Beampattern synthesis is a key problem in many wireless applications. With the increasing scale of MIMO antenna array, it is highly desired to conduct beampattern synthesis on a sparse array to reduce the power and hardware cost. In this paper, we consider conducting beampattern synthesis and sparse array construction jointly. In the formulated problem, the beampattern synthesis is designed by minimizing the matching error to the beampattern template, and the Shannon entropy function is first introduced to impose the sparsity of the array. Then, for this nonconvex problem, an iterative method is proposed by leveraging on the alternating direction multiplier method (ADMM) and the majorization minimization (MM). Simulation results demonstrate that, compared with the benchmark, our approach achieves a good trade-off between array sparsity and beampattern matching error with less runtime.