Strong NP-Hardness for Sparse Optimization with Concave Penalty Functions

We show that finding a global optimal solution for the regularized Lq-minimization problem (q ≥ 1) is strongly NP-hard if the penalty function is concave but not linear in a neighborhood of zero and satisfies a very mild technical condition. This implies that it is impossible to have a fully polynomial-time approximation scheme (FPTAS) for such problems unless P = NP. This result clarifies the complexity for a large class of regularized optimization problems recently studied in the literature.