Quantile regression (QR) can be used to describe the comprehensive relationship between a response and predictors. Prior domain knowledge and assumptions in application are usually formulated as constraints of parameters to improve the estimation efficiency. This paper develops methods based on multi-block ADMM to fit general penalized QR with linear constraints of regression coefficients. Different formulations to handle the linear constraints and general penalty are explored and compared. The most efficient one has explicit expressions for each parameter and avoids nested-loop iterations in some existing algorithms. Additionally, parallel ADMM algorithm for big data is also developed when data are stored in a distributed fashion. The stopping criterion and convergence of the algorithm are established. Extensive numerical experiments and a real data example demonstrate the computational efficiency of the proposed algorithms. The details of theoretical proofs and different algorithm variations are presented in Appendix.

翻译：分位数回归（QR）可用于描述响应变量与预测变量之间的综合关系。应用中的先验领域知识和假设通常被表述为参数约束，以提高估计效率。本文开发了基于多块ADMM的方法，用于拟合具有回归系数线性约束的一般惩罚分位数回归。探索并比较了处理线性约束和一般惩罚的不同形式。其中最有效的一种形式为每个参数提供了显式表达式，避免了现有某些算法中的嵌套循环迭代。此外，针对分布式存储的数据，还开发了适用于大数据的并行ADMM算法。建立了算法的停止准则和收敛性。大量数值实验和一个实际数据示例证明了所提算法的计算效率。理论证明的细节及不同算法变体见附录。