A strong sequential optimality condition for cardinality-constrained optimization problems

In this paper, we consider the continuous relaxation reformulation of cardinality-constrained optimization problems that has become more popular in recent years and propose a new sequential optimality condition (approximate stationarity) for cardinality-constrained optimization problems, which is proved to be a genuine necessary optimality condition that does not require any constraint qualification to hold. We compare this condition with the rest of the sequential optimality conditions and prove that our condition is stronger and closer to the local minimizer. A problem-tailored regularity condition is proposed, and we show that this regularity condition ensures that the approximate stationary point proposed in this paper is the exact stationary point and is the weakest constraint qualification with this property. Finally, we apply the results of this paper to safeguarded augmented Lagrangian method and prove that the algorithm converges to the approximate stationary point proposed in this paper under mild assumptions, the existing theoretical results of this algorithm are further enhanced.