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

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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.