## 6 Optimization Scenarios for 6G MCNs
### Energy-Efficient Communication for UAVs and AUVs  
UAVs and AUVs play critical roles in surveillance, data relaying, monitoring, and exploration. However, both UAVs and AUVs are subject to strict energy constraints stemming from limited onboard battery capacity and the high cost of recharging or recovery, especially in long-duration, remote, or autonomous missions [89]. Consequently, there is a need to design communication strategies that minimize energy consumption while ensuring that data transmission requirements are met [90; 91]. These strategies must intelligently adjust transmission power, data rate, or active/sleep scheduling, often under uncertain and dynamic network conditions. **Optimization Problem Formulation:** Let \(\mathcal{N}\) be the set of UAVs and AUVs, each needing to transmit a certain amount of data to a destination node (e.g., a base station, a satellite, or another vehicle). For each node \(i\in\mathcal{N}\), we can define: \(P_{i}\) as the transmission power of node \(i\) (decision variable); \(R_{i}(P_{i})\) as the data rate of node \(i\) as a function of \(P_{i}\); \(E_{i}\) as the energy consumed from node \(i\) (typically proportional to transmission power); \(R_{min,i}\) as the minimum required rate (QoS constraint); and \(P_{max,i}\) as the maximum allowed transmit power. Thus, we aim to minimize the total communication energy while satisfying rate and power constraints, as reflected in the following problem:  
\[ \begin{split}\min_{P_{i}}\sum\limits_{i\in\mathcal{N}}\frac{P_{i }}{\eta_{i}}\\ \text{s.t. (C1)}& R_{i}(P_{i})=B_{i}\text{log}_{2}\bigg{(}1 +\frac{h_{i}P_{i}}{N_{0}B_{i}}\bigg{)}\geq R_{min,i},\ \forall i\in\mathcal{N}\\ \eqref{eq:P_{i}}& 0\leq P_{i}\leq P_{max,i},\ \forall i\in\mathcal{N}\end{split}  \tag{4}\]  
where \(\eta_{i}\in(0,1]\) is the power amplifier efficiency of node \(i\). The objective function represents the energy required for transmission over a given duration (e.g., normalized to 1 time unit for simplicity). The solution to this problem regulates the transmitting power of nodes so as to ensure minimized energy consumption while respecting the constraints (C1) and (C2). The first constraint guarantees a minimum data rate QoS, assuming that \(B_{i}\) is the bandwidth for transmission, \(h_{i}\) channel gain (may vary based on altitude, sea clutter, or mobility) and \(N_{0}\) noise power spectral density. The second constraint ensures that power levels are non-negative and upper-bounded by the nodes' power budget. **Potential Solution Approaches:** Problem (4) can be solved by different optimization algorithms to find the power allocation that minimizes nodes' energy consumption. Below, we summarize some of the potential solvers that can be used to tackle problem (4):  
1. _Convex Optimization with Dual Decomposition_[92]: When the data rate function \(R_{i}(P_{i})\) is convex and differentiable (as in log-based Shannon capacity models), the optimization problem is convex and can be solved using Lagrangian duality [93], dual decomposition, or Water-filling algorithm (in multi-channel cases) [94]. Lagrangian duality introduces Lagrange multipliers for the QoS constraints and is solved via the Karush-Kuhn-Tucker (KKT) conditions [93]. In dual decomposition methods, we separate the global problem into per-node subproblems and solve them in parallel. Finally, multi-channel Water-filling solutions can be used to allocate more power to better links while ensuring the total energy is minimized. These approaches can be easily implemented and compared with each other, and they are applicable in centralized MCNs (such as when a Sea BS controls a group of AUVs/UAVs). 2. _DRL for Distributed Energy Optimization:_ When system dynamics are non-stationary or partially observable (e.g., unknown channel gains, time-varying loads, or node failures), DRL offers a powerful model-free solution [95]. Each UAV/AUV can be modeled as an agent that interacts with its environment.