  This paper deals with the grouped variable selection problem. A widely used
strategy is to augment the negative log-likelihood function with a
sparsity-promoting penalty. Existing methods include the group Lasso, group
SCAD, and group MCP. The group Lasso solves a convex optimization problem but
is plagued by underestimation bias. The group SCAD and group MCP avoid this
estimation bias but require solving a nonconvex optimization problem that may
be plagued by suboptimal local optima. In this work, we propose an alternative
method based on the generalized minimax concave (GMC) penalty, which is a
folded concave penalty that maintains the convexity of the objective function.
We develop a new method for grouped variable selection in linear regression,
the group GMC, that generalizes the strategy of the original GMC estimator. We
present an efficient algorithm for computing the group GMC estimator and also
prove properties of the solution path to guide its numerical computation and
tuning parameter selection in practice. We establish error bounds for both the
group GMC and original GMC estimators. A rich set of simulation studies and a
real data application indicate that the proposed group GMC approach outperforms
existing methods in several different aspects under a wide array of scenarios.
