## 3 The Proposed Scheme for WSP Minimization
### Discussion of the Initial Value Selection of [PERSON]'s Method and Weight Adjustment  
The rate of convergence indicates how quickly a real-valued sequence \(\{x_{k}\}\) can approach its local optimum \(x^{*}\), which is essential for evaluating the performance of an iterative method. [PERSON]'s method has a quadratic rate of convergence with order of convergence (time complexity) \(O(\log\log(1/\varepsilon))\) and is extremely fast. However, [PERSON]'s method does not necessarily converge. Depending on where it starts, it may \"quadratically quickly\" either converge or diverge. In practice, \(x_{0}\) must be sufficiently close to \(x^{*}\). Otherwise, it may result in a large number of iterations or even fail to converge. To avoid this situation, we should carefully select its initial value \(x_{0}\). Although there are sufficient conditions to ensure the convergence of [PERSON]'s method, selecting initial values based on them is often quite complex. Therefore, this paper adopts a simplified method as follows.  
First, when the spectrum sharing begins, we select an \(x_{0}\), which fulfills  
\[w^{\prime\prime\prime}(x_{0})\
eq 0. \tag{25}\]  
and  
\[\left|w^{\prime\prime}(x_{0})\right|^{2}>\left|w(x_{0})w^{\prime\prime}(x_{0} )\right|/2. \tag{26}\]  
This operation ensures [PERSON]'s method converges in most cases [35].  
Then, if the wireless environment changes slowly, we can use the optimal value of \(\theta\) in the previous frame as the initial value of \(\{x_{k}\}\).  
Through the proposed joint spectrum allocation, power control, and link matching described above, the system WSP can be effectively minimized within each frame. Then, we will discuss how to extend the lifetime of the entire network, including all the PLs and SLs, as much as possible.  
The minimum QoS requirement ensures the priority level of a link in terms of SE. On the other hand, it is essential to clarify the role of link weights in WSP. In WSEE maximization, the weights assigned to links determine their priority in terms of EE. Specifically, when we choose WSEE as the optimization objective, a link with a higher weight indicates a higher expected EE after resource allocation. Similarly, once the proposed WSP minimization scheme is implemented, a link with a higher weight will consume less energy. By reasonably setting their weights within each transmission frame, not only will the system power consumption be reduced but the energy consumption among different links will also be balanced.  
As analyzed earlier, in WSP minimization, the weight of a link reflects the willingness to use its energy budget. Many studies are dedicated to extending network lifetime in the fields of wireless sensor networks and unmanned aerial vehicle (UAV)-assisted wireless networks. Most solutions, however, tend to have high computational complexity, and some require ML approaches [39]. However, there are some heuristic weight dynamic adjustment strategies. For example, we can set the weight of each link as inversely proportional to its residual energy at the beginning of the current transmission frame. By performing it this way, different links' energy consumption rates can be balanced. In future research, we will jointly optimize weights and wireless resources to maximize the network lifetime.  
Table 3 shows the proposed algorithm with the heuristic initial value selection strategy.