I am trying to solve the following optimization problem for the vector $ y $, where $ A\_i $ are some given matrix (maybe low rank) and $ x\_i $ are unconstrained
$$ \min\_{y, x\_i} \sum\_{i=1}^J || y - A\_i x\_i ||\_2^2, \;\;\;\;\text{ subject to } ||y||\_2 = 1, \;\; || A\_i x\_i ||\_2 = 1 \;\; \forall i$$


I believe that solving the following problem, where I drop the constraints on the projections, would give the same solution for $ y $ but I am having a hard time proving it. 


$$ \min\_{y, x\_i} \sum\_{i=1}^J || y - A\_i x\_i ||\_2^2, \;\;\;\;\text{ subject to } ||y||\_2 = 1$$
To be clear, I only care about the value of $ y $ and not about the actual values of $ x\_i $. I know that the values of $ x\_i $ are going to be different in the two problem but is there a proof/disproof that the values of $ y $ are going to be the same ?



Prove that the following two optimization problems are equivalent
The optimal values of the two optimization problems are not identical. Consider for example $J=3$, $A\_1=\begin{pmatrix} 1 \\0 \end{pmatrix}$ and $A\_2=A\_3=\begin{pmatrix} 0 \\1 \end{pmatrix}$. The optimal values for the first optimization problem are $y=\frac{1}{\sqrt{5}} \begin{pmatrix}\pm 1\\\pm 2\end{pmatrix}$. The optimal values for the second optimization problem however are $y=\pm \begin{pmatrix}0\\ 1\end{pmatrix}$.


