A novel transcendental metaphor metaheuristic algorithm based on power method.

Area of Science:

• Computational Mathematics
• Optimization Theory
• Algorithm Development

Background:

• Complex optimization problems and eigenvalue problems in large sparse matrices present significant computational challenges.
• Existing metaheuristic algorithms often struggle with balancing exploration and exploitation, leading to local optima.
• The power iteration method provides a foundation for developing novel optimization strategies.

Purpose of the Study:

• To introduce the Power Method Algorithm (PMA), a novel metaheuristic algorithm inspired by the power iteration method.
• To evaluate the performance of PMA on benchmark functions and real-world engineering optimization problems.
• To demonstrate PMA's effectiveness in solving complex optimization tasks and its balance between exploration and exploitation.

Main Methods:

• The Power Method Algorithm (PMA) simulates the dominant eigenvalue and eigenvector computation process.
• PMA incorporates stochastic angle generation and adjustment factors for enhanced problem-solving.
• The algorithm was rigorously tested on 49 benchmark functions from CEC 2017 and CEC 2022 test suites and eight real-world engineering problems.

Main Results:

• PMA outperformed nine state-of-the-art metaheuristic algorithms across various dimensions, evidenced by low average Friedman rankings (3, 2.71, 2.69).
• Statistical tests (Wilcoxon rank-sum, Friedman) confirmed PMA's robustness and reliability.
• PMA consistently delivered optimal solutions for real-world engineering optimization problems, showcasing high convergence efficiency and avoidance of local optima.

Conclusions:

• The Power Method Algorithm (PMA) is a competitive and practically valuable metaheuristic for interdisciplinary complex optimization tasks.
• PMA effectively balances exploration and exploitation, ensuring efficient convergence and optimal solution discovery.
• The algorithm's performance on benchmark and real-world problems highlights its potential for addressing eigenvalue problems in large sparse matrices.