Semismooth Newton Augmented Lagrangian Algorithm for Adaptive Lasso Penalized Least Squares in Semiparametric Regression

This paper is concerned with a partially linear semiparametric regression model containing an unknown regression coefficient, an unknown nonparametric function, and an unobservable Gaussian distributed random error. We focus on the case of simultaneous variable selection and estimation with a divergent number of covariates under the assumption that the regression coefficient is sparse. We consider the applications of the least squares to semiparametric regression and particularly present an adaptive lasso penalized least squares (PLS) method to select the regression coefficient. We note that there are many algorithms for PLS in various applications, but they seem to be rarely used in semiparametric regression. This paper focuses on using a semismooth Newton augmented Lagrangian (SSNAL) algorithm to solve the dual of PLS which is the sum of a smooth strongly convex function and an indicator function. At each iteration, there must be a strongly semismooth nonlinear system, which can be solved by semismooth Newton by making full use of the penalized term. We show that the algorithm offers a significant computational advantage, and the semismooth Newton method admits fast local convergence rate. Numerical experiments on simulated and real data have demonstrated the effectiveness of the PLS method and the progressiveness of the SSNAL algorithm.