On the Abadie and Guignard constraint qualifications for Mathematical Programmes with Vanishing Constraints

We consider a special class of optimization problems that we call a Mathematical Programme with Vanishing Constraints. It has a number of important applications in structural and topology optimization, but typically does not satisfy standard constraint qualifications like the linear independence and the Mangasarian–Fromovitz constraint qualification. We therefore investigate the Abadie and Guignard constraint qualifications in more detail. In particular, it follows from our results that also the Abadie constraint qualification is typically not satisfied, whereas the Guignard constraint qualification holds under fairly mild assumptions for our particular class of optimization problems.