Mathematical Programs with Equilibrium Constraints: Solution Techniques from Parametric Optimization

Equilibrium constrained problems form a special class of mathematical programs where the decision variables satisfy a finite number of constraints together with an equilibrium condition. Optimization problems with a variational inequality constraint, bilevel problems and semi-infinite programs can be seen as particular cases of equilibrium constrained problems. Such models appear in many practical applications. Equilibrium constraint problems can be written in bilevel form with possibly a finite number of extra inequality constraints. This opens the way to solve these programs by applying the so-called Karush-Kuhn-Tucker approach. Here the lower level problem of the bilevel program is replaced by the Karush-Kuhn-Tucker condition, leading to a mathematical program with complementarity constraints (MPCC). Unfortunately, MPCC problems cannot be solved by classical algorithms since they do not satisfy the standard regularity conditions. To solve MPCCs one has tried to conceive appropriate modifications of standard methods. For example sequential quadratic programming, penalty algorithms, regularization and smoothing approaches. The aim of this thesis is twofold. First, as a basis, MPCC problems will be investigated from a structural and generical viewpoint. We concentrate on a special parametric smoothing approach to solve these programs. The convergence behavior of this method is studied in detail. Although the smoothing approach is widely used, our results on existence of solutions and on the rate of convergence are new. We also derive (for the first time) genericity results for the set of minimizers (generalized critical points) for one-parametric MPCC. In a second part we will consider the MPCC problem obtained by applying the KKT-approach to equilibrium constrained programs and bilevel problems. We will analyze the generic structure of the resulting MPCC programs and adapt the related smoothing method to these particular cases. All corresponding results are new.