# Energy-Efficient BWP Configuration for Multi-Slice Users
## 5 Centralized Approach: Optimization Problem
### The Objective Function  
The optimization problem is the following:  
\[ \begin{split}&\min_{A1,A2}\sum_{u\in\mathcal{P}}C_{u,MN}(A1)+\sum_ {u\in\mathcal{P}}C_{u,SN}(A2)\\ &\text{subject to:}\\ & a_{1u},a_{2u}\in\{0,1\},\forall u\in\mathcal{P}\\ & a_{1u}+a_{2u}=1,\forall u\in\mathcal{P}\end{split}  \tag{6}\]  
where  
* \(A1=(a_{1u})_{u\in\mathcal{P}}\) is the vector of binary decision variable \(a_{1u}\) which is equal to 1 when UE \(u\) has the \(MN\) strategy configured and equal to 0 otherwise. * \(A2=(a_{2u})_{\forall u\in\mathcal{P}}\) is the vector of binary decision variable \(a_{2u}\) which is equal to 1 when UE \(u\) has the \(SN\) strategy configured and equal to 0 otherwise. * \(C_{u,MN}\) and \(C_{u,SN}\) are the cost functions when UE \(u\) is configured with the \(MN\) and \(SN\) strategy, respectively, and are determined in the below Equations (7) and (8). \[ \begin{split} C_{u,MN}(A1)&=a_{1u}\cdot\frac{\alpha _{u,eb}\cdot V_{u,eb}\cdot D1\cdot(\sum_{k\
eq u}a_{1k}+1)}{\text{TotPRB}_{ BWP1}\cdot\text{Bits}\_per\_PRB_{u}}\\ &\quad+a_{1u}\cdot\frac{\alpha_{u,uc}\cdot V_{u,uc}\cdot D2\cdot( \sum_{k\
eq u}a_{1k}+1)}{\text{TotPRB}_{BWP2}\cdot\text{Bits}\_per\_PRB_{u}}\\ &\quad\quad+a_{1u}\cdot(\theta_{MN}\cdot SwDl_{u}+\beta_{u}\cdot B _{MN})\\ & C_{u,SN}(A2)&=a_{2u}\cdot\frac{\alpha_{u,eb}\cdot V _{u,eb}\cdot D3\cdot(\sum_{k\
eq u}a_{2k}+1)}{\text{TotPRB}_{BWP3}\cdot\text{ Bits}\_per\_PRB_{u}}\\ &\quad+a_{2u}\cdot\frac{\alpha_{u,uc}\cdot V_{u,uc}\cdot D3\cdot( \sum_{k\
eq u}a_{2k}+1)}{\text{TotPRB}_{BWP3}\cdot\text{Bits}\_per\_PRB_{u}}\\ &\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+a_{2u }\cdot(\beta_{u}\cdot B_{SN})\end{split}  \tag{7}\]  
with \(D1\), \(D2\) and \(D3\) representing the duration of the TTI when UE \(u\) is served by either BWP 1, 2 or 3, respectively. Equations (7) and (8) are derived from Equations (1) and (3)-(5) after replacing each term. To note that the number of users attached to each BWP is determined by the sum of the binary decision variables which leads to \(N_{users,BWP1}=N_{users,BWP2}=\sum_{k\
eq u}a_{1k}+1\) and \(N_{users,BWP3}=\sum_{k\
eq u}a_{2k}+1\). Therefore, the aim of this optimization problem is to minimize the cost function of all users globally by choosing for every user the adequate binary variables \(a_{1u}\) and \(a_{2u}\). Note that the constraint \(a_{1u}+a_{2u}=1,\forall u\in\mathcal{P}\) is added to limit the user's selection to only one strategy. The optimization problem defined in Equation (6) is non-convex, since it is an integer (binary variables) non-linear problem due to the product of the decision variables with \(a_{1u}\cdot(\sum_{k\
eq u}a_{1k}+1)\) and \(a_{2u}\cdot(\sum_{k\
eq u}a_{2k}+1)\). Therefore, an exhaustive search may be used to solve this problem. With exhaustive search, all possible combinations for user distributions among the BWPs are assessed to choose the one with the lowest cost function, as seen in the below Algorithm 2. Nonetheless, this approach is highly time consuming with a complexity of \(2^{N_{users}}\). For this reason, we transform the non-linear integer optimization problem in (6) to a linear one by replacing the multiplicative decision variables in what follows. ```
Input: Cost function for all \((a_{1u},a_{2u})_{\forall u\in\mathcal{P}}\) combinations possible (\(2^{N_{users}}\) combinations) Output: Combination of \((a_{1u},a_{2u})_{\forall u\in\mathcal{P}}\) with the lowest cost function for all users repeat  Calculation of BWP configuration cost function for all users \(\sum_{u\in\mathcal{P}}C_{u,MN}+C_{u,SN}\); Store the combination of \((a_{1u},a_{2u})_{\forall u\in\mathcal{P}}\) with the calculated BWP configuration cost function for all users; until Every combination of \((a_{1u},a_{2u})_{\forall u\in\mathcal{P}}\) is executed;  Select \((a_{1u},a_{2u})_{\forall u\in\mathcal{P}}\) with the lowest BWP configuration cost function for all users. ```  
**Algorithm 2**BWP Configuration Selection Exhaustive Search Algorithm