  Local superlinear convergence of the semismooth Newton method usually
requires the uniform invertibility of the generalized Jacobian matrix, e.g.
BD-regularity or CD-regularity. For several types of nonlinear programming and
composite-type optimization problems -- for which the generalized Jacobian of
the stationary equation can be calculated explicitly -- this is characterized
by the strong second-order sufficient condition. However, general
characterizations are still not well understood. In this paper, we propose a
strong second-order sufficient condition (SSOSC) for composite problems whose
nonsmooth part has a generalized conic-quadratic second subderivative. We then
discuss the relationship between the SSOSC and another second order-type
condition that involves the generalized Jacobians of the normal map. In
particular, these two conditions are equivalent under certain structural
assumptions on the generalized Jacobian matrix of the proximity operator. Next,
we verify these structural assumptions for $C^2$-strictly decomposable
functions via analyzing their second-order variational properties under
additional geometric assumptions on the support set of the decomposition pair.
Finally, we show that the SSOSC is further equivalent to the strong metric
regularity condition of the subdifferential, the normal map, and the natural
residual. Counterexamples illustrate the necessity of our assumptions.
