from $N+L$ to $N+r\leq N+M$. Therefore, the resulting problem dimension depends only on $M$ and $N$. Both the SDPT3 and ADMM implementations of RAM above can be reasonably fast when $M$ and $N$ are small though they may not possess good scalability, especially for SDPT3. An application at hand is DOA estimation in which the array size $M$ and aperture $N$ are typically small (on the order of 10) though $L$ can be a few hundred or even greater. Note also that the dimension reduction technique takes $O\sbra{M^2L}$ flops in DOA estimation according to Remark \[rem:sameusolution\]. Extensive numerical simulations will be provided in Section \[sec:simulation\] to demonstrate usefulness of our method.

Remarks on Algorithm Implementation
-----------------------------------

According to the discussions in Section \[sec:WAN\_SDP\], we can always start with $\m{u}_0=\m{0}$ and the first iteration coincides with ANM. When $L$ is large, we can also implement a weighting function in the first iteration inspired by Capon’s beamforming for faster convergence (see an example in Section \[sec:simulation\_DOA\]). Moreover, we gradually decrease $\epsilon$ during the algorithm and define the weighting function using the latest solution for avoiding local minima (note that the first iteration with $\m{u}_0=\m{0}$ essentially corresponds to $\epsilon=+\infty$ following from Theorem \[thm:epsilontoinf\]). 