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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.