# AI Optimization-Based Heterogeneous Approach for Green Next-Generation Communication Systems
## 3 Proposed SDR-Based LTE-LAA Architecture and Its Resource Management Scheme
### Resource Management Problem Formulation
* Equation (24) ensures that the number of users associated with any BS should be equal to or less than the number of licensed carriers. This is to meet the LTE-LAA requirement [54]. * Equation (25) ensures that unlicensed carriers are equally distributed between the connected users. * Equation (26) denotes that the power required by a user \(j\) to become connected to a BS \(i\) should not exceed the maximum transmission power \(P_{j_{max}}\) which the user can afford based on its available battery power. * Equation (27) ensures that each BS's traffic load is less than a maximum threshold \(\varphi_{i_{max}}\) which the BS can afford. * Equation (28) ensures that the SINR between the user and the assigned BS is the best for better seamless communication. * Equation (29) makes sure that the total data demands of the SUs do not exceed the total idle capacity of the PBS (\(\eta\)). We solved this optimization problem using the IBM CPLEX studio to find the optimum solution. The optimization problem in Equation (20) is an integer linear programming (ILP) problem, which is challenging to solve in real time. We prove that the above problem is NP-hard. Accordingly, the real-time application is not practical, especially for mobile networks which cover large areas with a huge number of users and BSs. Lemma 1: _The optimization problem in Equation (20) can be considered an NP-hard problem._  
Proof: Assume that we have two BSs which are located at the same point, and hence, a user's traffic load toward any of them is the same. In order to prove that Equation (20) is NP-hard, its congruous decision problem needs to be NP-complete. The decision problem could be written as follows:  
\[\text{maximize }z=\sum_{i\in I}\sum_{j\in I}(D_{ij}^{eff}+\beta\cdot(1-\frac{P_{ ij}}{P_{j_{max}}})) \tag{20}\]1  
where \(b\) is a positive value. Can we then obtain an appropriate solution where \(x=\{x_{ij}|j\in J,i\in I\}\) such that the constrains in Equations (21)-(23), (25), (26), (28), and (29) are satisfied? To prove that Equation (30) is NP-complete, we consider only two small-cell BSs installed at the same location, and they have the same maximum traffic load threshold. In other words, we have  
\[\text{maximize }z=\sum_{i\in I}\sum_{j\in I}(D_{ij}^{eff}+\beta\cdot(1-\frac{P_{ ij}}{P_{j_{max}}})) \tag{20}\]2  
where \(\epsilon\) is considered a small positive value. Hence, we have  
\[\text{maximize }z=\sum_{i\in I}\sum_{j\in I}(D_{ij}^{eff}+\beta\cdot(1-\frac{P_{ ij}}{P_{j_{max}}})) \tag{20}\]3  
Moreover, suppose that \(b\) equals \(+\infty\). Consequently, Equation (30) is always satisfied under all values of \(x\). Additionally, to meet the constraint in Equation (26), it is required to guarantee the following:  
\[\text{maximize }z=\sum_{i\in I}\sum_{j\in I}(D_{ij}^{eff}+\beta\cdot(1-\frac{P_{ ij}}{P_{j_{max}}})) \tag{20}\]4  
Thus, a partition problem could be generated from the previous decision problem, where the set of users is divided into two separate sets in order to make the traffic load equal for the two sets. Consequently, this partitioning problem can be reduced to Equation (30). Because the partition problem is considered an NP-complete problem, Equation (20) presents an NP-complete formula. Hence, it is considered an NP-hard problem. To address the problem in real time, in the next section, we propose a heuristic solution.