Consider the equation $$\pmb{c}+\text{sign}(G\pmb{c}) = L$$
$\pmb{c}$ is a $n\times1$ matrix.
$G$ is a $n\times n$ matrix which is also positive definite.
matrices $G$ and $c$ are real.
$L$ is a $n\times 1$ matrix whose entries are from the set $\{-1,1\}$.
Can this equation be solved for the matrix $\pmb{c}$? Suppose there is no signum function there, then the solution is $$\pmb{c} = (G+I_n)^{-1}L$$ and as $G$ is positive definite, there exists a unique solution. But with the presence of the $\text{sign}$ function, the problem doesn't seem to belong to linear algebra. I request for a reference to any subject or book for this type of equations. Does this belong to linear programming?(I don't know anything about it, so I am hoping it is related).
PS: please tag appropriately.
PS 2: $\text{sign}([a_{i,j}]_{m\times n}) = [\text{sign}(a_{i,j})]_{m\times n}$.
There are $3^n$ possible choices for the vector $h=\mathop{\rm sign}(Gc)$, since each entry must be in $\{-1,0,1\}$. This gives $3^n$ possible values for $c=L-h$, and one can simply check each such $c$ to see if $\mathop{\rm sign}(Gc)=h$.