We consider cardinality-constrained optimization problems (CCOPs), which are general nonlinear programs with an additional constraint limiting the number of nonzero continuous variables.
The continuous reformulation of CCOPs involves complementarity constraints, which pose significant theoretical and computational challenges.
To address these difficulties, we propose and analyze two numerical solution approaches: a general penalty method and a general regularization method. Both approaches generate a sequence of easier to solve problems, and we show convergence of the corresponding KKT points against an M-stationary point under CC-MFCQ.
Both methods rely on structural properties of the penalty and regularization functions, which we introduce and illustrate with examples.
Finally, we present comprehensive numerical experiments to assess the practical performance of the proposed methods and to compare them with established approaches.