## 4 Uplink Reception With Proposed Cooperation
Then, we consider the following conditions obtained by replacing the second and first terms on the RHSs of Equations (19) and (20), respectively, with their first-order Taylor approximations with the reference points \(\mathbf{\omega}^{\prime}\triangleq\{\omega_{i,r,j}^{\prime}\}_{i\in\{1,2\},r\in \mathcal{M},j\in\{1,2\}}\):  
\[R_{\text{sum},i}\leq\log_{2}\det\left(\mathbf{I}+P\left(P\tilde{ \mathbf{G}}_{i}\tilde{\mathbf{G}}_{i}^{H}+\sigma_{z}^{2}\mathbf{I}+\tilde{ \mathbf{\Omega}}_{i}\right)^{-1}\tilde{\mathbf{H}}_{i}\tilde{\mathbf{H}}_{i}^{H} \right),\ \forall i\in\{1,2\}, \tag{18b}\]0 \[R_{\text{sum},i}\leq\log_{2}\det\left(\mathbf{I}+P\left(P\tilde{ \mathbf{G}}_{i}\tilde{\mathbf{G}}_{i}^{H}+\sigma_{z}^{2}\mathbf{I}+\tilde{ \mathbf{\Omega}}_{i}\right)^{-1}\tilde{\mathbf{H}}_{i}\tilde{\mathbf{H}}_{i}^{H} \right),\ \forall i\in\{1,2\}, \tag{18b}\]1 \[R_{\text{sum},i}\leq\log_{2}\det\left(\mathbf{I}+P\left(P\tilde{ \mathbf{G}}_{i}\tilde{\mathbf{G}}_{i}^{H}+\sigma_{z}^{2}\mathbf{I}+\tilde{ \mathbf{\Omega}}_{i}\right)^{-1}\tilde{\mathbf{H}}_{i}\tilde{\mathbf{H}}_{i}^{H} \right),\ \forall i\in\{1,2\}, \tag{18b}\]2  
\[R_{\text{sum},i}\leq\log_{2}\det\left(\mathbf{I}+P\left(P\tilde{ \mathbf{G}}_{i}\tilde{\mathbf{G}}_{i}^{H}+\sigma_{z}^{2}\mathbf{I}+\tilde{ \mathbf{\Omega}}_{i}\right)^{-1}\tilde{\mathbf{H}}_{i}\tilde{\mathbf{H}}_{i}^{H} \right),\ \forall i\in\{1,2\}, \tag{18b}\]3 \[R_{\text{sum},i}\leq\log_{2}\det\left(\mathbf{I}+P\left(P\tilde{ \mathbf{G}}_{i}\tilde{\mathbf{G}}_{i}^{H}+\sigma_{z}^{2}\mathbf{I}+\tilde{ \mathbf{\Omega}}_{i}\right)^{-1}\tilde{\mathbf{H}}_{i}\tilde{\mathbf{H}}_{i}^{H} \right),\ \forall i\in\{1,2\}, \tag{18b}\]4  
Note that the feasible set for the constraints in Equations (21) and (22) is convex and not empty, since the non-convexity-inducing terms in Equations (19) and (20) have been linearized, and at least the point \(\omega^{\prime}\) belongs to the feasible space of Equations (21) and (22). Moreover, the constraints in Equations (21) and (22) are stricter than those in Equations (19) and (20) due to the following inequalities:  
\[R_{\text{sum},i}\leq\log_{2}\det\left(\mathbf{I}+P\left(P\tilde{ \mathbf{G}}_{i}\tilde{\mathbf{G}}_{i}^{H}+\sigma_{z}^{2}\mathbf{I}+\tilde{ \mathbf{\Omega}}_{i}\right)^{-1}\tilde{\mathbf{H}}_{i}\tilde{\mathbf{H}}_{i}^{H} \right),\ \forall i\in\{1,2\}, \tag{18b}\]5 \[R_{\text{sum},i}\leq\log_{2}\det\left(\mathbf{I}+P\left(P\tilde{ \mathbf{G}}_{i}\tilde{\mathbf{G}}_{i}^{H}+\sigma_{z}^{2}\mathbf{I}+\tilde{ \mathbf{\Omega}}_{i}\right)^{-1}\tilde{\mathbf{H}}_{i}\tilde{\mathbf{H}}_{i}^{H} \right),\ \forall i\in\{1,2\}, \tag{18b}\]6  
\[R_{\text{sum},i}\leq\log_{2}\det\left(\mathbf{I}+P\left(P\tilde{ \mathbf{G}}_{i}\tilde{\mathbf{G}}_{i}^{H}+\sigma_{z}^{2}\mathbf{I}+\tilde{ \mathbf{\Omega}}_{i}\right)^{-1}\tilde{\mathbf{H}}_{i}\tilde{\mathbf{H}}_{i}^{H} \right),\ \forall i\in\{1,2\}, \tag{18b}\]7  
This means that, if we find a solution to the _convex_ problem, which is obtained by replacing Equations (18b) and (18c) with Equations (21) and (22) in Equation (18), the resulting solution, denoted by \(\omega^{\prime\prime}\triangleq\{\omega_{i,r,j}^{\prime\prime}\}_{i\in\{1,2 \},j\in\mathcal{M},j\in\{1,2\}}\), will achieve a sum-rate larger than or equal to that of \(\omega^{\prime}\), while satisfying all the constraints of the original problem in Equation (18). If the sum-rate improvement is not negligible, we do the same process after updating the reference point \(\omega^{\prime}\leftarrow\omega^{\prime\prime}\), and this process can be repeated until the sum-rate converges. Since the sum-rate monotonically increases with iterations and the optimal sum-rate of the problem in Equation (18) is finite, the convergence of the MM algorithm is guaranteed. We refer to [21; 22] for more formal proof of convergence and stability of the MM algorithms. The MM algorithm customized to solve our DC problem is described in Algorithm 1. **1. ** Initialize \(\mathbf{R}_{C}\) as arbitrary rates that satisfy the constraints in Equations (18d)-(18f). **2.