Project to nearest point in convex polytope
A quadratic program is an optimization problem where the goal is to minimize $y^T Q y + c^T y$ subject to $A y \leq b$. If $Q$ is positive definite, then this is a convex quadratic program and we can solve this problem in polynomial time using several methods, one being the ellipsoid method (originally due to Kozlov, Tarasov and Khachiyan&nbsp;[1]).

We can write $\|y - x\|_2^2$ as $y^Ty - 2x^Ty + x^Tx$. Thus an equivalent problem is
$$
\min_y\ (y^Ty - 2x^Ty) \text{ such that } A y\leq b.
$$
This is a convex quadratic program, since in this case $Q$ is the identity matrix, which is positive definite.

[1] The polynomial solvability of quadratic programming, USSR Computational Mathematics and Mathematical Physics, 20(5):223&ndash;228, 1980; Science Direct