I am trying to find the closest point on the following constrained hyperplane to a general point $\vec x$ :
$$ \vec \omega \!\cdot\! \vec 1 = 1 \ \ s.t \ \ \alpha_i \le\omega_i\leq\beta_i $$
$$ 0\leq\alpha_i\lt\beta_i\leq1$$
I have projected $\vec x$ onto the plane.
Next, I think I should find the midpoint of the constrained hyperplane and perform an iterative loop (am writing code) to get as close to the center until all the inequalities are satisfied.
Will this work? , and how can I find the midpoint of the constrained hyperplane?
I can find the midpoint in 2d and 3d but I can't see how it generalizes to higher dimensions.
