Robust fused lasso (RFlasso) estimation plays an important role in regression analysis because it can deal with variable selection problems more robust than fused lasso for the case containing non-Gaussian distribution outliers, especially when the noise exhibits a heavy-tailed distribution. The RFlasso with \(\ell _1\) loss function has three non-smooth blocks in objective function and are non-separated which makes it difficult to solve. Thus, in this paper, we first give its dual problem and design a symmetric Gauss-Seidel technique based on the semi-proximal alternating direction method of multipliers (sGS-sPADMM) for solving it. The proposed algorithm can reduce the computational burden by alternating minimizing the variables and ensuring all subproblems have a closed-form solution in simple form. Moreover, the proposed algorithm provides the convergence guarantee for solving this three-block optimization problem. Finally, numerical tests on both simulated and real datasets demonstrate the effectiveness of our proposed algorithm.
