Minimize frobenius norm of Gram matrix I have an $n\times n $ gram matrix, $G$  (symmetric and positive semidefinite) which captures correlation between set of vectors. The $i,j$ th element of $G$ is given by,
 $\langle a_i, a_j\rangle = \rho_{ij}$ with $\|a_i\|_2 =\|a_j\|_2 =1$. An interesting property of these vectors $a_i$ is that they have a common origin. But this origin can be modified to optimize values in $G$. Further, they have fixed end points. You can imagine this in 2D plane as set of vectors starting at the point  $ (x,y) $ and ending at fixed set of points.
Now, I need to minimize $\|G\|_{F}$, within a given range of $x$, $0<x<X$ for origin of those set of vectors. You can assume $y$ coordinate of origin of the vectors as fixed. I need to identify
$$\arg \min_{x} \|G\|_{F}.$$ If possible can someone please help me to solve this problem.  
Thank you!  

Minimizing $\| G \|_{F}$ is equivalent to minimizing $\| G \|_{F}^{2}$, and 
$G_{i,j}=(a_{i}-x)^{T}(a_{j}-x)$, so your problem can be formulated as 
$\min \sum_{i=1}^{n} \sum_{j=1}^{n} G_{i,j}^{2}$
or
$\min \sum_{i=1}^{n} \sum_{j=1}^{n} ((a_{i}-x)^{T}(a_{j}-x))^{2}$
or 
$\min \sum_{i=1}^{n} \sum_{j=1}^{n} (a_{i}^{T}a_{j}-a_{i}^{T}x-a_{j}^{T}x+x^{T}x)^{2}$
It doesn't appear that this objective will generally be convex.  
