Direct single shooting for dynamic optimization of differential-algebraic equation systems with optimization criteria embedded

•Nonsmooth DAE without approximation of complementarity constraints by reformulation.•Embedded NLP is replaced by its KKT conditions yielding a nonsmooth DAE system.•DAE solver provides adjoint sensitivities for optimization of upper-level problem.•Illustrative examples stemming from phase equilibrium assumption and dynamic flux balance analysis approach.•The accompanying software to the article is made available open-source. Differential-algebraic equation systems with embedded optimization criteria (DAEOs) arise in a number of applications, most prominently dynamic models of separation processes based on phase equilibrium and microbial transformation processes modeled via the dynamic flux balance analysis approach. We consider the optimization of DAEOs. To this end, we first reformulate the DAEO using first-order optimality conditions yielding a nonsmooth DAE system. This solution approach tracks a stationary point of the embedded optimization problem that may not be globally or locally optimal on the total time horizon for non-convex problems. We use adjoint sensitivity analysis for the resulting nonsmooth DAEs for gradient-based optimization in a direct single-shooting approach. We apply the solution method to the optimization of a single flash unit, calculation of an optimal start-up scenario of a rectification column, and the optimization of biomass growth in a microbial transformation process.