Emerging millimeter-wave (mmWave) MIMO radars combine the benefits of large bandwidth available at mmWave frequencies with the spatial diversity provided by MIMO architectures, significantly enhancing radar capabilities for automotive, surveillance, and imaging applications. However, deploying large numbers of antennas and transceivers at these high frequencies substantially increases chip complexity and hardware costs. In this paper, we address the design of sparse two-dimensional (2D) antenna arrays that retain the desirable beampattern characteristics of fully populated arrays – namely, narrow mainlobes and low sidelobes – while significantly reducing the required number of antenna elements. We formulate the sparse array design problem as a beampattern matching optimization, which selects optimal subsets of transmit and receive antenna positions from an initial dense grid. To efficiently solve this challenging nonconvex optimization problem, we introduce an iterative algorithm combining Majorization–Minimization (MM) and Alternating Optimization (AO) techniques. We provide theoretical guarantees for convergence to at least a local optimum. Additionally, we propose a weighting vector optimization step to further enhance sidelobe suppression. Numerical simulations confirm that the proposed method maintains angular resolution and Sidelobe Levels (SLLs) comparable to those of full arrays, while substantially reducing hardware complexity and cost. Performance comparisons against existing methods demonstrate notable improvements in sidelobe suppression and computational efficiency without compromising processing gain.