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

该项目的工作集中在发展一阶和二阶最优理论，以及设计和实现函数空间中某些类型的带平衡约束的数学规划(MPEC)的有效求解算法。MPEC的最优化理论处理是复杂的，因为约束集的简并性以及由此导致的用于表征最优解的相关概念中的歧义。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条件。由于离散的MPEC会导致大规模的问题，因此发展了基于自适应有限元方法、半光滑牛顿方法和多层技术的数值求解技术。所研究的问题类别具有重要意义，因为所涉及的约束条件可以是拟变分不等式或第二类变分不等式，涵盖了从宾汉流体或摩擦接触、第二类超导体的磁化或塑性扭转问题到静电学中的电离的广泛应用。相关的MPEC方案通常旨在最优地控制或设计基础系统。在项目工作中，还将研究这些应用程序。

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.