## 1 Introduction  
In recent years, global optimization problems have found extensive applications across various fields, including business, engineering, and science. These problems typically involve input variables that are nonconvex, discontinuous, or high-dimensional. Since the 1990s, bio-inspired computing algorithms (BCAs) [1], drawing on principles from bionics, mathematics, computer science, and other disciplines, have been extensively researched, particularly in addressing NP-hard problems [2]. The BCAs are generally categorized into three main types: evolutionary algorithms, physics-inspired algorithms, and swarm intelligence algorithms. The first category includes methods such as differential evolution [3] and genetic algorithms [4], with extensions such as the deep learning-enhanced differential evolution approach [5], triple competitive differential evolution [6], modified genetic algorithms [7], and the hybrid quantum genetic algorithm [8]. The second category encompasses algorithms that emulate physical and chemical laws observed in nature, including the car tracking optimization algorithm [9], earthworm optimization algorithm [10], lightning search algorithm [11], and atom search optimization [12]. In addition to classical swarm intelligence algorithms, such as particle swarm optimization [13], ant colony optimization [14], and artificial bee colony [15], some novel methods mimicking the social behaviors of animals have emerged, including pigeon-inspired optimization (PIO) [16], the gray wolf optimizer [17], and the whale optimization algorithm [18]. The pigeon-inspired optimization (PIO) algorithm, inspired by pigeons' homing behaviors, demonstrates strong performance across various applications, including UAV formation [19], image processing [20], path planning [21], optimal control [22], and other problem domains [23]. Numerous PIO variants have been developed, which can be grouped into four main categories [24]. (1) Component replacement: These variants introduce parameter regulation mechanisms into the PIO through coefficient modifications, such as the Gaussian mutation PIO [25] and Cauchy mutation PIO [26]. (2) Operation addition: This category includes hybrids that incorporate heterogeneous learning mechanisms, such as quantum entanglement PIO within comprehensive learning PIO variants [27; 28]. (3) Structure adjustment: The modifications address both the sequencing of operators and classified updating of individuals, such as the adjacent disturbances and integrated-dispatching PIO [29] and the social class PIO [30]. (4) Application expansion: For multi-objective PIO, pigeons are evaluated using non-dominated sorting and crowded comparison [31]. In discrete manufacturing, a discrete knowledge-based PIO has been applied to optimize energy efficiency [32]. In recent decades, UAVs have achieved remarkable advancements due to rapid technological progress, leading to successful deployment across various fields. Nevertheless, individual UAVs remain limited in adaptability and capability for executing complex tasks, making UAV cooperation a primary focus. As a significant and challenging research topic, cooperative path planning for UAVs and trajectory tracking control based on path planning results has garnered considerable attention from researchers [33]. Path-planning models and efficient algorithms are necessary, such as the A\({}^{*}\) algorithm [34], artificial potential field [35], Voronoi diagram [36], Dubins curve [37], and mesh grid models [38], are crucial to generate suitable paths. However, most of them are designed for single-UAV scenarios and may not be well suited for multi-UAV operations. Although the artificial potential field (APF) method is effective in achieving collision-free, cohesive UAV flights, its tendency toward local minima limits its application in complex environments [39]. Sequential convex programming (SCP) has been recognized as a primary solution, yet its high computational cost hinders practical application.