  Two important goals of high-dimensional modeling are prediction and variable
selection. In this article, we consider regularization with combined $L_1$ and
concave penalties, and study the sampling properties of the global optimum of
the suggested method in ultra-high dimensional settings. The $L_1$-penalty
provides the minimum regularization needed for removing noise variables in
order to achieve oracle prediction risk, while concave penalty imposes
additional regularization to control model sparsity. In the linear model
setting, we prove that the global optimum of our method enjoys the same oracle
inequalities as the lasso estimator and admits an explicit bound on the false
sign rate, which can be asymptotically vanishing. Moreover, we establish oracle
risk inequalities for the method and the sampling properties of computable
solutions. Numerical studies suggest that our method yields more stable
estimates than using a concave penalty alone.
