Convergence Analysis and a Multi-Objective Optimization Algorithm for Bee Colonies Utilizing the Nelder-Mead Method

The Swarm Optimization algorithm, a heuristic grounded in swarm intelligence, has progressively been employed in production scheduling, power and water resource optimization, biomedicine, and other fields. Although considerable advancements have been made for single-objective optimization problems, there exists a need to enhance its performance in an increasing number of multi-objective optimization scenarios and to analyze global convergence for improved applications. In this paper, we propose a novel multi-objective swarm optimization algorithm by integrating the Nelder-Mead algorithm with a non-dominated set classification strategy and an aggregation comparison method. This approach utilizes the Nelder-Mead algorithm to augment the local search capability of the swarm optimization algorithm and combines it with a streamlined computational procedure to demonstrate robust optimization performance. Leveraging Markov theory, we analyze the convergence of the method and show that the proposed algorithm converges with probability 1 to a minuscule set of elementary integers. Additionally, we compare the proposed algorithm with established multi-objective optimization algorithms, such as MOGA, NPGA, NSGA, and NSGA-II, using multi-objective evaluation functions. Simulation results indicate that our proposed method exhibits superior convergence speed and is situated closer to the Pareto optimal frontier.