Abstract: The numerical method for solving the L1 minimization problem is studied.The penalty method and the Gauss-Seidal iteration technique are adopted to develop the method. The L1 regularization problem is regarded as a penalized L1 minimizationproblem. Taking into account that the L1 norm function is nonsmooth, a smoothing function is adopted as an approximation to it. Then a smooth unconstrained optimization problem is solved inexactly via Gauss-Seidal iteration. At last,the penalty parameter is adjusted in an appropriate way so that the generated sequence of iterates converges to a solution of the L1 minimization problem. The proposed method is tested by numerical experiments, and its performance is compared with some existing methods. The results show that the proposed method is practically effective.