Fixed or experiential penalty parameter of the penalty-based boundary intersection (PBI) function method cannot simultaneously ensure the convergence and diversity for all shape of Pareto front (PF). Too large penalty parameter may lead to bad convergence while too small parameter can not ensure the diversity. Specially, if the penalty parameter is too small, some reference weight vectors may have no solution on it. This error is hard to be rectified. In this paper, we prove that the lower bound of the penalty parameter is determined by three factors. The first one is the shape of the PF. The second one is the cosine distance between two adjacent reference vectors. The third one is the number of objectives. We deduce the lower bound of the penalty parameter. Once the penalty parameter was calculated, an individual with minimal PBI function is attached to the corresponding reference vector. The minimal-PBI-function-first principle is used in the environmental selection to guarantee the wideness and uniformity of the solution set. The time complexity is low. The proposed method is compared with other three state-of-the-art many-objective evolutionary algorithms on the unconstrained test problems MaOP, DTLZ and WFG with up to fifteen objectives. The experimental results show the competitiveness and effectiveness of the proposed algorithm in both time efficiency and accuracy.

基于惩罚的边界相交（PBI）函数方法的固定或经验惩罚参数不能同时保证所有形状的Pareto前沿（PF）的收敛性和多样性。惩罚参数太大可能会导致收敛性不好，而参数太小又不能保证多样性。特别地，如果惩罚参数太小，某些参考权重向量可能无解。这个错误很难纠正。在本文中，我们证明了惩罚参数的下界是由三个因素决定的。第一个是PF的形状。第二个是两个相邻参考向量之间的余弦距离。第三个是目标的数量。我们推导出惩罚参数的下界。一旦计算出惩罚参数，具有最小 PBI 函数的个体就会被附加到相应的参考向量上。环境选择采用最小PBI功能优先原则，保证解集的广泛性和一致性。时间复杂度较低。在具有多达 15 个目标的无约束测试问题 MaOP、DTLZ 和 WFG 上，将所提出的方法与其他三种最先进的多目标进化算法进行了比较。实验结果表明了该算法在时间效率和准确性方面的竞争力和有效性。