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This paper is concerned with a partially linear semiparametric regressionmodel containing an unknown regression coefficient, an unknown nonparametricfunction, and an unobservable Gaussian distributed random error. We focus onthe case of simultaneous variable selection and estimation with a divergentnumber of covariates under the assumption that the regression coefficient issparse. We consider the applications of the least squares to semiparametricregression and particularly present an adaptive lasso penalized least squares(PLS) method to select the regression coefficient. We note that there are manyalgorithms for PLS in various applications, but they seem to be rarely used insemiparametric regression. This paper focuses on using a semismooth Newtonaugmented Lagrangian (SSNAL) algorithm to solve the dual of PLS which is thesum of a smooth strongly convex function and an indicator function. At eachiteration, there must be a strongly semismooth nonlinear system, which can besolved by semismooth Newton by making full use of the penalized term. We showthat the algorithm offers a significant computational advantage, and thesemismooth Newton method admits fast local convergence rate. Numericalexperiments on some simulation data and real data to demonstrate that the PLSis effective and the SSNAL is progressive.

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