Two-dimensional (2D) fully-addressed arrays can conveniently realize three-dimensional (3D) ultrasound imaging while fully controlled such arrays usually demands thousands of independent channels, which is costly. Sparse array technique using stochastic optimization methods is one of promising techniques to reduce channel counts while due to the stochastic nature of these methods, the optimized results are usually unstable. In this work, we introduce a sparse array design approach that formulates the synthesis problem of sparse arrays as second-order cone programming (SOCP) and a re-weighted L1 technique is implemented to sequentially optimize the SOCP. Based on this method, an on-grid quasi-flatten side-lobe (Q-Flats) 2D sparse array with side-lobe level (SLL) no more than -21.26 dB and 252 activated elements is designed, which aims to achieve as high contrast performance as possible under the limits of resolution and maximum number of independent channels (i.e., 256). The imaging performance of the Q-Flats array was compared with those of a corresponding dense array (Dense), a Fermat spiral array (Spiral) and a spatially 50%-Tukey tapered spiral array (Spiral-Taper) using Field II simulations in a multi-angle steered diverging wave transmission scheme. It was demonstrated that the Dense achieved the best resolution and contrast and the Spiral-Taper the worst. The Q-Flats showed better resolution (about 3%) but slightly worse contrast than the Spiral. All the results indicate the re-weighted L1 SOCP method is a promising and flexible method for seeking trade-offs among resolution, contrast, and number of activated elements.