## 3 Main Results
,N=\mathbf{I}.\]  
For given feasible matrices \(G^{\kappa},G^{\kappa}_{r}\), (10) is equivalent to  
\[\mathbb{L}+\mathrm{He}\{\mathbb{X}N\mathbb{Y}^{\kappa}+\mathbb{X}^{\kappa}N( \mathbb{Y}-\mathbb{Y}^{\kappa})\}+\mathrm{He}\{(\mathbb{X}-\mathbb{X}^{\kappa} )N(\mathbb{Y}-\mathbb{Y}^{\kappa})\}<\mathbf{0}, \tag{11}\]which contains the linear term \(\mathbb{L}+\text{He}\{\mathbb{X}N\mathbb{Y}^{\kappa}+\mathbb{X}^{\kappa}N(\mathbb{ Y}-\mathbb{Y}^{\kappa})\}\) and the bilinear term \(\text{He}\{(\mathbb{X}-\mathbb{X}^{\kappa})N(\mathbb{Y}-\mathbb{Y}^{\kappa})\}\). By performing the matrix decomposition [22], we have  
\[ \begin{split}&\text{He}\{(\mathbb{X}-\mathbb{X}^{\kappa})N(\mathbb{ Y}-\mathbb{Y}^{\kappa})\}\\ &=\begin{bmatrix}(\mathbb{X}-\mathbb{X}^{\kappa})^{T}\\ (\mathbb{Y}-\mathbb{Y}^{\kappa})\end{bmatrix}^{T}\begin{bmatrix}\mathbf{0}&N\\ N^{T}&\mathbf{0}\end{bmatrix}\begin{bmatrix}(\mathbb{X}-\mathbb{X}^{\kappa})^{T}\\ (\mathbb{Y}-\mathbb{Y}^{\kappa})\end{bmatrix}\\ &=\begin{bmatrix}(\mathbb{X}-\mathbb{X}^{\kappa})^{T}\\ (\mathbb{Y}-\mathbb{Y}^{\kappa})\end{bmatrix}^{T}\begin{bmatrix}\beta N&\beta N \\ U^{T}&-U\end{bmatrix}\begin{bmatrix}\frac{1}{\beta}(U+U^{T})^{-1}&\mathbf{0}\\ \mathbf{0}&-\frac{1}{\beta}(U+U^{T})^{-1}\end{bmatrix}\begin{bmatrix}\beta N&\beta N \\ U^{T}&-U\end{bmatrix}^{T}\begin{bmatrix}(\mathbb{X}-\mathbb{X}^{\kappa})^{T}\\ (\mathbb{Y}-\mathbb{Y}^{\kappa})\end{bmatrix}\\ &\leq\frac{1}{\beta}\Big{(}\beta(\mathbb{X}-\mathbb{X}^{\kappa})N+( \mathbb{Y}-\mathbb{Y}^{\kappa})^{T}U^{T}\Big{)}\Big{(}U+U^{T}\Big{)}^{-1}\Big{(} \beta(\mathbb{X}-\mathbb{X}^{\kappa})N+(\mathbb{Y}-\mathbb{Y}^{\kappa})^{T}U^ {T}\Big{)}^{T},\end{split}  \tag{12}\]  
where \(U\) is introduced as an auxiliary variable satisfying \(U+U^{T}>\mathbf{0}\). Denote \(\Delta\mathbb{X}^{\kappa}\triangleq\mathbb{X}-\mathbb{X}^{\kappa},\Delta \mathbb{Y}^{\kappa}\triangleq\mathbb{Y}-\mathbb{Y}^{\kappa}\). Replacing the bilinear term \(\text{He}\{(\mathbb{X}-\mathbb{X}^{\kappa})N(\mathbb{Y}-\mathbb{Y}^{\kappa})\}\) with its upper-approximation and applying the [PERSON] complement lemma yields  
\[ \begin{bmatrix}\mathbb{L}+\text{He}\{\mathbb{X}N\mathbb{Y}^{\kappa}+\mathbb{X }^{\kappa}N\Delta\mathbb{Y}^{\kappa}\}&\beta\Delta\mathbb{X}^{\kappa}N+(U \Delta\mathbb{Y}^{\kappa})^{T}\\ \star&-\beta(U+U^{T})\end{bmatrix} <\mathbf{0}, \tag{13}\]  
and we can reorganize (13) as  
\[ \begin{bmatrix}\mathbb{L}+\text{He}\{\mathbb{X}N\mathbb{Y}^{\kappa}+\mathbb{X }^{\kappa}N\Delta\mathbb{Y}^{\kappa}\}&\beta\Delta\mathbb{X}^{\kappa}N+(U^{ \kappa}\Delta\mathbb{Y}^{\kappa})^{T}\\ \star&-\beta(U+U^{T})\end{bmatrix} +\text{He}\bigg{\{} \begin{bmatrix}(\Delta \mathbb{Y}^{\kappa})^{T}\\ \mathbf{0}\end{bmatrix}\begin{bmatrix}\mathbf{0}&(\Delta U^{\kappa})^{T}\end{bmatrix}  \right\}<\mathbf{0}. \tag{14}\]  
Recalling that \(\text{He}\{\mathbb{A}\mathbb{B}\}\leq\Delta\mathbb{M}\mathbb{A}^{T}+\mathbb{B }^{T}\mathbb{M}^{-1}\mathbb{B}\), then (14) can be conservatively expressed as  
\[ \begin{bmatrix}\mathbb{L}+\text{He}\{\mathbb{X}N\mathbb{Y}^{\kappa}+\mathbb{X }^{\kappa}N\Delta\mathbb{Y}^{\prime}\}&\beta\Delta\mathbb{X}^{\kappa}N+(U^{ \kappa}\Delta\mathbb{Y}^{\kappa})^{T}\\ \star&-\beta(U+U^{T})\end{bmatrix}  \tag{15}\]  
By the Schur complement lemma, one obtains  
\[\Gamma^{\prime}= \begin{bmatrix}\Phi&\Psi^{\prime}\\ (\Psi^{\prime})^{T}&\Xi\end{bmatrix} <\mathbf{0}, \tag{16}\]  
where  
\[ \begin{split}\Psi^{\prime}&=\begin{bmatrix}\beta(G-G^{\kappa})^{T} &\mathbf{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\mathbf{0}\\ \bar{M}^{T}(G_{r}-G_{r}^{\kappa})\bar{F}^{T}(U^{\kappa})^{T}&\bar{M}^{T}(G_{r}-G_ {r}^{\kappa})\bar{F}^{T}&\mathbf{0}\\ \bar{N}^{T}(G_{r}-G_{r}^{\kappa})\bar{F}^{T}(U^{\kappa})^{T}&\bar{N}^{T}(G_{r}-G_ {r}^{\kappa})\bar{F}^{T}&\mathbf{0}\end{bmatrix},\\ \Xi^{\prime}&=\begin{bmatrix}-\beta(U+U^{T})&\mathbf{0}&U-U^{\kappa}\\ \star&-V^{-1}&\mathbf{0}\\ \star&\star&-V\end{bmatrix}.\end{split}  \tag{17}\]  
For a given \(V^{\kappa}\), we obtain \(-V^{-1}\leq-2V^{\kappa}+(V^{\kappa})^{-1}V(V^{\kappa})^{-1}\) according to the concavity of \(-V^{-1}\).