Summary: The classical synthetic aperture radar (SAR) imaging techniques based on matched filters are limited by data bandwidth, resulting in limited imaging performance with side lobes and speckles present. To address the high-resolution SAR imaging problem, sparse reconstruction has been extensively investigated. However, the state-of-the-art sparse recovery methods seldom consider the complex-valued reflectivity of the scene and only recover an approximated real-valued scene instead. Furthermore, iterative schemes associated with the sparse recovery methods demand a high computational cost, which limits the practical applications of these methods. In this paper, we establish a sphere-constrained magnitude-sparsity SAR imaging model, aiming at enhancing the SAR imaging quality with high efficiency. We propose a non-convex non-smooth optimization method, which can be accelerated by stochastic average gradient acceleration to be scalable with large-scale problems. Numerical experiments are conducted with point-target and extended-target simulations. On the one hand, the point-target simulation showcases the superiority of our proposed method over the classical methods in terms of resolution. On the other hand, the extended-target simulation with random phases is considered to be in line with the practical scenario, and the results demonstrate that our method outperforms the classical SAR imaging methods and sparse recovery without phase prior in terms of PSNR. Meanwhile, owing to the stochastic acceleration, our method is faster than the existing sparse recovery methods by orders of magnitude.