Many works on inverse problems in the imaging sciences consider regularization via one or more penalty functions or constraint sets. When the models/images are not easily described using one or a few penalty functions/constraints, additive model descriptions for regularization lead to better imaging results... These include cartoon-texture decomposition, morphological component analysis, and robust principal component analysis; methods that typically rely on penalty functions. We propose a regularization framework, based on the Minkowski set, that merges the strengths of additive models and constrained formulations. We generalize the Minkowski set, such that the model parameters are the sum of two components, each of which is constrained to an intersection of sets. Furthermore, the sum of the components is also an element of another intersection of sets. These generalizations allow us to include multiple pieces of prior knowledge on each of the components, as well as on the sum of components, which is necessary to ensure physical feasibility of partial-differential-equation based parameters estimation problems. We derive the projection operation onto the generalized Minkowski sets and construct an algorithm based on the alternating direction method of multipliers. We illustrate how we benefit from using more prior knowledge in the form of the generalized Minkowski set using seismic waveform inversion and video background-anomaly separation. (read more)