## 3 Decremental Antenna Selection Algorithms
### \(\ell_{0}\)-Norm Minimization (L0 NM) Algorithm  
In each step of the RDS and the MRDS algorithms, \(k\) antennas are randomly selected and dropped, and the beamforming vector is optimized given the selected antenna subset. In other words, we first fix the item \(P_{\text{cir}}\
ormnorm{\mathbf{w}}_{0}\) and then minimize the item \(\
orm{\mathbf{w}}_{2}^{2}/\eta\) as the \(\ell_{0}\)-norm and the \(\ell_{2}\)-norm cannot be minimized simultaneously. The main drawback of the two algorithms is the randomness of the antenna selection. In this subsection, we propose an \(\ell_{0}\)-norm minimization (L0 NM) algorithm with higher efficiency. Different from the RDS and the MRDS algorithms, L0 NM selects the antennas through minimizing the \(\ell_{0}\)-norm instead of random selection. That is, we first fix the item \(\
orm{\mathbf{w}}_{2}^{2}/\eta\) and then minimize the item \(P_{\text{cir}}\
orm{\mathbf{w}}_{0}\). In the \(r\)-th step, we first calculate the minimum transmit power \(\
orm{\mathbf{w}_{(r-1)}}_{2}^{2}\) by the SCA approach for the previous subset \(S_{(r-1)}\) whose \(\ell_{0}\)-norm is \(\
orm{\mathbf{w}_{(r-1)}}_{0}\). Then, we set a positive value \(\theta\), if the new subset satisfies the following requirement, _i.e._,  
\[\frac{\
orm{\mathbf{w}_{(r)}}_{2}^{2}-\
orm{\mathbf{w}_{(r-1)}}_{2}^{2}}{\eta} \leq\theta\leq P_{\text{cir}}(\
orm{\mathbf{w}_{(r-1)}}_{0}-\
orm{\mathbf{w}_ {(r)}}_{0}) \tag{6}\]  
which implies  
\[\frac{\
orm{\mathbf{w}_{(r)}}_{2}^{2}}{\eta}+P_{\text{cir}}\
orm{\mathbf{w}_{( r)}}_{0}\leq\frac{\
orm{\mathbf{w}_{(r-1)}}_{2}^{2}}{\eta}+P_{\text{cir}} \
orm{\mathbf{w}_{(r-1)}}_{0} \tag{7}\]  
The subset \(S_{(r)}\) is superior to the subset \(S_{(r-1)}\). The requirement Equation (7) is equivalent to minimize \(\
orm{\mathbf{w}_{(r)}}_{0}\) and the minimal value of \(\
orm{\mathbf{w}_{(r)}}_{0}\) should be less than \(\
orm{\mathbf{w}_{(r-1)}}_{0}-\theta/P_{\text{cir}}\). So, we have the following optimization problem,  
\[\mathcal{P}_{3}:\min_{\mathbf{w}_{(r)}\in\mathbb{C}^{N}} \
orm{\mathbf{w}_{(r)}}_{0}\] s. t. \[\
orm{\mathbf{w}_{(r)}}_{2}^{2}\leq\
orm{\mathbf{w}_{(r-1)}}_{2}^ {2}+\eta\theta\] \[\
orm{\mathbf{w}_{(r)}^{H}\mathbf{h}_{i}^{(r-1)}}^{2}\geq\gamma_{ i},i=1,\cdots,M \tag{8}\]  
The problem \(\mathcal{P}_{3}\) is a minimum \(\ell_{0}\)-norm problem where both the objective function and the first constraint are non-convex. Since the \(\ell_{1}\)-norm is the closest convex approximation to the \(\ell_{0}\)-norm [3; 13], we replace the \(\ell_{0}\)-norm with \(\ell_{1}\)-norm which can also induces the sparsity. Then, the problem \(\mathcal{P}_{3}\) can be reformulated as  
\[\mathcal{P}_{4}:\min_{\mathbf{w}_{(r)}\in\mathbb{C}^{N}} \
orm{\mathbf{w}_{(r)}}_{1}\] s. t. \[\
orm{\mathbf{w}_{(r)}}_{2}^{2}\leq\
orm{\mathbf{w}_{(r-1)}}_{2}^ {2}+\eta\theta\] \[\
orm{\mathbf{w}_{(r)}^{H}\mathbf{h}_{i}^{(r-1)}}^{2}\geq\gamma_{ i},i=1,\cdots,M \tag{9}\]For the non-convexity of the SNR constraints, we adopt the iterative successive convex approximation method similarly as [5]. In each iteration step, the minimal transmit power increases by a fixed value \(\eta\theta\). Since the decreasing speed of the minimal transmit power becomes slower with the increase of \(N\), we drop more antennas in the first step to improve the efficiency of the L0 NM algorithm. The operation of the L0 NM algorithm is summarized as Algorithm 1. **Input:**\(M\), \(N\), \(\eta\), \(P_{\text{cir}}\), \(\widetilde{\gamma}_{i}\), \(i=1,...,M\) and \(\theta\). **Output:** The optimal subset \(S^{(r)}\) and the beamforming vector \(\mathbf{w}_{(r)}\). **Main procedure:**  
1. **Initialization** Given the initial set \(S_{(0)}\) including all the antennas and \(\left\|\mathbf{w}_{(0)}\right\|_{0}=N\). Set \(r=1\). 2. **Step r** Calculate the minimum transmit power \(\left\|\mathbf{w}_{(r-1)}\right\|_{2}^{2}\) for \(S^{(r-1)}\) according to the SCA approach. To solve the minimum \(\ell_{0}\)-norm problem \(\mathcal{P}_{3}\), the objective function is replaced by \(\ell_{1}\)-norm.