Mathematical programs with vanishing constraints: critical point theory

We study mathematical programs with vanishing constraints (MPVCs) from a topological point of view. We introduce the new concept of a T‐stationary point for MPVC. Under the Linear Independence Constraint Qualification we derive an equivariant Morse Lemma at nondegenerate T‐stationary points. Then, two basic theorems from Morse Theory (deformation theorem and cell‐attachment theorem) are proved. Outside the T‐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 T‐stationary level, the topology of the lower level set changes via the attachment of a q‐dimensional cell. The dimension q equals the stationary T‐index of the (nondegenerate) T‐stationary point. The stationary T‐index depends on both the restricted Hessian of the Lagrangian and the number of bi‐active vanishing constraints. Further, we prove that all T‐stationary points are generically nondegenerate. The latter property is shown to be stable under C ²‐perturbations of the defining functions. Finally, some relations with other stationarity concepts, such as strong, weak, and M‐stationarity, are discussed.