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Strong NP-Hardness for Sparse Optimization with Concave Penalty Functions
Yichen Chen · Dongdong Ge · Mengdi Wang · Zizhuo Wang · Yinyu Ye · Hao Yin

Sun Aug 06 06:24 PM -- 06:42 PM (PDT) @ Parkside 2
Consider the regularized sparse minimization problem, which involves empirical sums of loss functions for $n$ data points (each of dimension $d$) and a nonconvex sparsity penalty. We prove that finding an $\mathcal{O}(n^{c_1}d^{c_2})$-optimal solution to the regularized sparse optimization problem is strongly NP-hard for any $c_1, c_2\in [0,1)$ such that $c_1+c_2<1$. The result applies to a broad class of loss functions and sparse penalty functions. It suggests that one cannot even approximately solve the sparse optimization problem in polynomial time, unless P $=$ NP.