Preconditioned SQP solvers for nonlinear optimization problems with partial differential equations

偏微分方程（PDE）是描述自然科学和工程中无数现象的语言。这些过程的优化，或未知模型参数的识别，导致偏微分方程的优化问题。序列二次规划（SQP）算法是求解这类非线性问题的强大而广泛使用的方法。SQP算法的整体有效性取决于其全局和局部收敛性，以及在每次迭代中二次规划（QP）子问题的快速解。在建议的项目中，我们将研究一类新的预处理器，特别适合于有效的解决方案的子问题发生在流行的复合步骤信赖域SQP方法。因此，我们的研究将导致一个新的分支的预处理矩阵的非线性大规模优化问题的SQP求解器。由非线性，耦合和时间依赖的偏微分方程控制的并行应用将展示这些方法的潜力和局限性。

Partial differential equations (PDEs) are the language to describe countless phenomena in the natural sciences and engineering. The optimization of such processes, or the identification of unknown model parameters, lead to optimization problems with PDEs. Sequential quadratic programming (SQP) algorithms are powerful and widely used solution methods for nonlinear problems of this type. The overall effectiveness of an SQP algorithm depends on its global and local convergence properties, as well as on the fast solution of the quadratic programming (QP) subproblem in every iteration. In the proposed project, we will investigate a new class of preconditioners which are especially well suited for the efficient solution of the subproblems occuring in the popular composite-step trust-region SQP methods. Our research will thus lead to a new branch of preconditioned matrix-free SQP solvers for nonlinear large-scale optimization problems. Challenging applications governed by nonlinear, coupled and time-dependent PDEs will demonstrate the potential and limitations of these methods.