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We introduce a framework based on Rockafellar's perturbation theory toanalyze and solve general nonsmooth convex minimization and monotone inclusionproblems involving nonlinearly composed functions as well as linearcompositions. Such problems have been investigated only from a primalperspective and only for nonlinear compositions of smooth functions infinite-dimensional spaces in the absence of linear compositions. In the contextof Banach spaces, the proposed perturbation analysis serves as a foundation forthe construction of a dual problem and of a maximally monotone Kuhn--Tuckeroperator which is decomposable as the sum of simpler monotone operators. In theHilbertian setting, this decomposition leads to a block-iterative primal-dualalgorithm that fully splits all the components of the problem and appears to bethe first proximal splitting algorithm for handling nonlinear compositeproblems. Various applications are discussed.

A Perturbation Framework for Convex Minimization and Monotone Inclusion Problems with Nonlinear Compositions