Numerical Methods for Mixed-Integer Optimal Control with Combinatorial Constraints
Numerical Methods for Mixed-Integer Optimal Control with Combinatorial Constraints
复制标题
具有组合约束的混合整数最优控制的数值方法
DOI：
10.11588/heidok.00024070
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发表时间：
2018
期刊：
影响因子：
--
通讯作者：
F. Lenders
中科院分区：
文献类型：
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作者：
F. Lenders
本文研究了组合约束下混合最优控制问题的数值方法。本文建立了一个将组合约束下的混合凸最优控制问题与具有消失约束的连续松弛凸最优控制问题联系起来的逼近定理，为数值计算提供了基础。我们开发了一个关于四舍五入算法的消失约束来利用这种对应关系。 带消失约束的最优控制问题的直接离散化产生了一类带平衡约束的数学规划。具有平衡约束的数学规划由于其固有的非凸性和非光滑性而成为一类具有挑战性的问题。本文提出了一种求解平衡约束数学规划问题的有效集算法，并在适当的技术条件下证明了该算法的全局收敛性。 为了有效地计算最优控制问题的牛顿型步，我们建立了Hilbert空间中信赖域问题的广义Lanczos方法。为了保证跟踪型拉格朗日目标在线最优控制应用的实时可行性，我们开发了一个高斯-牛顿预条件的信赖域问题的迭代求解方法。 我们实现了所提出的方法，并证明了它们的适用性和有效性的几个基准问题。
This thesis is concerned with numerical methods for Mixed-Integer Optimal Control Problems with Combinatorial Constraints. We establish an approximation theorem relating a Mixed-Integer Optimal Control Problem with Combinatorial Constraints to a continuous relaxed convexified Optimal Control Problems with Vanishing Constraints that provides the basis for numerical computations. We develop a a Vanishing- Constraint respecting rounding algorithm to exploit this correspondence computationally. Direct Discretization of the Optimal Control Problem with Vanishing Constraints yield a subclass of Mathematical Programs with Equilibrium Constraints. Mathematical Programs with Equilibrium Constraint constitute a class of challenging problems due to their inherent non-convexity and non-smoothness. We develop an active-set algorithm for Mathematical Programs with Equilibrium Constraints and prove global convergence to Bouligand stationary points of this algorithm under suitable technical conditions. For efficient computation of Newton-type steps of Optimal Control Problems, we establish the Generalized Lanczos Method for trust region problems in a Hilbert space context. To ensure real-time feasibility in Online Optimal Control Applications with tracking-type Lagrangian objective, we develop a Gauß-Newton preconditioner for the iterative solution method of the trust region problem. We implement the proposed methods and demonstrate their applicability and efficacy on several benchmark problems.