We investigate the application of full discretization and a nonsmooth Newton method to large-scale optimal control problems. Based on a first discretize, then optimize approach, we discretize the state and control variables in time following a collocation method. Then, a nonsmooth Newton method combined with a line search globalization strategy is used to find a solution to the resulting finite-dimensional nonlinear optimization problem. In order to reduce the computational effort of solving the linear systems that arise from the application of the nonsmooth Newton method, we propose a structure exploitation strategy that results in a sparse banded matrix. We propose as well a substructure exploitation strategy based on a block LU decomposition. The different exploitation strategies combined with the use of appropriate linear solvers are demonstrated and compared for a quadratic 2D heat equation control problem discretized with the method of lines, and the approach that proved to be the most efficient is applied to a nonlinear version of the problem.

我们研究了完全离散化和非光滑牛顿法在大规模最优控制问题中的应用。基于先离散化，然后优化方法，我们按照搭配方法及时离散状态和控制变量。然后，将非光滑牛顿法与线搜索全球化策略相结合，对由此产生的有限维非线性优化问题求解。为了减少求解因应用非光滑牛顿法而产生的线性系统的计算工作量，我们提出了一种结构开发策略，可产生稀疏带状矩阵。我们还提出了一种基于块 LU 分解的子结构开发策略。针对使用线的方法离散的二次 2D 热方程控制问题，演示并比较了不同的开发策略与适当线性求解器的使用相结合，