Extended ADMM for general penalized quantile regression with linear constraints in big data

Abstract: 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.