Elliptic Mathematical Programs with Equilibrium Constraints (MPECs) in function space: optimality conditions and numerical realization

函数空间中具有平衡约束（MPEC）的椭圆数学规划：最优性条件和数值实现

• 批准号: 132218111
• 负责人: Professor Dr. Michael Hintermüller
• 金额: --
• 依托单位: Forschungsgruppe Nichtglatte Variationsprobleme und Operatorgleichungen
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2009
• 资助国家: 德国
• 起止时间: 2008-12-31 至 2014-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/132218111?language=en
• 关键词: Elliptic; Mathematical; Programs; Equilibrium; Constraints

The project work concentrates on the development of a first and second order optimality theory as well as the design and implementation of efficient solution algorithms for certain classes of mathematical programs with equilibrium constraints (MPECs) in function space. The optimization-theoretic treatment of MPECs is complicated by the degeneracy of the constraint set and the resulting ambiguities in associated concepts for characterizing optimal solutions. In this respect, the project work develops new mathematical technqiues for deriving and categorizing such optimality conditions. Since discretized MPECs result in large scale problems, tailored numerical solution techniques relying on adaptive finite element methods, semismooth Newton and multilevel techniques are developed.The problem class under investigation is of importance as the involved constraints, which are either quasi-variational inequalities or variational inequalities of the second kind, cover a wide range of applications from Bingham fluids or contact with friction, the magnetization of type-II superconductors or torsion problems in plasticity to the ionization in electrostatics. The associated MPEC formulation typically aims at optimally controlling or designing the underlying system. Within the project work these applications will be studied as well.

该项目的工作集中在一阶和二阶最优性理论的发展，以及设计和实施有效的解决方案算法的某些类的数学规划与平衡约束（MPEC）在函数空间。MPECs的优化理论处理是复杂的退化的约束集和由此产生的模糊性在相关的概念来表征最优解。在这方面，项目工作开发了新的数学技术，用于推导和分类这种最优性条件。由于离散化后的MPEC会导致大规模的问题，因此发展了依赖于自适应有限元方法、半光滑牛顿法和多层技术的定制数值求解技术。所研究的问题类是重要的，因为所涉及的约束要么是拟变分不等式，要么是第二类变分不等式，涵盖了从宾汉流体到摩擦接触的广泛应用，第二类超导体的磁化或塑性中的扭转问题到静电学中的电离。相关的MPEC公式通常旨在最佳地控制或设计基础系统。在项目工作中，也将研究这些应用。