## Appendix C Proof of Lemma 2  
The Schur-concavity can be proved from _[PERSON]'s condition_, presented in the following lemma. **Lemma A2** ([PERSON]'s condition). : _[_32_]_ _(Ch.3.A.4)_ _Let the function \(\varphi(\mathbf{a}):\mathcal{A}\subset\mathbb{R}^{n}\to\mathbb{R}\) be continuously differentiable. Then, \(\varphi\) is Schur-concave on \(\mathcal{A}\) if and only if \(\varphi\) is permutation-invariant,_i.e., \(\varphi(\mathbf{a})=\varphi(\mathbf{\Pi}_{n}\mathbf{a})\), and \((a_{m}-a_{t})\Big{(}\frac{\partial\varphi}{\partial a_{m}}-\frac{\partial\varphi }{\partial a_{t}}\Big{)}\leq 0\) for all \(1\leq m,l\leq n\); \(\mathbf{\Pi}_{n}\) is any \(n\times n\) permutation matrix._  
We first show the permutation-invariant property of \(g^{\star}\), i.e., \(g^{\star}(\mathbf{x}_{T},\mathbf{x}_{Q})=g^{\star}\)\((\mathbf{\Pi}_{M_{\mathrm{t}}\mathbf{x}_{T}},\mathbf{\Pi}_{L}\mathbf{x}_{Q})\). Let us define the diagonal matrices \(\mathbf{X}_{T}\) and \(\mathbf{X}_{Q}\) as \(\mathbf{X}_{T}=\mathrm{Diag}(x_{T,1},\cdots,x_{T,M_{t}})\) and \(\mathbf{X}_{Q}=\mathrm{Diag}(x_{Q,1},\cdots,x_{Q,L})\), respectively. Then, we have  
\[g^{\star}(\mathbf{\Pi}_{M_{\mathrm{t}}}\mathbf{x}_{T},\mathbf{\Pi }_{L}\mathbf{x}_{Q})=\min_{\mathbf{d}_{p}\in\mathcal{D}_{p}}g(\mathbf{d}_{p}, \mathbf{\Pi}_{M_{\mathrm{t}}}\mathbf{x}_{T},\mathbf{\Pi}_{L}\mathbf{x}_{Q}) \tag{11a}\] \[=\min_{\mathbf{T}(\mathbf{D}_{p}\mathbf{D}_{p}^{\mathsf{T}})\leq \mathcal{D}_{p}\mathbf{D}_{p}^{\mathsf{T}}\geq\mathbf{0}}\beta\mathbf{T}\Big{[} (\mathbf{\Pi}_{M_{\mathrm{t}}}\mathbf{X}_{T}^{-1}\mathbf{\Pi}_{M_{\mathrm{t}} }^{\mathsf{T}}+\mathbf{D}_{p}\mathbf{\Pi}_{L}\mathbf{X}_{Q}^{-1}\mathbf{\Pi}_ {L}^{\mathsf{T}}\mathbf{D}_{p}^{\mathsf{T}})^{-1}\Big{]}\] (11b) \[=\min_{\mathbf{T}(\mathbf{D}_{p}\mathbf{D}_{p}^{\mathsf{T}})\leq \mathcal{D}_{p}\mathbf{D}_{p}^{\mathsf{T}}\geq\mathbf{0}}\beta\mathbf{T}\Big{[} (\mathbf{X}_{T}^{-1}+\mathbf{D}_{p}\mathbf{X}_{Q}^{-1}\mathbf{\Pi}_{p}^{ \mathsf{T}})^{-1}\Big{]}\] (11c) \[=\min_{\mathbf{T}(\mathbf{D}_{p}\mathbf{D}_{p}^{\mathsf{T}})\leq \mathcal{D}_{p}\mathcal{D}_{p}\mathbf{D}_{p}^{\mathsf{T}}\geq\mathbf{0}}\beta \mathbf{T}\Big{[}(\mathbf{X}_{T}^{-1}+\mathbf{D}_{p}\mathbf{X}_{Q}^{-1} \mathbf{D}_{p}^{\mathsf{T}})^{-1}\Big{]}\] (11d) \[=\min_{\mathbf{d}_{p}\in\mathcal{D}_{p}}g(\mathbf{d}_{p},\mathbf{ x}_{T},\mathbf{x}_{Q})=g^{\star}(\mathbf{x}_{T},\mathbf{x}_{Q}), \tag{11e}\]  
where the equality in (11b) follows from the equivalence of the problem in (11a) and (11b) and the fact that \(\mathbf{\Pi}_{M_{\mathrm{t}}\mathbf{x}_{T}}=\mathbf{\Pi}_{M_{\mathrm{t}}}^{ \mathsf{T}}\mathbf{X}_{T}\mathbf{\Pi}_{M_{\mathrm{t}}}\) and \(\mathbf{\Pi}_{L}\mathbf{x}_{Q}=\mathbf{\Pi}_{L}^{\mathsf{T}}\mathbf{X}_{Q} \mathbf{\Pi}_{L}\). The equality in (11c) follows from the change of the variable from \(\mathbf{D}_{p}\) to \(\mathbf{\tilde{D}}_{p}=\mathbf{\Pi}_{M_{\mathrm{t}}}^{\mathsf{T}}\mathbf{D}_{ p}\mathbf{\Pi}_{L}\). The equality in (11e) follows from the equivalence of the problem in (11d) and (11e). From the above property, without loss of generality, it is assumed that the elements of \(\mathbf{x}_{T}\) and \(\mathbf{x}_{Q}\) are arranged in decreasing and increasing orders, respectively. Then, for given \(\mathbf{x}_{T}\) and \(\mathbf{x}_{Q}\), the solution to the inner minimization problem of (20) can be computed from the KKT optimality conditions as  
\[d_{p,j}^{\star}= \begin{cases}\Big{(}\mathcal{P}_{T}+\sum_{j=1}^{r}\frac{x_{Qj }}{x_{T,j}}\Big{)}\frac{\sqrt{x_{Qj}}}{\sum_{j=1}^{r}\sqrt{x_{Qj}}}-\frac{x_{Qj }}{x_{T,j}},&i=1,\cdots,r,\\ 0,&i=r+1,\cdots,\
u,\end{cases}  \tag{12}\]  
where \(r=\max\{i\in\{1,\cdots,\
u\}:d_{p,i}^{\star}>0\}\) denotes the largest \(i\) such that \(d_{p,j}^{\star}>0\).