## 4 Problem Formulation  
Let \(\mathbf{W}=[\mathbf{w}_{1},\,\cdots,\,\mathbf{w}_{K}]\in C^{M\times K}\), \(\mathbf{H}_{r}=[\mathbf{h}_{r,1},\,\cdots,\,\mathbf{h}_{r,K}]\in C^{N\times K}\), and \(\mathbf{H}_{d}=[\mathbf{h}_{d,1},\,\cdots,\,\mathbf{h}_{d,K}]\in C^{M\times K}\). The rate maximisation problem is considered for the user with the worst wireless channel condition under total power constraint. The objective is to maximise the achievable rate by jointly optimising the transmit beamforming at the AP and the reflected beamforming direction at the IRS under individual SINR constraints for all users. Thus, the optimisation problem can be formulated as:  
\[(\text{P0}):\underset{\mathbf{W},\mathbf{\theta}}{\text{max}} \quad\underset{\mathbf{k}}{\text{min}}\quad\text{R}_{\mathbf{k}}\] (4) s.t. \[\quad\frac{|(\mathbf{h}_{r,k}^{H}\mathbf{\Theta}\mathbf{G}+ \mathbf{h}_{d,k}^{H})\mathbf{w}_{k}|^{2}}{\sum_{j\
eq k}^{K}|(\mathbf{h}_{r,k} ^{H}\mathbf{\Theta}\mathbf{G}+\mathbf{h}_{d,k}^{H})\mathbf{w}_{j}|^{2}+\sigma _{k}^{2}}\geq\gamma_{k},\forall k, \tag{5}\] \[\quad\sum_{k=1}^{K}||\mathbf{w}_{k}||^{2}\leq P_{\Upsilon},\] (6) \[\quad 0\leq\theta_{n}\leq 2\pi,\quad n=1,...,N. \tag{7}\]  
Here, (5) is the QoS SINR constraint with \(\gamma_{k}>0\) as the minimum required SINR of user \(k\), and (6) is the total power constraint. The above problem is strictly non-convex due to the coupled variables in (4) and (5). This problem is solved by solving the corresponding power minimisation problem. The corresponding power minimisation solution can be formulated for the given optimal objective value \(R_{k}^{*}\) of the rate maximisation problem (P0) as:  
\[(\text{P1}):\underset{\mathbf{W},\mathbf{\theta}}{\text{min}} \quad\sum_{k=1}^{K}||\mathbf{w}_{k}||^{2}\] (8) s.t. \[\quad\frac{|(\mathbf{h}_{r,k}^{H}\mathbf{\Theta}\mathbf{G}+ \mathbf{h}_{d,k}^{H})\mathbf{w}_{k}|^{2}}{\sum_{j\
eq k}^{K}|(\mathbf{h}_{r,k} ^{H}\mathbf{\Theta}\mathbf{G}+\mathbf{h}_{d,k}^{H})\mathbf{w}_{j}|^{2}+\sigma _{k}^{2}}\geq\gamma_{k}^{*},\forall k, \tag{9}\] \[\quad 0\leq\theta_{n}\leq 2\pi,n=1,...,N, \tag{10}\]  
where \(\log_{2}(1+\gamma_{k}^{*})=R_{k}^{*}\). Note that problem (P1) is a variation of problem (P0). Interestingly, it can be seen later that problem (P1) is easier to solve compared to problem (P0). The problem (P1) will be solved below and a solution will be established for correspondence between (P1) and (P0). Problem (P1) aims at minimising the total transmit power subject to the minimum required SINR \(\gamma_{k}^{*}\) which is directly related to the achievable rate \(R_{k}^{*}\).  
**Theorem 1**.: _Any optimal solution to the power minimisation problem (P1) is also optimal for problem (P0)._  
Proof.: See Appendix A.  
Let us now consider a single user system for simplicity, i.e., \(K=1\). Simplifying the problem (P1), we get  
\[(\text{P2}):\min_{\mathbf{w},\mathbf{\theta}} ||\mathbf{w}||^{2}\] (11) s.t. \[|(\mathbf{h}_{r}^{H}\mathbf{\Theta}\mathbf{G}+\mathbf{h}_{d}^{H}) \mathbf{w}|^{2}\geq\gamma\sigma^{2}, \tag{12}\] \[0\leq\theta_{n}\leq 2\pi,\quad n=1,...,N. \tag{13}\]  
Even after simplifying the problem as much as possible, the problem still remains a non-convex optimisation problem since the left-hand side of the above problem is not jointly concave with respect to \(\omega\) and \(\theta\). This challenge is overcome by the use of alternating optimisation and SDR.