## 4 Formulation and Policy
### Policy Analysis
#### 4.3.2 Feasible Solution  
Regarding the proposed Optimal Eavesdropping and Jamming Problem, we will discuss whether it has the feasible solution or not. Based on Reference [40], the optimization model that has the feasible solution should satisfy three constraints: (a) The variable is effective collection based on the constraints in the optimization model, (b) the objective of the optimization model is the continuous function, and (c) the objective of the optimization model is a convex function. We will prove these three properties in this part. First, we will discuss the variable's effective collection under the constraints in our proposed optimization model. The constraints \(0\leq\sum_{x=1}^{m}P_{L}(x)\leq P_{L}^{total}\) and \(P_{L}(x)\leq P_{L}^{max}\) (\(\forall x,\ x=1,2,\ldots,m\)) relates to the practice in the reality, which defines \(P_{L}(x)\)'s maximums of upper and lower bound. The last two constraints should be proved, satisfying the effective collection. They make further definition of \(P_{L}(x)\)'s upper and lower bound, furthermore, the relationship between \(\frac{H_{s}(x)\cdot K_{2}^{-1}\ln\frac{K_{1}}{\epsilon}}{\delta^{2}}-N_{0}\) and \(\frac{(2^{\textit{max}}-1)H_{s}(x)\cdot K_{2}^{-1}\ln\frac{K_{1}}{\epsilon}}{ \delta^{2}}-N_{0}\) should be considered. In fact, the parameters \(H_{s}(x)\), \(K_{2}^{-1}\), and \(\delta^{2}\) are larger than zero. \(K_{1}\) is larger than \(\epsilon\), which means that \(\ln\frac{K_{1}}{\epsilon}>0\), then the last two constraints can be transformed into:  
\[1\leq\frac{\delta^{2}P_{L}(x)}{H_{s}(x)\ K_{2}^{-1}\ln\frac{K_{1}}{\epsilon}} \leq 2^{\textit{\rho}_{max}}-1\]  
\(\rho_{max}\) is a parameter that is larger than 1. Therefore, the variable \(P_{L}(x)\) has the effective collection under the four constraints in the optimization model. Second, we will discuss the objective's consecutiveness in the optimization model. Obviously, the objective is a composite function, which uses the constant function, power function, exponential function and the logarithmic function based on \(P_{L}(x)\), \(H_{s}(x)\), \(H_{e}(x)\), and \(H_{j}(x)\). It is easy to prove that the functions of \(P_{L}(x)\), \(H_{s}(x)\), \(H_{e}(x)\), and \(H_{j}(x)\) are all continuous functions. Moreover, the sum function does not affect the function's consecutiveness. Therefore, the objective in our proposed Optimal Eavesdropping and Jamming Problem is a continuous function. Finally, we will discuss whether the objective in our proposed Optimal Eavesdropping and Jamming Problem is a convex function or not. In order to simplify, we define the objective function as \(\mathrm{G}(\mathrm{x})\), where  
\[\mathrm{G}(\mathrm{x})=\left(1-\frac{1}{2}exp^{\beta_{1}-\beta_{0}\delta\sqrt {\frac{H_{e}(x)+H_{j}(x)}{H_{s}(x)}\cdot(1+\frac{P_{j}(x)}{N_{0}})}}\right)^{ 8(2f-l)}\]  
We have proved that the objective is a continuous function in the above paragraph, and then the convex property can be proved by the second derivation, which is denoted as:  
\[G^{{}^{\prime\prime}}(x) = -bln[8(2f-l)]\cdot\frac{1}{2}\exp\left(\beta_{1}-\beta_{0}\delta \sqrt{\frac{H_{e}(x)+H_{j}(x)}{H_{s}(x)}\cdot\left(1+\frac{P_{L}(x)}{N_{0}} \right)}\right)\] \[\cdot ln\frac{1}{2}\left[\left(\frac{H_{e}(x)+H_{j}(x)}{H_{s}(x)} \right)^{{}^{\prime\prime}}+\left(\frac{P_{L}(x)}{N_{0}}\frac{H_{e}(x)+H_{j}( x)}{H_{s}(x)}\right)^{{}^{\prime\prime}}\right]\]  
According to the non-negativity of exponential function, the second term of \(G^{{}^{\prime\prime}}(x)\) will be larger than zero. Regarding the first term of \(G^{{}^{\prime\prime}}(x)\), the preamble size \(l\) is always smaller than the frame size \(f\) in the practice, then the result of \(8(2f-l)\) will be larger than 1, thus the first term is smaller than 0. Regarding the third term of \(G^{\prime\prime}(x)\), which is denoted as:  
\[