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 $ell_1$-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.
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