In this paper, we propose two novel Alternating Direction Method of Multipliers (ADMM) algorithms for the sparse portfolio problem via sorted ℓ1-norm penalization (SLOPE). The first algorithm (FADMM) is presented by adding a prediction-correction step to the classic ADMM framework. Since the problem is not strongly convex, the second fast ADMM (FADMMR) is proposed by utilizing both prediction-correction step and restarting rules. Numerical experiments show that the FADMMR algorithm converges faster than the FADMM algorithm and ADMM algorithm when tuning parameters are relatively small. On the other hand, when tuning parameters are relative large, the FADMM algorithm performs better than the FADMMR algorithm and ADMM algorithm. The FADMM algorithm and FADMMR algorithm converge faster than the ADMM algorithm in terms of convergence time for different sizes of tuning parameters. For large-scale portfolio problem, the proposed algorithms have highly performance as well. Finally, empirical analysis on five datasets of stocks index show that the proposed algorithms are efficient and superior for solving sparse portfolio problems via SLOPE.
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