Interior point methods provide an attractive class of approaches for solving linear, quadratic and nonlinear programming problems, due to their excellent efficiency and wide applicability. In this paper, we consider PDE-constrained optimization problems with bound constraints on the state and control variables, and their representation on the discrete level as quadratic programming problems. To tackle complex problems and achieve high accuracy in the solution, one is required to solve matrix systems of huge scale resulting from Newton iteration, and hence fast and robust methods for these systems are required. We present preconditioned iterative techniques for solving a number of these problems using Krylov subspace methods, considering in what circumstances one may predict rapid convergence of the solvers in theory, as well as the solutions observed from practical computations.

中文翻译：

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偏微分方程约束优化引起的二次规划问题的快速内点解

内点方法由于其出色的效率和广泛的适用性，为解决线性、二次和非线性规划问题提供了一类有吸引力的方法。在本文中，我们考虑 PDE 约束优化问题，在状态和控制变量上具有边界约束，并将它们在离散级别上表示为二次规划问题。为了解决复杂的问题并在解决方案中实现高精度，需要解决由牛顿迭代产生的大规模矩阵系统，因此需要针对这些系统的快速而稳健的方法。我们提出了使用 Krylov 子空间方法解决许多这些问题的预处理迭代技术，考虑在什么情况下可以在理论上预测求解器的快速收敛，