## 4 Joint User Scheduling and Hybrid Beamforming Design for Maximizing EE
### DC Programming  
Since the constraint \(C_{1}\) is a Boolean constraint and the constraint \(C_{2}\) is a nonconvex constraint, the problem \(Q_{2}\) is a nonconvex and nonsmooth combinatorial optimization problem. To handle this challenge, we can transform the problem \(Q_{2}\) into a DC programming problem [33]. Therefore, the relaxation variable \(\zeta_{k,l}\) is introduced as the lower bound of the SINR of the \(k\)th user in the \(l\)th multicast group, \(\zeta=[\zeta_{1},\zeta_{2},\dots,\zeta_{L}]\), \(\zeta_{l}=[\zeta_{l,1},\zeta_{l,2},\dots,\zeta_{l,|U_{l}|}]^{T}\). The problem \(Q_{2}\) can be equivalently converted to:  
\[Q_{3}:\max_{\eta,W\text{SINR}^{\min},\zeta}\mathrm{EE}=\frac{\frac{B\sum\limits _{l=1}^{L}\log_{2}\big{(}1+\text{SINR}^{\min}_{l}\big{)}}{\sum\limits_{l=1}^{ L}Tr(\mathbf{W}_{l})+P_{0}}}, \tag{38}\]  
\[s.t.\ C_{1},\ C_{3},\ C_{4},\ C_{5},C_{6},C_{7}\ in\ Q_{1}, \tag{39}\]  
\[C_{2}:\text{SINR}_{k,l}\geq\zeta_{k,l},\forall k,l, \tag{40}\]  
\[C_{8}:\zeta_{k,l}\geq\eta_{k,\text{SINR}^{\min}_{l}},\forall k,l, \tag{41}\]  
where the constraint \(C_{2}\) can be equivalently converted to:  
\[C_{2}\Rightarrow 1+\text{SINR}_{k,l}\geq 1+\zeta_{k,l}, \tag{42}\]  
To further express (42) in the form of DC programming, we introduce new function variables \(\Gamma_{k,l}(\mathbf{W})\) and \(\mathrm{I}_{k,l}(\mathbf{W},\zeta_{k,l})\):  
\[\Gamma_{k,l}(\mathbf{W})=\sigma^{2}+\sum\limits_{j=1,j\
eq l}^{L}Tr\big{(}\mathbf{H}_{ k,l}\mathbf{W}_{j}\big{)}, \tag{43}\]  
\[\mathrm{I}_{k,l}(\mathbf{W},\zeta_{k,l})=\frac{\sigma^{2}+\sum\limits_{j=1}^{L}Tr \big{(}\mathbf{H}_{k,l}\mathbf{W}_{j}\big{)}}{1+\zeta_{k,l}}, \tag{44}\]  
Therefore, the constraint \(C_{2}\) can be rewritten as:  
\[C_{2}\Rightarrow\Gamma(\mathbf{W})-\mathrm{I}(\mathbf{W},\zeta_{k,l})\leq 0, \tag{45}\]In (45), \(\Gamma_{k,l}(\mathbf{W}_{l})\) is the affine function of \(\mathbf{W}\), \(I_{k,l}(\mathbf{W},\xi_{k,l})\) is the concave function of \(\mathbf{W}\) and \(\zeta_{k,l}\). The transformed constraint \(C_{2}\) is a typical DC constraint. Similarly, the constraint \(C_{8}\) can be equivalently converted into the following DC form:  
\[C_{8}\Rightarrow 4\zeta_{k,l}+\left(\eta_{k,l}-\text{SINR}_{l}^{\text{min}} \right)^{2}\geq\left(\eta_{k,l}+\text{SINR}_{l}^{\text{min}}\right)^{2}, \tag{46}\]  
In (38), the objective function in the problem \(Q_{3}\) is a fractional programming problem with the sum-of-ratios form. To handle this problem, we invoke the quadratic transformation algorithm [34] and convert the problem \(Q_{3}\) into the following form:  
\[Q_{4}:\max_{\mathbf{\eta},W,SINR_{l}^{\text{min}},\zeta}\text{EE} =2q\left(B\sum_{l=1}^{L}\Upsilon_{l}(\text{SINR}_{l}^{\text{min}}) \right)^{\frac{1}{2}}-q^{2}\left(\sum_{l=1}^{L}Tr(\mathbf{W}_{l})+P_{0}\right), \tag{47}\] \[s.t.\ C_{1},\ C_{3},\ C_{4},\ C_{5},C_{6},C_{7}\ in\ Q_{1}, \tag{39}\]0  
where \(q\) is the introduced auxiliary variable, and \(\Upsilon_{l}(\text{SINR}_{l}^{\text{min}})\) is the introduced auxiliary function, which can be expressed as:  
\[s.t.\ C_{1},\ C_{3},\ C_{4},\ C_{5},C_{6},C_{7}\ in\ Q_{1}, \tag{39}\]1  
\[s.t.\ C_{1},\ C_{3},\ C_{4},\ C_{5},C_{6},C_{7}\ in\ Q_{1}, \tag{39}\]2  
In addition, for the nonsmooth combinatorial optimization problem caused by constraint \(C_{1}\), we invoke a relaxation and penalty algorithm. Firstly, we relax constraint \(C_{1}\) into \(C_{1}\Rightarrow 0\leq\eta_{k,l}\leq 1,\ \forall k,l\).