Differential-algebraic equations with embedded optimization criteria (DAEO) are a class of mathematical models for underdetermined differential-algebraic equation (DAE) systems with less algebraic equations than algebraic variables. The algebraic variables may be calculated as the solution of an embedded (non)linear program, yielding a DAEO system. An example for DAEOs is the dynamic flux balance analysis (DFBA) approach, where the formulation of metabolic reaction networks leads to an underdetermined equation system for the intracellular fluxes that are assumed to behave optimally with respect to some cell-specific optimization criterion.

We present a toolbox that allows formulation of DAEOs in the object-oriented Modelica modeling language. The solution method is based on substituting the embedded optimization problem with its first-order Karush-Kuhn-Tucker conditions to obtain a nonsmooth DAE system that can be simulated by a root-finding DAE solver. One nonlinear example and two examples based on DFBA demonstrate the performance of the toolbox.

具有嵌入式优化标准（DAEO）的微分代数方程是一类数学模型，用于欠定的微分代数方程（DAE）系统，其代数方程比代数变量少。可以将代数变量计算为嵌入式（非线性）程序的解，从而得出DAEO系统。DAEO的一个例子是动态通量平衡分析（DFBA）方法，其中代谢反应网络的形成导致细胞内通量的方程组尚未确定，假定该方程组相对于某些特定于细胞的优化标准表现出最佳性能。

我们提供了一个工具箱，可以使用面向对象的Modelica建模语言来编写DAEO。该解决方案方法是将嵌入式优化问题替换为其一阶Karush-Kuhn-Tucker条件，以获得可以由寻根DAE求解器模拟的非平滑DAE系统。一个非线性示例和两个基于DFBA的示例演示了工具箱的性能。