@bhaskar2013atomic; @yang2015gridless], which is a first-order algorithm and guarantees global optimality. To derive the ADMM algorithm, we reformulate the SDP in (\[formu:weightedSDP\]) as follows: which is very similar to the SDP solved in [@yang2015gridless]. Then we can write the augmented Lagrangian function and iteratively update $\sbra{\m{u},\m{X},\m{Y}}$, $\m{\cQ}$ and the Lagrangian multiplier in closed-form expressions until convergence. We omit the details since all the formulae and derivations are similar to those in [@yang2015gridless], to which interested readers are referred. We mention that an eigen-decomposition of a matrix of order $N+L$ (the order of $\m{\cQ}$) is required at each iteration. Note that the ADMM converges slowly to an extremely accurate solution while moderate accuracy is typically sufficient in practical applications [@boyd2011distributed].

Dimension Reduction for Large $L$
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The number of measurement vectors $L$ can be large, possibly with $L\gg M$, in DOA estimation, which increases considerably the computational workload. We provide the following result to reduce this number from $L$ to $\rank\sbra{\m{Y}_{\m{\Omega}}^o}\leq \min(L,M)$.

Let $r=\rank\sbra{\m{Y}_{\m{\Omega}}^o}\leq \min(L,M)$. Find a unitary matrix $\m{Q}\in\bC^{L\times L}$ (for example, by QR decomposition) satisfying that $\m{Y}_{\m{\Omega}}^{o}\m{Q}= \begin{bmatrix}\m{Y}_{\m{\Omega}}^{o}\m{Q}_1 & \m{0}\end{bmatrix}$, where $\m{Q}_1\in\bC^{L\times r}$. If we make the substitutions $\m{Y}_{\m{\Omega}}^{o}\rightarrow\m{Y}_{\m{\Omega}}^{o}\m{Q}_1$ and $\m{Y}\in\bC^{N\times L}\rightarrow\m{Z}\in\bC^{N\times r}$ in (\[formu:problem\_j\]) and denote by $\sbra{\m{Z}^*,\m{u}^*}$ 