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Optimality Conditions for Cardinality-Constrained Programs and a SCA Method

We study a cardinality-constrained optimization problem with nonnegative variables in this paper. This problem is often encountered in practice. 
Firstly we study some properties on the optimal solutions of this optimization problem under some conditions. An equivalent reformulation of the problem under consideration is proposed. Based on the reformulation, we present a successive convex approximation method for the cardinality constrained optimization problem. We prove that the method converges to a KKT point of the reformulation problem. Under some conditions, the KKT points of the reformulation problem are local optimizers of the original problem. Our numerical results on a limited diversified mean-variance portfolio selection problem demonstrate some promising results.