A hybrid pied kingfisher optimizer for constrained continuous optimization: application to 3D UAV path planning

Qiaoping Li; Jiahao Wan; Qiaoping Li; Jiahao Wan

doi:10.3934/math.20261123

AIMS Mathematics

2026, Volume 11, Issue 9: 28161-28201. doi: 10.3934/math.20261123

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Research article

A hybrid pied kingfisher optimizer for constrained continuous optimization: application to 3D UAV path planning

Qiaoping Li ^,,
Jiahao Wan

School of Economics, Zhengzhou University of Aeronautics, Zhengzhou 450046, China

Received: 08 May 2026 Revised: 12 August 2026 Accepted: 19 August 2026 Published: 03 September 2026
MSC : 90C26, 90C59, 68T20

In this paper, we proposed a hybrid pied kingfisher optimizer (HPKO) for constrained continuous optimization and applied it to three-dimensional unmanned aerial vehicle (3D UAV) path planning under terrain and threat constraints. Based on the original pied kingfisher optimizer (PKO), HPKO integrated tent chaotic initialization, refracted opposition-based learning (ROBL), and intelligent boundary control (IBC) to enhance population diversity, local optimum escape, and solution feasibility near variable bounds. HPKO was evaluated on the CEC2017 and CEC2022 benchmark suites and compared with 13 well-known optimization algorithms in terms of solution quality, convergence behavior, robustness, and statistical significance. Results showed that HPKO maintains strong overall competitiveness across search landscapes and dimensional settings. Ablation studies showed that ROBL provides the strongest standalone improvement across the four function categories, while the complete three-module configuration achieved the best overall rank. In the 3D UAV simulations, HPKO achieved the lowest composite cost among HPKO, PKO, PSO, and GWO in simple and complex terrain–threat environments, with a larger margin in the complex scenario. Weight sensitivity analysis further quantified the trade-offs among path length, external threat exposure, and altitude behavior. These results demonstrated that HPKO serves as an effective constrained optimization framework with practical value for 3D UAV path planning.
- constrained optimization,
- metaheuristic algorithm,
- boundary handling,
- opposition-based learning,
- 3D UAV path planning
Citation: Qiaoping Li, Jiahao Wan. A hybrid pied kingfisher optimizer for constrained continuous optimization: application to 3D UAV path planning[J]. AIMS Mathematics, 2026, 11(9): 28161-28201. doi: 10.3934/math.20261123

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Abstract

In this paper, we proposed a hybrid pied kingfisher optimizer (HPKO) for constrained continuous optimization and applied it to three-dimensional unmanned aerial vehicle (3D UAV) path planning under terrain and threat constraints. Based on the original pied kingfisher optimizer (PKO), HPKO integrated tent chaotic initialization, refracted opposition-based learning (ROBL), and intelligent boundary control (IBC) to enhance population diversity, local optimum escape, and solution feasibility near variable bounds. HPKO was evaluated on the CEC2017 and CEC2022 benchmark suites and compared with 13 well-known optimization algorithms in terms of solution quality, convergence behavior, robustness, and statistical significance. Results showed that HPKO maintains strong overall competitiveness across search landscapes and dimensional settings. Ablation studies showed that ROBL provides the strongest standalone improvement across the four function categories, while the complete three-module configuration achieved the best overall rank. In the 3D UAV simulations, HPKO achieved the lowest composite cost among HPKO, PKO, PSO, and GWO in simple and complex terrain–threat environments, with a larger margin in the complex scenario. Weight sensitivity analysis further quantified the trade-offs among path length, external threat exposure, and altitude behavior. These results demonstrated that HPKO serves as an effective constrained optimization framework with practical value for 3D UAV path planning.

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