## 3 Localization Method
### Initial Solution
The constraint (26b) takes on an equivalent form as follows:  
\[\mathbf{\mathsf{H}}((k-1)M+i,(k-1)M+i)=\mathbf{\mathsf{h}}((k-1)M+i)^{2}=\mathbf{\mathsf{q}} ^{T}\mathbf{\mathsf{Y}}\mathbf{\mathsf{q}}-2\mathbf{\mathsf{q}}^{T}\mathbf{\mathsf{X}}^{T}\mathbf{ \mathsf{s}}_{i,k}+\mathbf{\mathsf{s}}_{i,k}^{T}\mathbf{\mathsf{s}}_{i,k}, \tag{28}\]  
for \(i=1,\cdots,M,k=1,\cdots,K\), where \(\mathbf{\mathsf{q}}=\left[1,k-1\right]^{T}\) thus \(\mathbf{\mathsf{u}}_{k}=\mathbf{\mathsf{X}}\mathbf{\mathsf{q}}\). Applying the Cauchy-Schwartz inequality to (26b) obtains  
\[\mathbf{\mathsf{H}}(l_{1},l_{2})\geq|\mathbf{\mathsf{q}}_{1}^{T}\mathbf{\mathsf{Y}}\mathbf{ \mathsf{q}}_{2}-\mathbf{\mathsf{q}}_{1}^{T}\mathbf{\mathsf{X}}^{T}\mathbf{\mathsf{s}}_{i_{2},k _{2}}-\mathbf{\mathsf{q}}_{2}^{T}\mathbf{\mathsf{X}}^{T}\mathbf{\mathsf{s}}_{i_{1},k_{1}}+ \mathbf{\mathsf{s}}_{i_{1},k_{1}}^{T}\mathbf{\mathsf{s}}_{i_{2},k_{2}}|, \tag{29}\]  
where \(\mathbf{\mathsf{q}}_{1}=\left[1,k_{1}-1\right]^{T}\), \(\mathbf{\mathsf{q}}_{2}=\left[1,k_{2}-1\right]^{T}\). The variables \(i_{1}\), \(i_{2}\), \(k_{1}\), and \(k_{2}\) satisfy \(1\leq l_{1}<l_{2}\leq MK\) where \(l_{1}=\left(k_{1}-1\right)M+i_{1},l_{2}=\left(k_{2}-1\right)M+i_{2}\). Considering the difficulty in formulating constraint (26c) in a convex Thus, the nonconvex constraint above can be expressed as  
\[\mathbf{H}((k-1)M+i,(k-1+K)M+i)\] \[\mathbf{\mathsf{d}}=[d_{1,1},d_{2,1},\cdots,d_{M,1},\cdots,d_{M,K}]^{T}, \tag{25}\]0  
Additionally, applying the Cauchy-Schwartz inequality to (26c) obtains an additional constraint,  
\[\mathbf{\mathsf{d}}=[d_{1,1},d_{2,1},\cdots,d_{M,1},\cdots,d_{M,K}]^{T}, \tag{25}\]1  
Squaring both sides of (32) results in  
\[\mathbf{\mathsf{d}}=[d_{1,1},d_{2,1},\cdots,d_{M,1},\cdots,d_{M,K}]^{T}, \tag{25}\]2  
At this stage, two nonconvex constraints remain: \(\mathbf{H}=\mathbf{h}\mathbf{h}^{T}\) and \(\mathbf{Y}=\mathbf{X}^{T}\mathbf{X}\). The SDR method can be employed [49] to relax these constraints into convex inequalities \(\mathbf{H}\succeq\mathbf{h}\mathbf{h}^{T}\) and \(\mathbf{Y}\succeq\mathbf{X}^{T}\mathbf{X}\), which can be expressed as Linear Matrix Inequalities (LMIs):  
\[ \begin{bmatrix}1&\mathbf{h}^{T}\\ \mathbf{h}&\mathbf{H}\end{bmatrix}  \succeq\mathbf{0}, \tag{34}\] \[ \begin{bmatrix}\mathbf{I}_{N}&\mathbf{X}\\ \mathbf{X}^{T}&\mathbf{Y}\end{bmatrix}  \succeq\mathbf{0}, \tag{35}\]  
where the rank-1 constraints are ignored. Since the rank of \((\mathbf{A}_{1}^{T}\mathbf{Q}^{-1}\mathbf{A}_{1}+\mathbf{A}_{2}^{T}\mathbf{Q}^ {-1}\mathbf{A}_{2})\) is equal to \(2K(M-1)\), the convex optimization formulation (27)-(29), (31), and (32) may exhibit ambiguities. To address this, two additional penalties \(\eta_{1}\operatorname{tr}(\mathbf{H}(1:MK,1:MK))\) and \(\eta_{2}\operatorname{tr}(\mathbf{H}(MK+1:2 MK,MK+1:2 MK))\) need to be introduced to tighten the constraints, where \(\eta_{1}\) and \(\eta_{2}\) are penalty factors. Additionally, second-order cone (SOC) constraints [50]  
\[\left\|\mathbf{X}(:,1)\mathbf{q}-\mathbf{s}_{i,k}\right\|\leq\mathbf{h}((k-1) M+i),\quad i=1,\cdots,M, \tag{36}\]  
are incorporated to enhance the location accuracy. The discussions above lead to the proposed SDP problem as follows,  
\[\min_{\mathbf{h},\mathbf{H},\mathbf{X},\mathbf{Y}} \operatorname{tr}[(\mathbf{A}_{1}^{T}\mathbf{Q}^{-1}\mathbf{A}_{1}+ \mathbf{A}_{2}^{T}\dot{\mathbf{Q}}^{-1}\mathbf{A}_{2})\mathbf{H}]-2\mathbf{H }^{T}(\mathbf{A}_{1}^{T}\mathbf{Q}^{-1}\mathbf{r}+\mathbf{A}_{2}^{T}\dot{ \mathbf{Q}}^{-1}\dot{\mathbf{r}})\] \[\quad+\eta_{1}\operatorname{tr}(\mathbf{H}(1:MK,1:MK))+\eta_{2} \operatorname{tr}(\mathbf{H}(MK+1:2 MK,MK+1:2 MK)),\] s.t.