# Find a dual problem with one dual decision variable to the problem of finding the orthogonal projection of a given vector

Given the set $$T_{\alpha}=\{x\in\mathbb{R}^n:\sum x_i=1,0\leq x_i\leq \alpha\}$$
For which $$\alpha$$ the set is non-empty?
Find a dual problem with one dual decision variable to the problem of finding the orthogonal projection of a given vector $$y ∈\mathbb{R}^n$$ onto $$T_α$$.

I found that $$\alpha\geq1/n$$ and when i tried to calculate the dual function I did the following.
$$\min$$ $$||x-y||^2$$
s.t
$$\sum x_i=1$$
$$0\leq x_i\leq\alpha\to X=\{x:0\leq x_i\leq\alpha\}$$
Setting the lagrangian to be $$L(x,\lambda)=||x-y||^2+2\lambda(\sum x_i-1)$$,we would like to minimize the lagrangian w.r.t x then :
$$\underset{x\in X}{\min}L(x,\lambda)=\sum x_i^2+2(\lambda-y_i)x_i+||y||^2-2\lambda$$
$$\frac{\partial L}{\partial x_i}=2x_i+2(\lambda-y_i)=0\to x_i^*=\left\{\begin{array}{rcl} y_i-\lambda&0\leq y_i-\leq\alpha\\ \alpha&y_i-\lambda>\alpha\\ 0&y_i-\lambda<0\end{array}\right.$$
and the dual function is $$q(\lambda)=-2\lambda+||y||^2+\left\{\begin{array}{rcl}n\alpha^2+2n\alpha\lambda-2\alpha\sum y_i&y_i-\lambda>\alpha\\ -\sum [y_i-\lambda]_+^2&\mbox{else}\end{array}\right.$$ and the derivative of $$q$$ is:
$$q'(\lambda)=-2+\left\{\begin{array}{rcl}2n\alpha &y_i-\lambda>\alpha\\ -2\sum[y_i-\lambda]_+^2&\mbox{else}\end{array}\right.$$

I'm not sure if I've found the right dual function and if I've found the right derivative, and I need to write a Matlab code that solves the dual problem and afterwards returns x