Bilevel Optimal Control: Theory, Algorithms, and Applications

具有两个决策层且至少有一个决策者需要解决最优控制问题的分层优化问题被称为双层最优控制问题（bocp）。这种结构的模型通常来自于与能源市场定价、过程控制中的参数估计或数据压缩相关的实际应用。bocp本质上是非光滑的、无限维的程序，具有隐含的约束，受到固有的不规则性的影响。这使得这个问题类相当具有挑战性。在这个后续项目中，我们计划深入分析偏微分方程的bocp，重点关注潜在的最优性条件和求解算法。因此，我们将利用层次模型的两种不同的单级替代品：利用较低级参数优化问题的最优值函数的最优值变换和用一阶最优性条件取代较低级问题的Karush-Kuhn-Tucker- (KKT-)变换。利用最优值函数的适当上估计（无论该函数是凸还是凹），迭代改进最优值变换的松弛替代问题可行集的算法是可以想象的。在SPP的第一阶段，我们导出了在全凸数据情况下的这种类型的解方法。在这里，得到的最优值函数的凸性是必不可少的。现在，我们要研究最优值函数是凹的情况，这在参数重构的背景下是很自然的。此外，我们将进行一些数值分析，以证明在适当的假设下，我们的算法在离散阶段计算的解收敛于函数空间设置中的解。注意到具有低层次不等式约束的BOCP的kkt变换是函数空间中的互补约束优化问题（MPCC），后一类问题将被仔细研究。我们计划基于点特征以及相关的约束条件推导出新的问题定制的平稳性概念。进一步，导出了后一类问题的二阶充分最优性条件。我们的目的是构造有限维和无限维mpcc数值解的活动集方法。最后，我们希望将我们所有的发现应用于双层编程的原型应用，即参数估计问题以及使用双层测量方法的数据压缩问题。这些模型将从最优性条件和求解算法的角度进行研究。我们希望建立一个相应的基准问题集合，以便将推导出的理论结果与数值方法进行比较。

Hierarchical optimization problems with two decision levels where at least one decision maker has to solve an optimal control problem are referred to as bilevel optimal control problems (BOCPs). Models of this structure typically arise from real-world applications which are related to e.g. pricing in energy markets, parameter estimation in process control, or data compression. BOCPs are inherently nonsmooth, infinite-dimensional programs with implicit constraints that suffer from inherent irregularity. This makes this problem class rather challenging.In this follow-up project, we plan to deepen our analysis on BOCPs of partial differential equations with a focus on potential optimality conditions and solution algorithms. Therefore, we are going to exploit two different single-level surrogates of the hierarchical model: The optimal-value-transformation which exploits the optimal value function of the lower level parametric optimization problem and the Karush-Kuhn-Tucker- (KKT-) transformation which replaces the lower level problem by first-order optimality conditions. Using appropriate upper estimates of the optimal value function which are available whenever this function is convex or concave, algorithms which iteratively refine the feasible set of relaxed surrogate problems of the optimal-value-transformation are imaginable. In the SPP's first stage, we derived a solution method of this type in the case of fully convex data. Here, the resulting convexity of the optimal value function was essential. Now, we want to investigate the situation where the optimal value function is concave which is natural in the context of parameter reconstruction. Furthermore, we are going to perform some numerical analysis in order to show that the solutions computed by our algorithms on the discretized stage converge to solutions in the function space setting under appropriate assumptions.Noting that the KKT-transformation of a BOCP with lower level inequality constraints is a complementarity-constrained optimization problem (MPCC) in function spaces, the latter problem class will be investigated carefully. We plan to derive new problem-tailored stationarity notions based on pointwise characterizations as well as associated constraint qualifications. Furthermore, second-order sufficient optimality conditions for the latter problem class will be derived. We aim for the construction of an active set method for the numerical solution of MPCCs in finite and infinite dimensions. Finally, we want to apply all our findings to prototypical applications from bilevel programming, namely parameter estimation problems as well as a problem of data compression using a bilevel measure approach. These models will be investigated from the viewpoint of optimality conditions and solution algorithms. We want to set up a corresponding collection of benchmark problems which allows a comparison of the derived theoretical results and numerical methods.