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This paper investigates quantile regression in the presence of non-convex andnon-smooth sparse penalties, such as the minimax concave penalty (MCP) andsmoothly clipped absolute deviation (SCAD). The non-smooth and non-convexnature of these problems often leads to convergence difficulties for manyalgorithms. While iterative techniques like coordinate descent and local linearapproximation can facilitate convergence, the process is often slow. Thissluggish pace is primarily due to the need to run these approximationtechniques until full convergence at each step, a requirement we term as a\emph{secondary convergence iteration}. To accelerate the convergence speed, weemploy the alternating direction method of multipliers (ADMM) and introduce anovel single-loop smoothing ADMM algorithm with an increasing penaltyparameter, named SIAD, specifically tailored for sparse-penalized quantileregression. We first delve into the convergence properties of the proposed SIADalgorithm and establish the necessary conditions for convergence.Theoretically, we confirm a convergence rate of $o\big({k^{-\frac{1}{4}}}\big)$for the sub-gradient bound of augmented Lagrangian. Subsequently, we providenumerical results to showcase the effectiveness of the SIAD algorithm. Ourfindings highlight that the SIAD method outperforms existing approaches,providing a faster and more stable solution for sparse-penalized quantileregression.

Smoothing ADMM for Sparse-Penalized Quantile Regression with Non-Convex Penalties