## 5 The Optimal Working Strategies for Wireless Rechargeable Sensor Networks
### The Simplification and Linearization of OPT-4  
The optimization problem formulated in the last subsection is not easy to solve for several reasons. First of all, the optimization problem has variables that are continuous over time. Secondly, the constraints of this problem contain integration terms and proportional terms, which make this problem nonlinear. Therefore, it is not an easy job to find an efficient algorithm to obtain the optimal solution. Instead of providing a heuristic algorithm, in this paper, we try to reach the optimal solution of OPT-4 by simplifying and linearizing it. First of all, we want to introduce two properties that are useful when transforming the original problem into a linear programming problem. **Property 1**. : _The energy of each node reaches its peak when the locomotive mounted with the wireless transfer device finishes recharging it. Moreover, its energy falls to the bottom when the locomotive arrives at this node._  
**Property 2**. : _If the batteries of nodes are fully recharged, that is the energy of each battery reached when the locomotive finishes recharging task, the optimal value of OPT-4 remains the same._  
With these two properties, we are able to slightly change the form of OPT-4. For the constraint (15), we can rewrite it as:  
\[E_{i}(t_{i})\geq E_{min},E_{i}(t_{i}+\tau_{i})=E_{max} \tag{18}\]  
where \(t_{i}\) denotes the arriving time of the recharging locomotive at the i-th node. Then, we have the new optimization problem, OPT-5, with identical optimality. \[\max \tau_{s}/\tau\] s.t. (12)-(18)  
We will then try to take the integration terms out to simplify OPT-5. The integration terms are introduced by Constraint (14). We first substitute Equation (14) with Equations (12) and (13), then we have:  
\[\rho\int_{\tau}r_{s}^{i}(t)dt+(\Phi_{1}+\Phi_{2}d_{i}^{k})\int_{ \tau}r_{\tau}^{i}(t)dt=U\tau_{i} \tag{19}\]  
According to the Lagrange mean value theorem, there exists \(r_{s}^{i}\) and \(r_{\tau}^{i}\), which satisfy:  
\[\rho r_{s}^{i}\tau+(\Phi_{1}+\Phi_{2}d_{i}^{k})r_{\tau}^{i}\tau=U \tau_{i} \tag{20}\]  
For Equation (12), we have,  
\[r^{i}+r_{s}^{i}=r_{\tau}^{i} \tag{21}\]  
Then, we have the optimization problem OPT-6,  
\[\max \tau_{s}/\tau\] s.t. (16)-(21)  
The constraint (14) is omitted since we plug it into Equation (20). In OPT-6, we use \(r_{k}^{i}\) and \(r_{\tau}^{i}\) instead of continuous variables. The next theorem will show the equal optimality of OPT-5 and OPT-6. **Theorem 3**. : _If we use discrete variables \(r_{k}^{i}\) and \(r_{\tau}^{i}\) instead of \(r_{k}^{i}(t)\) and \(r_{\tau}^{i}(t)\), the optimal values of both OPT-5 and OPT-6 are the same, that is OPT-5 and OPT-6 have equal optimality._  
**Proof. ** The proof of this theorem can be divided into two steps. On the one hand, since we use \(r_{{}_{R}}^{i}\) and \(r_{{}_{T}}^{i}\) instead of \(r_{{}_{R}}^{i}(t)\) and \(r_{{}_{T}}^{i}(t)\), the flexibility of variables is of course compromised to a certain extent. The optimal values of OPT-5 and OPT-6 are denoted as \(O_{5}\) and \(O_{6}\). Then, we have \(O_{5}\geq O_{6}\). On the other hand, for each feasible solution of OPT-5, we are always able to construct a corresponding solution to OPT-6 that yields the same objective value, which means \(O_{5}\leq O_{6}\). Therefore, we have \(O_{5}=O_{6}\). The next theorem will tell us the optimal traveling path that the locomotive should take to achieve the optimality. Theorem 4: _The optimal traveling path that the recharging locomotive should take is the shortest roaming path connecting all nodes and the sojourn spot._  
**Proof. ** The proof of this theorem is based on contradiction, that is if a solution to the optimization problem yields the optimal value while the locomotive does not follow the shortest roaming path, we can construct a new solution from it that yields a better objective value while the locomotive travels along the shortest roaming path. Intuitively, if the locomotive spends less time on traveling, it will be rewarded more time staying at the sojourn spot.