Random Walk on Bézier Curves for Global Optimization

2026-06-25

AI Summary

This study addresses the persistent challenge of balancing exploration and exploitation in metaheuristic optimization by proposing a geometrically grounded, interpretable framework that reformulates evolutionary search as adaptive trajectory construction in decision space. The approach integrates variable-order Bézier curve modeling with distance-aware random walks, enabling a smooth transition from global exploration to local exploitation through dynamic adjustment of the curve order. Comprehensive experiments on 41 benchmark functions from CEC2017 and CEC2022 suites across dimensions 10 to 100 demonstrate that the proposed method significantly outperforms 13 state-of-the-art optimizers. Furthermore, its efficacy and robustness are validated on five constrained engineering design problems, confirming its practical applicability beyond synthetic benchmarks.

Abstract

Balancing exploration and exploitation remains a central challenge in metaheuristic optimization. To address this issue, this paper proposes Bézier Walk Evolution (BWE), a geometry-driven optimization framework that reformulates evolutionary search as adaptive trajectory construction in the decision space. BWE integrates Bézier curve modeling with a distance-aware random walk mechanism to generate topology-guided search trajectories. By adaptively varying the curve order during evolution, the proposed method enables a smooth transition from diversified global exploration to refined local exploitation. Higher-order Bézier curves leverage multiple population-derived control points to enhance search diversity, while lower-order curves generate near-linear trajectories to improve convergence efficiency. This adaptive geometric search mechanism provides an interpretable alternative to conventional nature-inspired designs. Extensive experiments on 41 benchmark functions from the CEC2017 and CEC2022 suites, spanning dimensions from 10 to 100, show that BWE achieves strong overall performance and favorable scalability compared with 7 classical and 6 state-of-the-art optimizers, including L-SHADE and CMA-ES. Additional evaluations on five constrained engineering design problems further demonstrate the practical applicability and robustness of BWE.

Problem

• exploration-exploitation balance
• metaheuristic optimization
• global optimization
• evolutionary search
• trajectory construction

Innovation

• Bézier curves
• random walk
• adaptive trajectory
• global optimization
• exploration-exploitation balance