  The generalized minimax concave (GMC) penalty is a nonconvex sparse
regularizer which can preserve the overall-convexity of the regularized
least-squares problem. In this paper, we focus on a significant instance of the
GMC model termed scaled GMC (sGMC), and present various notable findings on its
solution-set geometry and regularization path. Our investigation indicates that
while the sGMC penalty is a nonconvex extension of the LASSO penalty (i.e., the
$\ell_1$-norm), the sGMC model preserves many celebrated properties of the
LASSO model, hence can serve as a less biased surrogate of LASSO without losing
its advantages. Specifically, for a fixed regularization parameter $\lambda$,
we show that the solution-set geometry, solution uniqueness and sparseness of
the sGMC model can be characterized in a similar elegant way to the LASSO model
(see, e.g., Osborne et al. 2000, R. J. Tibshirani 2013). For a varying
$\lambda$, we prove that the sGMC solution set is a continuous polytope-valued
mapping of $\lambda$. Most noticeably, our study indicates that similar to
LASSO, the minimum $\ell_2$-norm regularization path of the sGMC model is
continuous and piecewise linear in $\lambda$. Based on these theoretical
results, an efficient regularization path algorithm is proposed for the sGMC
model, extending the well-known least angle regression (LARS) algorithm for
LASSO. We prove the correctness and finite termination of the proposed
algorithm under a mild assumption, and confirm its
correctness-in-general-situation, efficiency, and practical utility through
numerical experiments. Many results in this study also contribute to the
theoretical research of LASSO.
