# Bi-Objective Power Optimization of Radio Stripe Uplink Communications
## 3 Uplink Power Optimization Problem
### Multi-Objective Optimization Based on DE  
In MO mathematical programming there are multiple OFs to be optimized. The solutions in the decision variable space are mapped onto the OF space, where their components are the corresponding values of each OF. In general, the purpose of MO optimization is to find efficient solutions in the decision variable space leading to non-dominated solutions in the OF space. These are characterized by the absence of another feasible solution that improves simultaneously all OFs, i.e., the improvement of an OF value can only be achieved by accepting that, at least, another OF value is degraded [23]. The non-dominated front includes the solutions from which a final choice (or ranking) should be made, considering the underlying trade-offs between the competing OFs [23].  
The DE algorithm can be employed in MO optimization with the same solution generation and selection mechanisms to those used in the SO case. However, the OF assignment and archiving functions will require specific structure. The OF assignment must guarantee a correct ordering between non-dominated solutions according to their preference relations.  
In this work, the ranking approach is completed through a front index similar to the one employed in the non-dominated sorting genetic algorithm II (NSGA-II) [24]. In this approach, the index 0 is assigned to all non-dominated solutions in the population, forming the non-dominated front. The set of solutions with index 0 is denoted as \(\mathcal{G}_{0}\). A new front is created from the remaining solutions, where the non-dominated solutions relative to this set are determined and assigned index 1, \(\mathcal{G}_{1}\). The subsequent front, \(\mathcal{G}_{2}\), is assigned with index 2, and so on, increasing the index by one unit for each new front, until all points of the population are labeled. The set \(\mathcal{G}_{j}\) denotes non-dominated front that contains all solutions that are dominated by at least one solution from set \(\mathcal{G}_{j-1}\).  
In order to generate a representative distribution of the non-dominated set it is also necessary to add a factor to untie the solutions that have the same index, while preserving the relative order between the solutions that have different indices. The NSGA-II algorithm [24] proposed the use of crowding distance, \(\varrho\), in which the ordering of solutions with the same index, \(\mathcal{G}_{j}\) with cardinality \(G_{j}=\mathrm{card}\left(\mathcal{G}_{j}\right)\), is performed in ascending order of one OF, being labeled with indices \(x=1\) to \(x=G_{j}\). Then, for each solution from \(x=2\) to \(x=G_{j}-1\), \(\varrho\) is obtained  
\[\varrho\left(\mathbf{p}^{(t,x)}\right)=C_{p}\left[f_{1}\left(\mathbf{p}^{(t,x -1)}\right)-f_{1}\left(\mathbf{p}^{(t,x+1)}\right)\right]+\left[f_{2}\left( \mathbf{p}^{(t,x+1)}\right)-f_{2}\left(\mathbf{p}^{(t,x-1)}\right)\right]. \tag{23}\]  
This factor is proportional to the perimeter of a rectangle whose opposite vertices are the neighboring solutions of \(\mathbf{p}^{(t,x)}\). \(C_{p}\) is a constant that takes into account the different orders of magnitude between the OFs. The larger \(\varrho\left(\mathbf{p}^{(t,x)}\right)\) the further away solution \(\mathbf{p}^{(t,x)}\) is from its neighbors, and therefore the more important is this solution to guarantee the representativeness of the set of solutions. Finally, the archiving operation leverages on the NSGA-II solution ordering mechanism by keeping a set of \(N_{arch}\) solutions from multiple non-dominated frontiers at each generation. This means that in every generation \(t\), the same ranking approach is performed on the set \(\left\{N_{arch},N_{pop}\right\}\), that contains both the \(N_{pop}\) new solutions and the \(N_{arch}\) non-dominated solutions saved in the archive from the previous generation \(t-1\). From this procedure, the \(N_{arch}\) best ranking solutions are kept in the archive in generation \(t\).