Orthogonal Projection onto a Polyhedron (Matrix Inequality)

Question

How to efficiently solve:

$$\begin{align*} \arg \min_{\boldsymbol{X}} \quad & \frac{1}{2} {\left\| \boldsymbol{x} - \boldsymbol{y} \right\|}_{2}^{2} \\ \text{subject to} \quad & \begin{aligned} \boldsymbol{A} \boldsymbol{x} & \leq \boldsymbol{b} \end{aligned} \end{align*}$$

Related to How to find projection to polyhedron which is stated the same yet not actually a projection (Or projection of the $\boldsymbol{0}$ vector).

Answer 1

Since each inequality $ \boldsymbol{a}_{i}^{T} \boldsymbol{x} \leq {b}_{i} $ is not a sub space it should be solved by Orthogonal Projection onto the Intersection of Convex Sets.

Each projection is given by Orthogonal Projection onto a Half Space.

The full code is available on my StackExchange Mathematics GitHub Repository (Look at the Mathematics\Q4938099 folder).