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Given an affine set $Ax=b$, the Projection operator to this set is $$P(z) = z - A^{T}(AA^{T})^{-1}(Az-b)$$ which is also affine.

How is this derived?

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$P(z)$ solves the problem $$ \min \|z -x\|^2 $$ subject to the constraint $$ Ax=b. $$ The KKT system is a necessary and sufficient optimality condition (why?): $$ Ax = b, \ A^T\mu = x-z. $$ Multiply the second equation by $A$, assume $(AA^T)^{-1}$ being invertible, then solve for $\mu$, then solve for $z$.

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  • $\begingroup$ Thanks for the reply. KKT is necessary and sufficient because strong duality obtains. $\endgroup$ – haripkannan May 16 '14 at 0:50

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