  In this paper, we consider the regularized multi-response regression problem
where there exists some structural relation within the responses and also
between the covariates and a set of modifying variables. To handle this
problem, we propose MADMMplasso, a novel regularized regression method. This
method is able to find covariates and their corresponding interactions, with
some joint association with multiple related responses. We allow the
interaction term between covariate and modifying variable to be included in a
(weak) asymmetrical hierarchical manner by first considering whether the
corresponding covariate main term is in the model. For parameter estimation, we
develop an ADMM algorithm that allows us to implement the overlapping groups in
a simple way. The results from the simulations and analysis of a
pharmacogenomic screen data set show that the proposed method has an advantage
in handling correlated responses and interaction effects, both with respect to
prediction and variable selection performance.

Methodology;Quantitative Methods;Machine Learning