Generalized ℓ1-penalized quantile regression with linear constraints

In many application areas, prior subject matter knowledge can be formulated as constraints on parameters in order to get a more accurate fit. A generalized ℓ1-penalized quantile regression with linear constraints on parameters is considered, including either linear inequality or equality constraints or both. It allows a general form of penalization, including the usual lasso, the fused lasso and the adaptive lasso as special cases. The KKT conditions of the optimization problem are derived and the whole solution path is computed as a function of the tuning parameter. A formula for the number of degrees of freedom is derived, which is used to construct model selection criteria for selecting optimal tuning parameters. Finally, several simulation studies and two real data examples are presented to illustrate the proposed method.