# An Improved Pied Kingfisher Optimizer for Maritime UAV Path Planning
## 2 Principle of the Improved PKO Algorithm
### Refracted Opposite-Based Learning  
Opposite-Based Learning (OBL) [24] is a widely applied technique in intelligent optimization algorithms, aimed at expanding the search space to enhance optimization performance. The fundamental concept involves computing the opposite solutions of the current position, enabling the algorithm to explore a broader search area and facilitate quicker discovery of the global optimum during the early stages of the problem-solving process. Specifically, OBL conducts searches based on the symmetric points derived from the current solutions, introducing solutions at symmetric locations within the search space to increase diversity. This approach effectively overcomes the limitations associated with local search in the initial algorithm stages, thereby improving overall performance. Nonetheless, despite its significant advantages in early-stage optimization, OBL's singular approach restricts the algorithm flexibility and renders it susceptible to local optima during later optimization stages. The Refracted Opposite-Based Learning (ROBL) method is introduced in this study to address the limitations of OBL in optimization algorithms more comprehensively. ROBL enhances traditional OBL by incorporating the law of refraction, thereby infusing the exploration of the solution space with increased depth and diversity. The law of refraction describes how light changes its propagation direction when passing through different media. This concept is applied to optimization algorithms to generate solutions that are more profound, enhancing both the thoroughness and accuracy of the search process. Specifically, the principles of OBL can be drawn as an analogy to the process of light propagating through various media. In nature, when light enters a different medium, its propagation path changes due to refraction. Similarly, in ROBL, the exploration path of solutions is adjusted based on specific rules, altering the original single-direction reverse search. These refraction points, determined by various refraction angles or medium characteristics, offer a wider range of search directions. The fundamental concept behind ROBL is to simulate the behavior of light refraction to enrich the process of generating opposite solutions. This procedure involves the following details: First, defining refraction rules to generate multiple potential refraction points from the current solution. Secondly, considering how refraction angles or medium characteristics influence the position and distribution of these points that, subsequently, affect search effectiveness. The basic principles of ROBL are illustrated in Figure 2. In the subsequent search process, the algorithm employs a specific selection mechanism to choose the optimal solution from these refractive points and proceed with the next step of the search. This refraction-based generation mechanism enables the exploration of a wider solution space, thus improving both the effectiveness and efficiency of optimization. The specific mathematical model is delineated as follows:  
\[\frac{h}{h^{\prime}}=\frac{\frac{lb+ub}{2}-x}{x^{\prime}-(\frac{lb+ub}{2})} \tag{12}\]  
where the search interval on the X-axis is considered to be bounded by the lower limit \(lb\) and the upper limit \(ub\), with the Y-axis depicting a convex lens. Suppose an incoming light ray \(l\) casts a projection \(x\) on the X-axis and has a height \(h\). Upon refraction through the lens, the emergent ray \(l^{\prime}\) exhibits a projection \(x^{\prime}\) on the X-axis. Let \(k=h/h^{\prime}\), the refracted opposite solution obtained after transformation is:  
\[x^{\prime}_{i,\ j}=\frac{lb_{j}+ub_{j}}{2}+\frac{lb_{j}+ub_{j}}{2k}-\frac{x_{i, \ j}}{k} \tag{13}\]  
where \(x^{\prime}_{i,\ j}\) represents the refracted opposite solution, while \(lb_{j}\) and \(ub_{j}\) denote the minimum and maximum values in the j-dimensional space. The value of \(k\) is not a constant; rather, it dynamically varies with the number of iterations.