Massive multiple-input multiple-output(MIMO)technology enables higher data rate transmis-sion in the future mobile communications.However,exploiting a large number of antenna elements at base station(BS)makes effective implementation of mas-sive MIMO challenging,due to the size and weight limits of the masssive MIMO that are located on each BS.Therefore,in order to miniaturize the massive MIMO,it is crucial to reduce the number of antenna elements via effective methods such as sparse array synthesis.In this paper,a multiple-pattern synthe-sis is considered towards convex optimization(CO).The joint convex optimization(JCO)based synthesis is proposed to construct a codebook for beamforming.Then,a criterion containing multiple constraints is de-veloped,in which the sparse array is required to full-fill all constraints.Finally,extensive evaluations are performed under realistic simulation settings.The re-sults show that with the same number of antenna el-ements,sparse array using the proposed JCO-based synthesis outperforms not only the uniform array,but also the sparse array with the existing CO-based syn-thesis method.Furthermore,with a half of the number of antenna elements that on the uniform array,the per-formance of the JCO-based sparse array approaches to that of the uniform array.