Stochastic Maximum Likelihood-Based Direction-of-Arrival Estimators via SDP and ADMM

Direction of arrival (DOA) estimation under the stochastic maximum likelihood (SML) criterion is, in essence, a constrained optimization problem. Its intractable high non-convexity is primarily attributed to the logarithmic-determinant (log-det) term. To handle the log-det term, representative strategies involve the majorization-minimization (MM) algorithm for non-convex upper bound approximation or Taylor expansion for local approximation. However, these approaches suffer from sensitivity to initialization and step-size, are prone to becoming trapped in local optima, and require iterative updates of the expansion point until convergence, which is computationally intensive. To address these limitations, we leverage variable splitting convex relaxation, the Schur complement lemma, and inequality theory to transform the entire constrained optimization problem into a convex one. Consequently, a one-step semidefinite programming (SDP) can directly yield a global optimality certificate, eliminating approximation errors and sensitivity issues. This method is therefore named SML-SDP. Moreover, the relaxed objective function explicitly incorporates a nuclear norm term to induce low-rank solutions, which enables SML-SDP to be applicable in scenarios involving rank constraints, such as when the number of sources is known as a priori. Since SDP is time-consuming, we further develop a more efficient version of SML-SDP using the alternating direction method of multipliers (ADMM), named SML-ADMM. Extensive simulation and experiments demonstrate the superiority of the proposed methods.