Constrained Hybrid Metaheuristic: A Universal Framework for Continuous Optimisation

Optimization is a foundational element across scientific and engineering disciplines, involving tasks such as designing efficient transportation networks, tuning hyperparameters in machine learning models, and scheduling resources in industrial processes. Many modern optimization problems are characterized by high dimensionality, nonlinearity, and non-convexity, leading to traditional optimization techniques often failing in these settings. Additionally, with the rise of applications such as deep learning, financial modeling, and network design, the need for algorithms that can handle noisy, incomplete, or non-differentiable objective functions is increasingly common. Recent developments in metaheuristic optimization have increasingly focused on enhancing algorithmic adaptability, scalability, and real-world applicability.

Many modern optimization problems are characterized by high dimensionality, nonlinearity, and non-convexity, leading to traditional optimization techniques often failing in these settings. Particularly for objective functions with complex, heterogeneous, or unknown properties, existing metaheuristic algorithms often need to be tailored to specific function classes or problem domains, lacking universality and adaptability. Additionally, the need for algorithms that can handle noisy, incomplete, or non-differentiable objective functions is increasingly common, posing higher demands on optimization algorithms.

The core innovation of the cHM algorithm lies in its universality and adaptability. Firstly, it employs a modular structure and two-phase operation, allowing dynamic adaptation to problem characteristics. Secondly, the cHM algorithm harnesses synergy between candidate solutions and metaheuristic strategies, applying the most appropriate search behavior during the optimization process, thus improving convergence and robustness. Unlike most algorithms based on specific natural phenomena, cHM does not rely on the mechanisms of a single model but provides a flexible framework capable of adapting to various problem structures.

• �� The cHM algorithm employs a modular structure and two-phase operation, allowing dynamic adaptation to problem characteristics.
• �� In the first phase, the cHM algorithm iterates over all available inner methods and probes each of them to find the best one for a given optimization phase.
• �� In the second phase, the cHM selects the best inner metaheuristics from the probing phase and performs the optimization.
• �� The algorithm harnesses synergy between candidate solutions and metaheuristic strategies, applying the most appropriate search behavior during the optimization process.
• �� The cHM algorithm is capable of handling complex, non-convex, and heterogeneous objective functions, providing a universal solution.

The cHM algorithm was evaluated on 28 benchmark functions, including various complexities and characteristics such as non-convexity, non-separability, and varying smoothness. Experiments were conducted on a 12th Gen Intel Core i5-12450H 2.00 GHz processor with 16GB of RAM. Each benchmark function was run 50 times, using absolute difference as the fitness function and calculating the Euclidean distance between the generated solution and the optimal one.

Extensive experimental evaluation on 28 benchmark functions demonstrates that the cHM algorithm consistently matches or outperforms traditional metaheuristics in terms of solution quality and convergence speed. For instance, on certain non-convex functions, cHM's convergence speed improved by approximately 30%. Additionally, in a practical application for feature selection, the cHM algorithm excelled in the context of data classification, with classification accuracy improved by approximately 15% over traditional methods. Ablation studies confirmed the contribution of synergy among different metaheuristic strategies in the cHM algorithm to optimization performance.

The cHM algorithm, as a versatile black-box optimizer, has broad application potential. It can be used in feature selection, data classification, bioinformatics, financial modeling, and other fields. Its modular and adaptive nature makes it widely applicable across various application scenarios, particularly in handling complex, non-convex, and heterogeneous objective functions.

The cHM algorithm may encounter performance bottlenecks when dealing with extremely high-dimensional problems, as its synergy mechanism requires frequent switching between multiple metaheuristic strategies. Additionally, for certain types of objective functions, the initial parameter selection of the cHM algorithm may affect its final optimization performance. In some cases, the computational cost of the cHM algorithm may be high, especially in complex problems requiring numerous function evaluations.