## 3 System Model and Problem Description
### UAV Trajectory Optimization Problem  
The corresponding solving variables for the UAV flight trajectory optimization problem under the condition of a known user device offloading rate are \(c_{(t)}=\left[q_{(t)},h_{(t)}\right]=\left[x_{(t)},y_{(t)},z_{t}\right]\), and this sub-problem corresponds to the constraint conditions C1\(\sim\)C5 in P1. The expression for question P4 is  
\[P4:\binom{max}{c_{(t)}}f_{2}(\gamma)\ \ \ s.t.\ C1\sim\text{C5} \tag{25}\]  
where \(f_{2}(\gamma)\) is related to \(\gamma\). \[f_{2}(\gamma)=-\gamma\left[\sum_{t=1}^{N}\sum_{u=1}^{U}\left(E_{u}^{rec}(t)+E_ {u}^{comp}(t)\right)+\sum_{t=1}^{N}E^{fly}(t)\right] \tag{26}\]  
It is obvious that problem P4 is nonlinear and complicated. In order to balance computational efficiency and solution quality, this paper uses the Arctic puffin optimization (APO) algorithm to solve this problem. The APO algorithm is a metaheuristic optimization algorithm proposed by [PERSON] et al. [29], inspired by the survival and predation behavior of Arctic puffins. Due to its unique algorithm structure and novel iterative method, it has strong adaptive optimization capabilities. The algorithm mainly consists of three stages: population initialization, aerial flight (global exploration), and underwater foraging (local development). The aerial flight stage utilizes the Levy flight and velocity factor mechanism to improve the global search ability, while the underwater foraging stage enhances the local development ability through synergistic effects and adaptive change factors. The following provides the biological mechanism and key description of using the APO algorithm to solve problem P4. Design of fitness function  
Based on the objective function \(f_{2}(\gamma)\) and corresponding constraint conditions, a penalty function is designed to jointly construct the fitness evaluation criteria \(f_{2}(\gamma)^{\prime}\). For solutions that violate the constraint conditions, the penalty term \(P(\gamma)\) is defined as  
\[P(\gamma)=\sum_{k=1}^{5}C_{k}*max(0,g_{k}(\gamma))^{2} \tag{27}\]  
where \(C_{k}(k=1,2,3,4,5)\) is the dynamic penalty coefficient used to adjust the punishment intensity, which increases with the number of iterations, and \(g_{k}(\gamma)\) is the standard form of the constraint function corresponding to the \(k\)-th constraint condition. The final fitness function can be expressed as  
\[f_{2}(\gamma)^{\prime}=f_{2}(\gamma)+P(\gamma) \tag{28}\]By balancing the value of the objective function with the degree of constraint violation, \(f_{2}(\gamma)^{{}^{\prime}}\) selects high-quality populations and eliminates inefficient solutions. (2) Population initialization  
The initialization population of the APO algorithm is an \(N_{p2}*D_{2}\) matrix \(X=\left[X_{1},X_{2},\ldots,X_{Np_{2}}\right]^{T}\), where \(N_{p2}\) is the population size, \(D_{2}\) is the variable dimension, and \(X_{i}\) is a \(D_{2}\) dimensional variable, where \(X_{i}=\left[X_{i}^{1},X_{i}^{2},\ldots,X_{i}^{D_{2}}\right]\) corresponds to the discrete coordinates of the 3D trajectory of the UAV in \(N\) time slots. In the classical APO algorithm, the initial individual's position is usually randomly generated within a pre-determined range. To improve algorithm efficiency, the search direction is guided by initial values to avoid complete randomness and accelerate algorithm convergence. Among them, the initial value is obtained based on prior knowledge of the problem (corresponding to the coordinates of the UAV trajectory in the previous iteration when the inner loop alternately optimizes the UAV trajectory and offloading rate in Algorithm 1), and then Gaussian random perturbations are applied to the initial value to guide the search direction, balancing diversity and convergence. (3) Aerial flight phase  
There are two main strategies used during the aerial flight phase. The first strategy is the aerial search, which simulates the low-altitude collaborative reconnaissance behavior of Arctic puffin populations to search for areas with abundant prey and fewer natural enemies.