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
