Solving programmatically a least squares problem with one constrain I need to solve the following problem (preferably in python but any other suggestion is welcome) $$ \min_x||Ax - b||_2 $$ $$ s.t. \: Dx = Dy $$ everything except x is known. $A$ and $D$ are square sparse matrices, $x$,$y$ and $b$ are vectors. From what I understand, without the constrain the problem is solvable using the pseudo-inverse, however I am having trouble incorporating the constrain.

The problem is given by:

$$ \begin{alignat*}{3} \arg \min_{x} & \quad & \frac{1}{2} \left\| A x - b \right\|_{2}^{2} \\\ \text{subject to} & \quad & D x = e \end{alignat*} $$

Where $ e = D y $.

The Lagrangian is given by:

$$ L \left( x, \
u \right) = \frac{1}{2} \left\| A x - b \right\|_{2}^{2} + {\
u}^{T} \left( D x - e \right) $$

From KKT Conditions the optimal values of $ \hat{x}, \hat{\
u} $ obeys:

$$ \begin{bmatrix} {A}^{T} A & {D}^{T} \\\ D & 0 \end{bmatrix} \begin{bmatrix} \hat{x} \\\ \hat{\
u} \end{bmatrix} = \begin{bmatrix} {A}^{T} b \\\ e \end{bmatrix} $$

Now all needed is to solve the above with any Linear System Solver.