Preconditioned SQP solvers for nonlinear optimization problems with partial differential equations

用于偏微分方程非线性优化问题的预处理 SQP 求解器

• 批准号: 215680620
• 负责人: Professor Dr. Roland Herzog
• 金额: --
• 依托单位: Interdisziplinäres Zentrum für Wissenschaftliches Rechnen (IWR)
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2012
• 资助国家: 德国
• 起止时间: 2011-12-31 至 2015-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/215680620?language=en
• 关键词: Preconditioned; SQP; solvers; nonlinear; optimization

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.

偏微分方程（PDE）是描述自然科学和工程中无数现象的语言。此类过程的优化或未知模型参数的识别会导致偏微分方程的优化问题。顺序二次规划 (SQP) 算法是此类非线性问题的强大且广泛使用的求解方法。 SQP 算法的整体有效性取决于其全局和局部收敛特性，以及每次迭代中二次规划 (QP) 子问题的快速求解。在拟议的项目中，我们将研究一类新型预处理器，它们特别适合有效解决流行的复合步骤信赖域 SQP 方法中出现的子问题。因此，我们的研究将带来用于非线性大规模优化问题的预处理无矩阵 SQP 求解器的新分支。由非线性、耦合和瞬态偏微分方程控制的具有挑战性的应用将证明这些方法的潜力和局限性。

项目成果

Preconditioned Solution of State Gradient Constrained Elliptic Optimal Control Problems

• DOI: 10.1137/130948045
• 发表时间: 2016-03
• 期刊: SIAM J. Numer. Anal.
• 作者: R. Herzog;Susann Mach

Superlinear Convergence of Krylov Subspace Methods for Self-Adjoint Problems in Hilbert Space

Hilbert空间自伴问题的Krylov子空间方法的超线性收敛

• DOI: 10.1137/140973050
• 发表时间: 2015
• 期刊: SIAM J. Numer. Anal.
• 作者: Roland Herzog;Ekkehard Sachs
• 通讯作者: Ekkehard Sachs