I'm trying to find the local minima, if they exist, of,
$$G(r) = \sum_{n=1}^N \beta_n e^{{-(r-r_n)}^2}$$
Such that $r$, $r_n$, and $\beta_n \in \Bbb R^+$ are positive scalars.
Edit:
The $r_n$ are actually a condensed form of $r_{n,m}$, which are given by a triangulation pattern in $\Bbb R^2$
$$x_{n,m} = \alpha(n+\frac{|n|}{n}\frac{1+(-1)^{m+1}}{4})$$ $$y_{n,m} = \alpha m\frac{\sqrt{3}}{2}$$ $$r_{n,m} = \sqrt{x_{n,m}^2+y_{n,m}^2}=\alpha\sqrt{n^2+0.75m^2+(0.125+0.5|n|(1+(-1)^{m+1}))}$$
Where $\alpha\in\Bbb R^+$ is the length of the triangle, and $n,m\in\Bbb Z$ are the indices of the grid. When calculated this way, the sum becomes,
$$G_R(r)=\sum_{n,m\in\Bbb Z}^{n^2+m^2\leq R^2}\beta_{n,m}e^{-(r-r_{n,m})^2}$$
The $\beta_{n,m}$ are equal for radii of equal magnitude, i.e.,
$$\beta_{n,m} = \beta_{i,j} \text{ if } r_{n,m} = r_{i,j}$$
So that the $\beta$ are optimized by ring.