Nonlinear optimization: Optimizing a matrix to make its square is close to a given matrix. I'm trying to solve a  minimization problem whose purpose is to optimize a matrix whose square is close to another given matrix. But I can't find an effective tool to solve it.
Here is my problem:

Assume we have an unknown  Q with parameter $q11, q12,q14,q21,q22,q23,q32,q33,q34,q41,q43,q44$,  and a given matrix G, that is, 
  $Q=\begin{pmatrix}
 q11&q12 &0 &q14 \\q21&q22& q23&0\\ 0&q32& q33&q34\\ q41&0& q43&q44\\ 
 \end{pmatrix} $,   $G=\begin{pmatrix}
 0.48&0.24 &0.16 &0.12 \\ 0.48&0.24 &0.16 &0.12\\0.48&0.24 &0.16 &0.12\\0.48&0.24 &0.16 &0.12
 \end{pmatrix} $,
The problem is how to find the values of $q11, q12,q14,q21,q22,q23,q32,q33,q34,q41,q43,q44$ such that the square of $Q$ is very close to matrix $G$.

I choose to minimize the Frobenius norm of their difference, that is, 

$ Q* ={\arg\min}_{Q}   \| Q^2-G\|_F$
s.t.  $0\leq q11, q12,q14,q21,q22,q23,q32,q33,q34,q41,q43,q44 \leq 1$,$\quad$
     $\quad$ $q11+q12+q14=1$,
     $\quad$ $q21+q22+q23=1$,
     $\quad$ $q32+q33+q34=1$,
     $\quad$ $q41+q43+q44=1$.

During those days, I am frustrated to find a effective tool to execute the above optimization algorithm, can someone help me to realize it?

Nice answer,  @Samrat Mukhopadhyay
I would like to add something that can bring a supplementary light to the issue.
It is based on a property that has been overlooked, i.e., that $rank(P)=1$.
$P=\begin{pmatrix}
1 \\ 1 \\ 1 \\ 1
\end{pmatrix}
\begin{pmatrix}
0.48 & 0.24 & 0.16 & 0.12
\end{pmatrix}$
Said otherwise, the Singular Value Decomposition of $P$ is 
$P=\sigma_1 U_1 V_1^T=1.14543 \begin{pmatrix}
.5 \\ .5 \\ .5 \\ .5
\end{pmatrix}
\begin{pmatrix}
0.8381 & 0.4191 & 0.2794 & 0.2095
\end{pmatrix}$
Approximants we are looking for can always be decomposed into the form 
$$A=P+\sigma_2 U_2 V_2^T+\sigma_3 U_3 V_3^T+\sigma_4 U_4 V_4^T$$
The squared error to the target $\|A-P\|_F^2$ is thus $\sigma_2^2+\sigma_3^2+\sigma_4^2$ (F = Frobenius norm).
This is not at all independent of the ellipse Samrat Mukhopadhyai is speaking about.
More time and space would be necessary to detail all these aspects...
