  Linear Mixed-Effects (LME) models are a fundamental tool for modeling
correlated data, including cohort studies, longitudinal data analysis, and
meta-analysis. Design and analysis of variable selection methods for LMEs is
more difficult than for linear regression because LME models are nonlinear. In
this work we propose a relaxation strategy and optimization methods that enable
a wide range of variable selection methods for LMEs using both convex and
nonconvex regularizers, including $\ell_1$, Adaptive-$\ell_1$, CAD, and
$\ell_0$. The computational framework only requires the proximal operator for
each regularizer to be available, and the implementation is available in an
open source python package pysr3, consistent with the sklearn standard. The
numerical results on simulated data sets indicate that the proposed strategy
improves on the state of the art for both accuracy and compute time. The
variable selection techniques are also validated on a real example using a data
set on bullying victimization.
