## 2 Materials and Methods
In problems (9)-(11), the possibility of regulating the radius of coverage (10) and the level of the intersection of coverage areas (11) are taken into account. The result of solving this problem will be a set of Pareto-optimal pairs of radii of coverage and levels of the intersection of coverage areas with which the maximum coverage of the territory is achieved at the minimum cost of the network. Using the method of convolution of criteria, construct the objective function:  
\[F(r,c)=\alpha P(r,c)+(1-\alpha)S_{uncovered}(r,c), \tag{12}\]  
where \(\alpha\)--expert assessment of the importance of parameters. Using function (12) present problem (9)-(11) in the form:  
\[F(r,c)\rightarrow\min_{(r,c)\in G}, \tag{13}\]  
\[G:\left\{ \begin{array}{l}r\in[r_{min},r_{max}]\\ c\in[c_{min},r_{max}]\end{array} \right. \tag{14}\]  
Let us prove the existence of a solution to the described problem. **Lemma 2**. : _Objective function \(F(r,c)\) (13) reaches its minimum value in the domain \(G\)._  
**Proof of Lemma 2. ** The function \(F(r,c)\) is continuous as the sum of continuous functions (12). The domain \(G\) is bounded because all variables have minimum and maximum values \(r\in[r_{min},r_{max}]\) and \(c\in[c_{min},r_{max}]\). The domain \(G\) is also closed, since for the variables \(r\) and \(c\) the segments \([r_{min},r_{max}]\) and \([c_{min},r_{max}]\) are closed. Thus, the domain \(G\) is compact, and, according to [PERSON]'s theorem, the function \(F(r,c)\) reaches its minimum value in the domain \(G\). Lemma is proved. \(\square\)  
**Theorem 2**. : _Problems (13) and (14) has a solution._  
**Proof of Theorem 2. ** The function \(F(r,c)\) is a convex and linear combination of convex functions, continuous and differentiated. According to Lemma 2, the domain \(G\) is compact; then, according to the [PERSON] theorem, there is a saddle point of the Lagrange function and it is an extremum. Thus, problem (13) and (14) has a solution. The theorem is proved. \(\square\)The parameter \(\alpha\) can be included in the function \(F(r,c)\) as an unknown variable. Thus, we obtain the problem in the form (15) and (16):  
\[F_{1}(r,c,\alpha)\rightarrow\min_{(r,c,\alpha)\in G}, \tag{15}\]  
\[G:\left\{ \begin{array}{l}r\in[r_{min},r_{max}]\\ c\in[c_{min},r_{max}]\\ \alpha\in[0,1]\end{array} \right. \tag{16}\]  
Let us prove the existence of a solution to the described problem. **Lemma 3**. : _The function \(F_{1}(r,c,\alpha)\) (15) reaches its minimum value in the domain \(G\) (16)._  
Proof of Lemma 3.: The function \(F_{1}(\mathbf{r},c,\alpha)\) is continuous as the sum of continuous functions (12). The domain \(G\) is bounded because all variables have minimum and maximum values \(\mathbf{r}\in\ [r_{min},r_{max}]\), \(c\in[c_{min},r_{max}]\) and \(\alpha\in[\mathbf{0},\mathbf{1}]\). The domain \(G\) is also closed, since for the variables \(\mathbf{r}\), \(c\) and \(\alpha\) the segments \([r_{min},r_{max}]\), \([c_{min},r_{max}]\) and \([\mathbf{0},\mathbf{1}]\) are closed. Thus, the domain \(G\) is compact, and, according to [PERSON]'s theorem [18], the function \(F_{1}(\mathbf{r},c,\alpha)\) reaches its minimum value in the domain \(G\). Lemma is proved. \(\square\)  
**Theorem 3**. : _Problems (15)-(16) has a solution._  
Proof of Theorem 3.: The function \(F_{1}(\mathbf{r},c,\alpha)\) is a convex and linear combination of convex functions, continuous and differentiated. According to Lemma 3, the domain \(G\) is compact; then, according to the [PERSON] theorem [19], there is a saddle point of the Lagrange function and it is an extremum. Thus, problems (15) and (16) has a solution. The theorem is proved. \(\square\)  
To find the solutions of problems (13)-(16), use the gradient descent method. The solutions of the problems will be a pair of values \(\left(r_{opt},c_{opt}\right)\), which will be the value of the optimal radius of coverage and the value of the intersection of coverage areas where maximum coverage is achieved while minimizing energy consumption.