The Spreading Projection Algorithm for Rapid K-space sampLING, or SPARKLING, is an optimization-driven method that has been recently introduced for accelerated 2D T2* weighted magnetic resonance imaging~(MRI) using compressed sensing. It has then been extended to address 3D imaging using either stacks of 2D sampling patterns or a local 3D strategy that optimizes a single sampling trajectory at a time. 2D SPARKLING actually performs variable density sampling (VDS) along a prescribed target density while maximizing sampling efficiency and meeting the gradient-based hardware constraints. However, 3D SPARKLING has remained limited in terms of acceleration factors along the third dimension if one wants to preserve a peaky point spread function (PSF) and thus good image quality. In this paper, in order to achieve higher acceleration factors in 3D imaging while preserving image quality, we propose a new efficient algorithm that performs globally optimized 3D SPARKLING. Its computational complexity scales up with the number of k-space samples p at the rate of O(p log(p)) and allows for the optimization of all k-space trajectories simultaneously, thus yielding a full 3D VDS pattern. We compare multi-CPU and GPU implementations and demonstrate that the latter is optimal for 3D imaging in the high-resolution acquisition regime (600 um isotropic). Finally, we show that this novel globally optimized 3D SPARKLING approach outperforms stacking strategies through retrospective and prospective studies on NIST phantom and in vivo brain scans at 3 Tesla, respectively. Overall the proposed method allows for 5x shorter scan times compared to GRAPPA-4 parallel imaging acquisition at 3 Tesla.