  In this paper, we consider a large class of nonlinear equations derived from
first-order type methods for solving composite optimization problems.
Traditional approaches to establishing superlinear convergence rates of
semismooth Newton-type methods for solving nonlinear equations usually
postulate either nonsingularity of the B-Jacobian or smoothness of the
equation. We investigate the feasibility of both conditions. For the
nonsingularity condition, we present equivalent characterizations in broad
generality, and illustrate that they are easy-to-check criteria for some
examples. For the smoothness condition, we show that it holds locally for a
large class of residual mappings derived from composite optimization problems.
Furthermore, we investigate a relaxed version of the smoothness condition -
smoothness restricted to certain active manifolds. We present a conceptual
algorithm utilizing such structures and prove that it has a superlinear
convergence rate.
