  The curse of dimensionality is a recognized challenge in nonparametric
estimation. This paper develops a new L0-norm regularization approach to the
convex quantile and expectile regressions for subset variable selection. We
show how to use mixed integer programming to solve the proposed L0-norm
regularization approach in practice and build a link to the commonly used
L1-norm regularization approach. A Monte Carlo study is performed to compare
the finite sample performances of the proposed L0-penalized convex quantile and
expectile regression approaches with the L1-norm regularization approaches. The
proposed approach is further applied to benchmark the sustainable development
performance of the OECD countries and empirically analyze the accuracy in the
dimensionality reduction of variables. The results from the simulation and
application illustrate that the proposed L0-norm regularization approach can
more effectively address the curse of dimensionality than the L1-norm
regularization approach in multidimensional spaces.
