A proximal dual semismooth Newton method for computing zero-norm penalized QR estimator
Abstract
This paper is concerned with the computation of the high-dimensional zero-norm penalized quantile regression estimator, defined as a global minimizer of the zero-norm penalized check loss function. To seek a desirable approximation to the estimator, we reformulate this NP-hard problem as an equivalent augmented Lipschitz optimization problem, and exploit its coupled structure to propose a multi-stage convex relaxation approach (MSCRA\PPA), each step of which solves inexactly a weighted 1-regularized check loss minimization problem with a proximal dual semismooth Newton method. Under a restricted strong convexity condition, we provide the theoretical guarantee for the MSCRA\PPA by establishing the error bound of each iterate to the true estimator and the rate of linear convergence in a statistical sense. Numerical comparisons on some synthetic and real data show that MSCRA\PPA not only has comparable even better estimation performance, but also requires much less CPU time.