This paper investigates quantile regression in the presence of non-convex and non-smooth sparse penalties, such as the minimax concave penalty (MCP) and smoothly clipped absolute deviation (SCAD). The non-smooth and non-convex nature of these problems often leads to convergence difficulties for many algorithms. While iterative techniques like coordinate descent and local linear approximation can facilitate convergence, the process is often slow. This sluggish pace is primarily due to the need to run these approximation techniques until full convergence at each step, a requirement we term as a secondary convergence iteration. To accelerate the convergence speed, we employ the alternating direction method of multipliers (ADMM) and introduce a novel single-loop smoothing ADMM algorithm with an increasing penalty parameter, named SIAD, specifically tailored for sparse-penalized quantile regression. We first delve into the convergence properties of the proposed SIAD algorithm and establish the necessary conditions for convergence. Theoretically, we confirm a convergence rate of o k − 1 4 for the sub-gradient bound of augmented Lagrangian. Subsequently, we provide numerical results to showcase the effectiveness of the SIAD algorithm. Our findings highlight that the SIAD method outperforms existing approaches, providing a faster and more stable solution for sparse-penalized quantile regression.
This paper proposes an alternating direction method of multiplier (ADMM) based algorithm for solving the sparse robust phase retrieval with non-convex and non-smooth sparse penalties, such as minimax concave penalty (MCP). The accuracy of the robust phase retrieval, which employs an l1 based estimator to handle outliers, can be improved in a sparse situation by adding a non-convex and non-smooth penalty function, such as MCP, which can provide sparsity with a low bias effect. This problem can be effectively solved using a novel proximal ADMM algorithm, and under mild conditions, the algorithm is shown to converge to a stationary point. Several simulation results are presented to verify the accuracy and efficiency of the proposed approach compared to existing methods.
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