Traditional optimisation theory and -methods on the basis of the
Lagrangian function do not apply to objective or constraint functions
which are defined by means of a combinatorial selection structure. Such
selection structures can be explicit, for example in the case of "min",
"max" or "if" statements in function evaluations, or implicit as in the
case of inverse optimisation problems where the combinatorial structure is
induced by the possible selections of active constraints. The resulting
optimisation problems are typically neither convex nor smooth and do not
fit into the standard framework of nonlinear optimisation. Users typically
treat these problems either through a mixed-integer reformulation, which
drastically reduces the size of tractable problems, or by employing
nonsmooth optimisation methods, such as bundle methods, which are
typically based on convex models and therefore only allow for weak
convergence results. In this talk we argue that the classical Lagrangian
theory and SQP methodology can be extended to a fairly general class of
nonlinear programs with combinatorial constraints. The paper is available