On the Weak Stationarity Conditions for Mathematical Programs with Cardinality Constraints: A Unified Approach

Sequential optimality conditions for cardinality-constrained optimization problems with applications

“…In principle, these AKKT conditions can also be applied to the optimization problem (2.2) by viewing this program as an NLP. But then too many points satisfy the AKKT property, see [22,Thm. 4.1], so that the AKKT conditions turn out to be an optimality condition, which is too weak for this problem (i.e., besides the local minima, many other feasible points satisfy the standard AKKT conditions).…”

“…We stress that our definition is based on the original problem (1.1) in the x-space. The recent report [22] also introduces a sequential optimality condition for cardinality-constrained programs which, however, is essentially based on the reformulated problem (2.2) in the (x, y)-space. Nevertheless, it turns out that our formulation is, in some sense, equivalent to the notion from [22].…”

“…The recent report [22] also introduces a sequential optimality condition for cardinality-constrained programs which, however, is essentially based on the reformulated problem (2.2) in the (x, y)-space. Nevertheless, it turns out that our formulation is, in some sense, equivalent to the notion from [22]. Since this equivalence is not exploited in our subsequent analysis, we discuss the details in an appendix, see Sect.…”

“…In [22], a sequential optimality condition called AW-stationarity was introduced. This condition was based the relaxed reformulation of (1.1) from [12], i.e.…”

“…7. There is also an appendix where we compare our sequential optimality condition with an existing one from [22], which is formulated specifically for the continuous reformulation, see Sect. 1.…”