In this paper, we consider multiuser detection that copes with multiple access interference caused in low-rate wireless communication systems with a large number of nodes, such as machine-to-machine communications. We take account of discreteness of transmitted signals (e.g., binary signals taking values of $\pm 1$), which we utilize as prior information in detection. We first formulate the detection problem as a maximum a posteriori estimation problem. Since this problem is highly nonconvex and discontinuous, we propose convex relaxation based on the sum-of-absolute-values (SOAV) optimization, which can be efficiently solved by proximal splitting. We also derive an upper bound of the relaxation error by the SOAV optimization in noisy cases based on the restricted isometry property. Numerical simulations are shown to illustrate the effectiveness of the proposed approach.