In recent years, the sparse and group sparse optimization problem has attracted extensive attention due to its wide applications in statistics, bioinformatics, signal interpretation and machine learning, which yields the sparsity both in group-wise and element-wise. In this paper, the sparse and group sparse optimization problem with a nonsmooth loss function is considered, where the sparsity and group sparsity are induced by a penalty composed of a combination of \(\ell _1\) norm and \(\ell _{2,1}\) norm, so it is called the sparse group Lasso (SGLasso) problem. To solve this problem, the nonsmooth loss function is smoothed first. Then, based on the smooth approximation of the loss function, a smoothing composite proximal gradient (SCPG) algorithm is proposed. It is showed that any accumulation point of the sequence generated by SCPG algorithm is a global optimal solution of the problem. Moreover, it is proved that the convergence rate of the objective function value is \(O(\frac{1}{k^{1-\sigma }})\) where \(\sigma \in (0.5,1)\) is a constant. Finally, numerical results illustrate that the proposed SCPG algorithm is effective and robust for sparse and group sparse optimization problems. Especially, compared with some popular algorithms, SCPG algorithm has obvious advantages in anti-outlier.
