projection onto affine image set

Question

A={$y: y=Ax+b,A\in R^{n\times m}, x\in R^m$}
I want to compute the projection onto A from a point z$\in R^n$,is there a close formulation to do that? Thanks.

Answer 1

A "closed form" solution is available, but I do not know how helpful it is (it requires solving a linear system of equations).

The projection problem is $$ \min_{x, y} \quad \frac{1}{2}\|y - z\|^2 \quad \text{s.t.} \quad y = A x + b $$ Substituting $y$ into the objective function, we obtain the unconstrained linear least-squares problem $$ \min_{x} \|A x + (b - z)\|^2 $$ Least squares problems have a closed form solution which you can find in any text-book, or in Wikipedia. after finding the optimal $x^*$, your projection is $y^* = A x^* + b$. Have fun.