to be considered a costraint and what an objective or cost. Can somebody point me in the right direction?