After a model my problem I found a rectangular linear system : $$Ax=b$$ I can easely solve it with a least square with QR/SVD... But the model include constrains for each solution $x_i$, the $\vec{x}$ have n elements. ($1\leqslant i\leqslant n$)
For each $x_i$ I have a min and max: $$ x_i \geqslant min_i $$ $$ x_i \leqslant max_i $$ And exist some couple constraint with $(i,j)$ as: $$ (i,j) \in \left (\frac{\mathbb{Z}}{n\mathbb{Z}} \right )^2 | i \neq j $$ $i$ and $j$ is on $\mathbb{Z}$ and is included on $[1,n]$
I have 2 others kinds of constraints $L^p_{ij}$ and $E^q_{ij}$ on "99.9%" of my cases : $$ L^p_{ij} = L^p_{ji} $$ $$ E^q_{ij} = E^q_{ji} $$ I have N ($1 \leqslant p \leqslant N$) contrains L and K ($1 \leqslant q \leqslant K$) constains E.
$$ L^p_{ij} : x_i + x_j = 1 \space \mathbf{or} \space 0 $$ $$E^p_{ij} : x_i > 0 \Rightarrow x_j = 0$$ I can rewrite $E^q_{ij}$ like : $$ \begin{matrix} x_i+x_j=\max(x_i,x_j) \Rightarrow \\ x_i + x_j = \frac{1}{2}\left(x_i + x_j + \left | x_i - x_j \right |\right) \Rightarrow \\ x_i + x_j = \frac{1}{2}\left (x_i + x_j + \sqrt{ (x_i - x_j)^2 }\right ) \end{matrix} $$
I search a way to minimize $\|Ax=b\|$ with my $min_i$, $max_i$, $L^p_{ij}$, $E^q_{ij}$. All $x_i$ have one $min_i$ and $max_i$, but for $L^p_{ij}$ and $E^q_{ij}$ I have no control on number or "topology" constrains.
As you see I have an $\mathbf{or}$ on $L^p_{ij}$ and the derivation of $E^q_{ij}$ is not continue so I can't simply use the Lagrange Multiplier.
Any comment or solutions are welcome.
Thanks
$Ax = b$ with bounds on $x$ is dealt with in linear programming. Your $L$ and $E$ constraints will take you into mixed integer linear programming.