We propose a joint sparse maximum a posteriori (MAP) estimator for DOA estimation from multiple snapshots, reformulated as a mixed-integer semidefinite program (MISDP). This enables efficient computation of globally optimal solutions using off-the-shelf MISDP solvers based on the branch-and-bound method. Unlike other nonconvex approaches for joint sparse recovery, such as the greedy methods and sparse Bayesian learning techniques, it provides a solution with an optimality assessment even with early termination. Additionally, we present a more scalable approximate solution approach for the MISDP problem based on randomized rounding. Numerical simulations demonstrate the improved threshold behavior, resolution, and robustness of our proposed method against popular DOA estimation methods. In particular, the proposed method applied with the randomized rounding algorithm exhibits a superior estimation performance at a significantly reduced running time, compared to the deterministic maximum likelihood (DML) estimator.