## 2 System Model  
Figure 1 depicts a system model where an access point (AP) sends a multicast message to \(K\) user terminals (UTs), while an IRS assists the direct AP transmission by reflecting the signal from the AP toward the UTs. Each UT receives signals that include a common message simultaneously from the AP and the IRS. The AP and each UT have \(M\) antennas and a single antenna, respectively, while the IRS is equipped with \(N\) reflecting elements. The multiple input, multiple output (MIMO) channel from the AP to the IRS is denoted by an \(N\times M\) matrix \(\mathbf{H}_{0}\). The multiple input, single output (MISO) channel from the AP to the \(k\)th UT (i.e., \(UT_{k}\)) is denoted by an \(M\times 1\) vector \(\mathbf{g}_{k},\ k=1,\ldots,K\) and the MISO channel from the IRS to the \(j\)th UT (i.e., \(UT_{j}\)) is denoted by an \(N\times 1\) vector \(\mathbf{h}_{j},\ j=1,\ldots,K\). The elements of these matrices and vectors are independent and identically distributed as \(\mathcal{CN}(0,1)\). The AP transmits signal \(\mathbf{w}\mathbf{x}\) through \(M\) antennas, where \(\mathbf{w}\) is an \(M\times 1\) pre-coding vector with \(\|\mathbf{w}\|^{2}=1\) and \(x\) is the common multicast message signal for \(K\) UTs with the transmit power \(\mathbb{E}[|x|^{2}]=P\). The transmitted signal from AP is delivered to the UTs through the direct link from the AP and the secondary link through the IRS, where the two signals are added up at the UTs. The phase-shifting at IRS is modeled as an \(N\times N\) complex diagonal matrix \(\mathbf{\Phi}\), where the \(i\)-th diagonal element \(\phi_{i},\ i=1,\ldots,N\) with \(|\phi_{i}|=1\). The received signals of \(UT_{k},\ k=1,\ldots,K\) are written as follows:  
\[y_{k}=\mathbf{g}_{k}^{T}\mathbf{w}x+\beta\mathbf{h}_{k}^{T}\mathbf{\Phi} \mathbf{H}_{0}\mathbf{w}x+n_{k}, \tag{1}\]  
where \(\beta\) represents the attenuation at the IRS which is uniformly applied to all the IRS elements. The rate of \(x\) is \(R\) bits per second per channel use; \(n_{k}\) is the additive noise signals at \(UT_{k}\) with \(\mathcal{CN}(\mathbf{0},\mathbf{1})\) distribution. Without loss of generality, we can set \(\beta=1\). The signal-to-noise ratio \(\gamma_{k}\) at \(UT_{k}\) is then given as  
\[\gamma_{k}=|(\mathbf{g}_{k}^{T}+\mathbf{h}_{k}^{T}\mathbf{\Phi}\mathbf{H}_{0}) \mathbf{w}|^{2}P, \tag{2}\]  
Figure 1: Multicast channel with an AP, an intelligent reflecting surface (IRS) and \(K\) user terminals (UTs). with the mutual information (MI) at \(UT_{k}\) as \(\log(1+\gamma_{k})\). For the MC transmission, the user with minimum MI among the \(K\) MC users determines the rate, and the logarithmic function possesses monotonic relation with its argument. Therefore, the MC transmission rate \(R\) can be maximized by maximizing \(\min_{k\in\{1,\ldots,K\}}\gamma_{k}\) or, equivalently,  
\[(\mathbf{w},\mathbf{\Phi})=\max_{k\in\{1,\ldots,K\}}|(\mathbf{g}_ {k}^{T}+\mathbf{h}_{k}^{T}\mathbf{\Phi}\mathbf{H}_{0})\mathbf{w}|,\] \[\|\mathbf{w}\|^{2}=1,\ |\phi_{i}|=1,\ i=1,\ldots,N. \tag{3}\]  
The optimization of the objective function in (3) involves the vector \(\mathbf{w}\) and the diagonal matrix \(\mathbf{\Phi}\), where the unit magnitude constraint of \(\phi_{i}\) makes the problem non-convex and demanding at the same time. Problems with sets of parameters, as in this case, are often handled by alternating approaches, where each set of parameters is optimized with the other sets being fixed. With \(\mathbf{\Phi}\) fixed, the optimization of \(\mathbf{w}\) is well-studied in the context of multicast beamforming, and the near-optimal solution can be found by an SDR-based method [1; 2]. Although the SDR approach is widely applied, providing effective solutions to many classes of problems, it requires demanding computational complexity.