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This paper is concerned with the computation of the high-dimensionalzero-norm penalized quantile regression estimator, defined as a globalminimizer of the zero-norm penalized check loss function. To seek a desirableapproximation to the estimator, we reformulate this NP-hard problem as anequivalent augmented Lipschitz optimization problem, and exploit its coupledstructure to propose a multi-stage convex relaxation approach (MSCRA\_PPA),each step of which solves inexactly a weighted $\ell_1$-regularized check lossminimization problem with a proximal dual semismooth Newton method. Under arestricted strong convexity condition, we provide the theoretical guarantee forthe MSCRA\_PPA by establishing the error bound of each iterate to the trueestimator and the rate of linear convergence in a statistical sense. Numericalcomparisons on some synthetic and real data show that MSCRA\_PPA not only hascomparable even better estimation performance, but also requires much less CPUtime.

A proximal dual semismooth Newton method for computing zero-norm penalized QR estimator