Periodic systems are widely used in separation processes and in reaction engineering. They are designed for and operated at a cyclic steady state (CSS). Identifying and optimizing the CSS has proven to be computationally challenging. A novel framework for equation-oriented simulation and optimization of cyclic processes is introduced. A two-step reformulation of the process model is proposed, comprising, (1) a full discretization of the time and spatial domains and (2) recasting the discretized model as a differential-algebraic equation system, for which theoretical stability guarantees are provided. Additionally, a mathematical, structural connection between the CSS constraints and material recycling is established, which allows us to deal with these conditions via a "tearing" procedure. These developments are integrated in a pseudo-transient design optimization framework and two extensive case studies are presented: a simulated moving bed chromatography system and a pressure swing adsorption process. (C) 2017 American Institute of Chemical Engineers