We describe an optimization methodology for the antenna positions and tapering weights of a MIMO 3Tx/4Rx sparse array front-end. The goal is to improve the resolution of a uniform array of equal number of virtual elements, while maintaining a similar or lower sidelobe level (SLL). We describe the metric in terms of the Mainlobe width (MLW) and SLL, and quantify the noise amplification incurred by the amplitude tapering using the concept of equivalent noise bandwidth (ENBW). For the global optimization we use simulated annealing, providing practical insights about the constraints, and assessment of solutions and trade-offs. We include a parametric trade-off of MLW versus SLL for different field-of-views, with and without joint optimization with amplitude tapering, and in terms of number of antenna elements. Results are validated with real measurements comparing two prototype 3Tx/4Rx boards, for uniform and sparse optimized arrays, processing signals with different sparse reconstruction algorithms using 1-snapshot after preliminary range-Doppler processing.