Linear Array Synthetic Aperture Radar (LASAR) is a novel and promising radar imaging technique. In recent years, Compressed Sensing (CS) sparse recovery has been a research focus for high-resolution three-Dimensional (3-D) LASAR imaging. Compared with the traditional two-Dimensional (2-D) SAR imaging, LASAR suffers from many problems, including under-sampling data and multi-dimensional and higher-order phase errors due to its sparse Linear Array Antenna (LAA) and the joint 2-D motions of the platform and LAA. The conventional autofocusing methods of 2-D SAR may be not suitable for CS-based LASAR 3-D sparse autofocusing. To address the multi-dimensional and higher-order phase errors in LASAR 3-D imaging with respect to under-sampling data, in this paper, we propose a sparse autofocusing algorithm based on semidefinite programming for CS-based LASAR imaging. First, by combining CS-based imaging theory, image maximum sharpness, and the minimum square error principle, we construct a LASAR phase-error estimation model based on under-sampled data. Next, we use semi-definite programming relaxation to estimate the phase errors. Lastly, we employ an iterated approximation method to improve the precision of the phase-error estimation and achieve the final CS-based LASAR autofocusing. To further improve the efficiency of the algorithm, we select only the dominant scattering areas for LASAR phase-error estimation. We present our simulation and experimental results to confirm the effectiveness of out proposed algorithm.