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Nowadays, L0 optimization model has shown its superiority when pursuingsparsity in many areas. For this nonconvex problem, most of the algorithms canonly converge to one of its critical points. In this paper, we consider ageneral L0 regularized minimization problem, where the L0 ''norm'' iscomposited with a continuous map. Under some mild assumptions, we show thatevery critical point of this problem is a local minimizer, which improves theconvergence results of existing algorithms. Surprisingly, this conclusion doesnot hold for low rank minimization, a natural matrix extension of L0 ''norm''of a vector.

Every critical point of an L0 composite minimization problem is a local minimizer