Suppose we are given two convex polyhedra $\mathcal{C}_1, \mathcal{C}_2 \subset \mathbb{R}^n$ with non-empty intersection $\mathcal{C}_1 \cap \mathcal{C}_2 \neq \emptyset$.
For the orthogonal projection of a point $x\in \mathbb{R}^n$ onto the intersection $\mathcal{C}_1 \cap \mathcal{C}_2$, we would in general have to solve the quadratic program $$\textrm{argmin}_z ||x - z||_2^2 \; s.t. \; z \in \mathcal{C}_1 \cap \mathcal{C}_2 \; . $$
Now, if we additionally know that the point to be projected is already in one of the sets, say $x \in \mathcal{C}_1$, can one use this information to solve the problem more efficiently compared to solving a full quadratic program?
