  We introduce a framework based on Rockafellar's perturbation theory to
analyze and solve general nonsmooth convex minimization and monotone inclusion
problems involving nonlinearly composed functions as well as linear
compositions. Such problems have been investigated only from a primal
perspective and only for nonlinear compositions of smooth functions in
finite-dimensional spaces in the absence of linear compositions. In the context
of Banach spaces, the proposed perturbation analysis serves as a foundation for
the construction of a dual problem and of a maximally monotone Kuhn--Tucker
operator which is decomposable as the sum of simpler monotone operators. In the
Hilbertian setting, this decomposition leads to a block-iterative primal-dual
algorithm that fully splits all the components of the problem and appears to be
the first proximal splitting algorithm for handling nonlinear composite
problems. Various applications are discussed.
