The second volume of the Journal of Nonsmooth Analysis and Optimization (2021)
Hegerhorst-Schultchen, Lisa C. ; Kirches, Christian ; Steinbach, Marc C..
This work is part of an ongoing effort of comparing non-smooth optimization
problems in abs-normal form to MPCCs. We study the general abs-normal NLP with
equality and inequality constraints in relation to an equivalent MPCC
reformulation. We show that kink qualifications and MPCC constraint
qualifications of linear independence type and Mangasarian-Fromovitz type are
equivalent. Then we consider strong stationarity concepts with first and second
order optimality conditions, which again turn out to be equivalent for the two
problem classes. Throughout we also consider specific slack reformulations
suggested in [9], which preserve constraint qualifications of linear
independence type but not of Mangasarian-Fromovitz type.
Hegerhorst-Schultchen, Lisa C. ; Kirches, Christian ; Steinbach, Marc C..
This work continues an ongoing effort to compare non-smooth optimization
problems in abs-normal form to Mathematical Programs with Complementarity
Constraints (MPCCs). We study general Nonlinear Programs with equality and
inequality constraints in abs-normal form, so-called Abs-Normal NLPs, and their
relation to equivalent MPCC reformulations. We introduce the concepts of
Abadie's and Guignard's kink qualification and prove relations to MPCC-ACQ and
MPCC-GCQ for the counterpart MPCC formulations. Due to non-uniqueness of a
specific slack reformulation suggested in [10], the relations are non-trivial.
It turns out that constraint qualifications of Abadie type are preserved. We
also prove the weaker result that equivalence of Guginard's (and Abadie's)
constraint qualifications for all branch problems hold, while the question of
GCQ preservation remains open. Finally, we introduce M-stationarity and
B-stationarity concepts for abs-normal NLPs and prove first order optimality
conditions corresponding to MPCC counterpart formulations.