A Joint Least Squares and Least Absolute Deviation Model

We propose a joint least squares and least absolute deviations (JOLESALAD) model, show that the proposed model can cover least absolute shrinkage and selection operator (LASSO) and two of its variants, namely the generalized LASSO (gLASSO) and the constrained LASSO (cLASSO), and prove that under a full rank condition, the JOLESALAD can be transformed into cLASSO. Based on this equivalency, rich tools currently available for LASSO models can be applied to solve gLASSO and cLASSO. As an example of illustration, we demonstrate an application of the proposed model to the restoration of a noisy ramp signal that needs proper use of penalty term.