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.

具有嵌入式优化标准 (DAEO) 的微分代数方程系统出现在许多应用中，最突出的是基于相平衡的分离过程动态模型和通过动态通量平衡分析方法建模的微生物转化过程。

我们考虑 DAEO 的优化。为此，我们首先使用产生非光滑 DAE 系统的一阶最优条件重新制定 DAEO。该解决方案方法跟踪嵌入式优化问题的驻点，该驻点在非凸问题的总时间范围内可能不是全局或局部最优的。

我们对产生的非平滑 DAE 使用伴随灵敏度分析，以在直接单次射击方法中进行基于梯度的优化。我们将求解方法应用于单个闪蒸单元的优化、精馏塔最佳启动方案的计算以及微生物转化过程中生物质生长的优化。