I'm having difficulties in proving that the projection of $$(s,y)\in R \times R^{n}$$ onto the second-order cone $$Q^{n+1} = \{(t,x) \in R \times R^n : \|x\|\_2 \leq t \}$$ is $$ \frac{s+\|y\|\_2}{2\|y\|\_2} (\|y\|\_2,y)$$ when $\|y\|\_2 > s $ and $ \|y\|\_2 > -s $.


I tried to first show that to minimize the distance, $x$ must be parallel to $y$, then I construct $$\alpha (\|y\|\_2 + \epsilon,y)$$ and minimize the distance over $\alpha$ and $\epsilon$.


I indeed come up with two quadratic functions individually attains its minimum when $$\epsilon = 0$$ and $$\alpha = \frac{s+\|y\|\_2}{2\|y\|\_2}$$ but I still have $$-\frac{s^2(\|y\|\_2+\epsilon)}{2\|y\|\_2} - \frac{{\|y\|\_2}^3}{2\|y\|\_2 + \epsilon}$$ which attains its maximum when $$\epsilon = 0$$ As a result, I can't say the distance attains its minimum accordingly.
Is there any other method or elementary method to prove the optimal solution is $$ \frac{s+\|y\|\_2}{2\|y\|\_2} (\|y\|\_2,y)$$ when $\|y\|\_2 > s $ and $ \|y\|\_2 > - s $ ?



Projection onto the second-order cone
The proof can be found in [H.H. Bauschke](https://people.ok.ubc.ca/bauschke/)'s 1996 doctoral dissertation: [Projection Algorithms and Monotone Operators (p. 40, Theorem 3.3.6)](http://summit.sfu.ca/system/files/iritems1/7015/b18025766.pdf).


P.S. I wonder what the downvote is for.


