Variables selection using L0 penalty

The determination of a tuning parameter by the generalized information criterion (GIC) is considered an important issue in variable selection. It is shown that the GIC and the G0 penalized objective functions are equivalent, leading to a new G0 penalized maximum likelihood method for high-dimensional generalized linear models in this article. Based on the technique of the well-known discrete optimization problem in theoretical computer science, a two-step algorithm for local solutions is proposed. The first step optimizes the G0 penalized objective function under a given model size, where only a maximum likelihood algorithm is needed. The second step optimizes the G0 penalized objective function under a candidate set of model sizes, where only the GIC is needed. As the tuning parameter can be fixed, the selection of the tuning parameter can be ignored in the proposed method. The theoretical study shows that the algorithm is polynomial and any resulting local solution is consistent. Thus, it is not necessary to use the global solution in practice. The numerical studies show that the proposed method outperforms its competitors in general.