In this work we develop and analyze a semi-smooth Newton method for the general non-
linear conic programming problem. In particular, we study the problem with a generalized
simplicial cone, i.e., the image of a symmetric cone under a linear mapping. We generalize
Robinson’s normal equations to a conic setting, yielding what we call the conic projection equa-
tions. The resulting system is equivalent to the KKT conditions associated with the nonlinear
conic programming problem. A semi-smooth Newton iteration is proposed for solving it, and
local quadratic convergence is established. We study properties of generalized simplicial cones
and prove strong semi-smoothness of the projection operator onto them. Numerical experi-
ments compare the method against a recent smoothing Newton approach on the circular cone
programming problem, and we also apply it to the low-rank matrix completion problem.