A Hybrid CMA-ES–GWO Algorithm for Solving Optimization Problems

Abstract

Metaheuristic algorithms often suffer from an imbalance between exploration and exploitation, which can lead to premature convergence. In this paper, a hybrid optimization algorithm is proposed by integrating the Grey Wolf Optimizer (GWO) and the Covariance Matrix Adaptation Evolution Strategy (CMA-ES), termed the Hybrid CMA-ES–GWO Algorithm (HCG). The proposed hybridization is motivated by the strong exploration capability of GWO and the efficient exploitation of CMA-ES. The HCG algorithm adopts a multi-population framework in which GWO is first employed to explore the search space. Subsequently, CMA-ES is applied to exploit these regions through covariance-based sampling, enabling accurate convergence toward optimal solutions. This cooperative mechanism allows HCG to effectively balance global search and local refinement while maintaining population diversity. To evaluate the performance of the algorithm, experiments were conducted on 19 benchmark functions, including unimodal, multimodal, and fixed-dimension multimodal problems. The performance of HCG was compared with the original CMA-ES and GWO algorithms. The experimental results demonstrate that HCG consistently achieves superior optimization accuracy compared to the standalone algorithms. These findings confirm that the proposed hybrid strategy effectively overcomes the individual limitations of CMA-ES and GWO, making HCG a reliable method for solving complex continuous optimization problems.