A smaller number of transmit antennas can reduce the array system cost, while sparse antenna design can provide additional degrees of freedom to beampattern synthesis. It is also known that the min-max beampattern matching design criterion can synthesize beampattern with small mainlobe ripple, low peak sidelobe level, and specified null/notch. Based on this design metric, we formulate a unified sparse array design framework for beampattern synthesis. Our framework includes the colocated multiple-input multiple-output (MIMO) radar transmit beampattern synthesis, phase-only beampattern synthesis, and phased-array beampattern synthesis as special cases. In the design formulation, the antenna positions and phase-only waveforms (for MIMO radar beampattern) or array weight vector (for phase-only and traditional phased-array beampattern) are jointly determined. The resultant problem is challenging because the nonconvex nonsmooth min-max objective function, and phase-only/unimodular requirements make it become typically NP-hard, and the antenna position selection results in a difficult Boolean constraint. Nevertheless, we are able to find a local optimal solution efficiently by applying the Lawson approximation and majorization-minimization. Numerical examples show that the devised approach can achieve the desired transmit beampattern by jointly optimizing the phase-only waveforms/weight vector and antenna locations.