MPCC: Critical Point Theory

We study mathematical programs with complementarity constraints (MPCC) from a topological point of view. Special focus will be on C-stationary points. Under the linear independence constraint qualification (LICQ) we derive an equivariant Morse lemma at nondegenerate C-stationary points. Then, two basic theorems from Morse theory (deformation theorem and cell-attachment theorem) are proved. Outside the C-stationary point set, continuous deformation of lower level sets can be performed. As a consequence, the topological data (such as the number of connected components) then remain invariant. However, when passing a C-stationary level, the topology of the lower level set changes via the attachment of a q-dimensional cell. The dimension q equals the stationary C-index of the (nondegenerate) C-stationary point. The stationary C-index depends on both the restricted Hessian of the Lagrangian and the Lagrange multipliers related to biactive complementarity constraints. Finally, some relations with other stationarity concepts, such as W-, A-, M-, S-, and B-stationarity, are discussed.