  In this paper, we extend and investigate the properties of the semi-smooth
Newton method when applied to a general projection equation in finite
dimensional spaces. We first present results concerning Clarke's generalized
Jacobian of the projection onto a closed and convex cone. We then describe the
iterative process for the general cone case and establish two convergence
theorems. We apply these results to the constrained quadratic conic programming
problem, emphasizing its connection to the projection equation. To illustrate
the performance of our method, we conduct numerical experiments focusing on
semidefinite least squares, in particular the nearest correlation matrix
problem. In the latter scenario, we benchmark our outcomes against previous
literature, presenting performance profiles and tabulated results for clarity
and comparison.
