Basis pursuit de-noising (BPDN) problem is considered to be an important model encountered in the sparse signal reconstruction problem. A lot of numerical algorithms about solving the BPDN problems have been extensively developed, but the solving speed and accuracy are still need improved when the dimension increases greatly. Therefore, it is an important and challenging problem to construct an optimization algorithm with faster solution speed and higher accuracy for solving BPDN problems. Based on splitting the decision variable into two nonnegative auxiliary variables in the BPDN problems, and transform BPDN problems into linear complementarity problems. Based on a special iterative format, we present a new projection algorithm without any line search to find a suitable step size and computation of the subdifferential of the absolute value function, which needs only one value of the mapping per iteration and only one projection onto the nonnegative quadrant. At the same time, the global convergence of iterative sequences generated by the new algorithm is established in detail. Numerical experiments on the BPDN problems with different cases are also given to verify the effectiveness of the proposed method. Thus, the proposed new method improves the efficiency and accuracy of solving the high dimensional the BPDN problems, and have certain theoretical value and practical significance.