Maximum A Posteriori Direction-of-Arrival Estimation via Mixed-Integer Semidefinite Programming

In this paper, we consider the maximum a posteriori (MAP) estimation for the multiple measurement vectors (MMV) problem with application to direction-of-arrival (DOA) estimation, which is classically formulated as a regularized least-squares (LS) problem with an $\ell_{2,0}$-norm constraint, and derive an equivalent mixed-integer semidefinite program (MISDP) reformulation. The proposed MISDP reformulation can be exactly solved by a generic MISDP solver using a semidefinite programming (SDP) based branch-and-bound method, which, unlike other nonconvex approaches for the MMV problem, such as the greedy methods and sparse Bayesian learning techniques, provides a solution with an optimality assessment even with early termination. We also present an approximate solution approach based on randomized rounding that yields high-quality feasible solutions of the proposed MISDP reformulation at a practically affordable computation time for problems of extremely large dimensions. Numerical simulations demonstrate the improved error performance of our proposed method in comparison to several popular DOA estimation methods. In particular, compared to the deterministic maximum likelihood (DML) estimator, which is often used as a benchmark, the proposed method applied with the randomized rounding algorithm exhibits a superior estimation performance at a significantly reduced running time.