The second volume of the Journal of Nonsmooth Analysis and Optimization (2021)
Lisa C. Hegerhorst-Schultchen ; Christian Kirches ; Marc C. Steinbach.
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.
Section:
Original research articles
Lisa C. Hegerhorst-Schultchen ; Christian Kirches ; Marc C. Steinbach.
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.
Section:
Original research articles
Nguyen Duy Cuong ; Alexander Y. Kruger.
We propose a unifying general (i.e. not assuming the mapping to have any
particular structure) view on the theory of regularity and clarify the
relationships between the existing primal and dual quantitative sufficient and
necessary conditions including their hierarchy. We expose the typical sequence
of regularity assertions, often hidden in the proofs, and the roles of the
assumptions involved in the assertions, in particular, on the underlying space:
general metric, normed, Banach or Asplund. As a consequence, we formulate
primal and dual conditions for the stability properties of solution mappings to
inclusions
Section:
Original research articles
Matúš Benko.
In this paper, we study continuity and Lipschitzian properties of set-valued
mappings, focusing on inner-type conditions. We introduce new notions of inner
calmness* and, its relaxation, fuzzy inner calmness*. We show that polyhedral
maps enjoy inner calmness* and examine (fuzzy) inner calmness* of a multiplier
mapping associated with constraint systems in depth. Then we utilize these
notions to develop some new rules of generalized differential calculus, mainly
for the primal objects (e.g. tangent cones). In particular, we propose an exact
chain rule for graphical derivatives. We apply these results to compute the
derivatives of the normal cone mapping, essential e.g. for sensitivity analysis
of variational inequalities.
Section:
Original research articles