Jiri Outrata, Institute of Information Theory and Automation, Prague
Abstract:
The aim of the lecture is to provide deeper insight into an important class of optimization problems which have been extensively studied especially in the past fifteen years. Our current model has grown up from Stackelberg games and encompasses a considerable number of mathematical programs possessing an equilibrium among the constraints. If this equilibrium can be described by a variational inequality or generalized equation, we speak about mathematical programs with equilibrium constraints and use the acronym MPEC. Our main attention is focused on optimality conditions, on two selected numerical approaches and on some research perspectives. The theory of optimality conditions for MPECs has benefited a lot from the recent developments of modern variational analysis. Conversely, theoretical questions arising in connection with MPECs have provided incentives to deep studies of calculus rules for nonsmooth and set-valued mappings. In the talk we employ above all the generalized differential calculus of B. Mordukhovich. This powerful tool enables us to develop optimality conditions for an MPEC in a very general form and to specialize these conditions to various concrete equilibria. The second part of the talk is devoted to two successful numerical approaches which can be applied in sufficiently large classes of MPECs. It is the NLP approach tailored to equilibria governed by complementarity problems and the implicit programming approach. The latter is illustrated by an MPEC in which one controls a unilateral contact problem obeying the Coulomb friction law. The last part of the lecture deals with possible topics of further research in this area. The attention is paid to a class of evolutionary equilibria and to variational inequalities with nonpolyhedral constraint sets.