The primary objective of multi-objective evolutionary algorithms (MOEAs) is to find a set of evenly distributed nondominated solutions that approximate the Pareto front (PF) of a multi-objective optimization problem (MOP) or a many-objective optimization problem (MaOP). This implies that the approximated solution set obtained by MOEAs should be as close to PF as possible while remaining diverse, adhering to criteria of convergence and diversity. However, existing MOEAs exhibit an imbalance between achieving convergence and maintaining diversity in the objective space. As far as the diversity criterion is concerned, it is still a challenge to achieve an evenly distributed approximation set with different sizes for a problem with a complicated PF shape. Furthermore, Pareto dominance has its own weaknesses as the selection criterion in evolutionary multiobjective optimization. Algorithms based on Pareto criterion (PC) can suffer from problems such as slow convergence to the optimal front and inferior performance on problems with many objectives. To effectively address these challenges, we propose a multi-objective bean optimization algorithm (MOBOA). Given that the selection of parent species, representing global optimal solutions, directly influences the convergence and diversity of the algorithm, MOBOA incorporates a preference order equilibrium parent species selection strategy (POEPSS). By extending the Pareto criterion with the preference order optimization criterion, the algorithm effectively enhances parent species selection pressure across multiple objectives. To balance convergence and diversity, MOBOA proposes a multi-population global search strategy explicitly maintaining an external archive during the search process. Leveraging the inherent multi-population advantages of bean optimization algorithm (BOA), the algorithm facilitates information sharing among the main population, auxiliary populations, and historical archive solution sets. Additionally, a diversity enhancement strategy is employed in the environmental selection stage, introducing the environmental selection strategy of the SPEA2 algorithm to generate a set of evenly distributed nondominated solutions. Experimental results on a series of widely used MOPs and MaOPs demonstrate that the proposed algorithm exhibits higher effectiveness and competitiveness compared to state-of-the-art algorithms.

多目标进化算法 (MOEA) 的主要目标是找到一组均匀分布的非支配解，这些解近似于多目标优化问题 (MOP) 或多目标优化问题 (MaOP) 的帕累托前沿 (PF) 。这意味着 MOEA 获得的近似解集应尽可能接近 PF，同时保持多样性，遵循收敛性和多样性的标准。然而，现有的 MOEA 在目标空间中实现收敛和保持多样性之间表现出不平衡。就多样性准则而言，对于具有复杂PF形状的问题实现不同大小的均匀分布的近似集仍然是一个挑战。此外，帕累托优势作为进化多目标优化的选择标准也有其自身的弱点。基于帕累托准则 (PC) 的算法可能会遇到诸如收敛到最优前沿速度缓慢以及在多目标问题上性能较差等问题。为了有效解决这些挑战，我们提出了一种多目标 bean 优化算法（MOBOA）。鉴于代表全局最优解的亲本物种的选择直接影响算法的收敛性和多样性，MOBOA采用了偏好顺序平衡亲本物种选择策略（POEPSS）。该算法通过将Pareto准则扩展为偏好顺序优化准则，有效增强了跨多个目标的亲本选择压力。为了平衡收敛性和多样性，MOBOA 提出了一种多群体全局搜索策略，在搜索过程中明确维护外部档案。 该算法利用Bean优化算法（BOA）固有的多群体优势，促进主群体、辅助群体和历史档案解集之间的信息共享。此外，在环境选择阶段采用多样性增强策略，引入SPEA2算法的环境选择策略，生成一组均匀分布的非支配解。在一系列广泛使用的 MOP 和 MaOP 上的实验结果表明，与最先进的算法相比，所提出的算法表现出更高的有效性和竞争力。