We consider the problem of synthesizing feasible signals in a Hilbert space in the presence of inconsistent convex constraints, some of which must imperatively be satisfied. This problem is formalized as that of minimizing a convex objective measuring the amount of violation of the soft constraints over the intersection of the sets associated with the hard ones. The resulting convex optimization problem is analyzed, and numerical solution schemes are presented along with convergence results. The proposed formalism and its algorithmic framework unify and extend existing approaches to inconsistent signal feasibility problems. An application to signal synthesis is demonstrated