The existing multiobjective evolutionary algorithms (EAs) based on nondominated sorting may encounter serious difficulties in tackling many-objective optimization problems (MaOPs), because the number of nondominated solutions increases exponentially with the number of objectives, leading to a severe loss of selection pressure. To address this problem, some existing many-objective EAs (MaOEAs) adopt Euclidean or Manhattan distance to estimate the convergence of each solution during the environmental selection process. Nevertheless, either Euclidean or Manhattan distance is a special case of Minkowski distance with the order ${P=2}$ or ${P=1}$ , respectively. Thus, it is natural to adopt Minkowski distance for convergence estimation, in order to cover various types of Pareto fronts (PFs) with different concavity–convexity degrees. In this paper, a Minkowski distance-based EA is proposed to solve MaOPs. In the proposed algorithm, first, the concavity–convexity degree of the approximate PF, denoted by the value of ${P}$ , is dynamically estimated. Subsequently, the Minkowski distance of order ${P}$ is used to estimate the convergence of each solution. Finally, the optimal solutions are selected by a comprehensive method, based on both convergence and diversity. In the experiments, the proposed algorithm is compared with five state-of-the-art MaOEAs on some widely used benchmark problems. Moreover, the modified versions for two compared algorithms, integrated with the proposed ${P}$ -estimation method and the Minkowski distance, are also designed and analyzed. Empirical results show that the proposed algorithm is very competitive against other MaOEAs for solving MaOPs, and two modified compared algorithms are generally more effective than their predecessors.

中文翻译：

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基于Minkowski距离的多目标优化进化算法。

现有的基于非支配排序的多目标进化算法（EA）在解决多目标优化问题（MaOP）时可能会遇到严重的困难，因为非支配解决方案的数量会随着目标数量的增加而呈指数增长，从而导致选择压力的严重损失。为了解决这个问题，一些现有的多目标EA（MaOEAs）采用欧氏距离或曼哈顿距离来估计环境选择过程中每个解决方案的收敛性。然而，欧氏距离或曼哈顿距离是Minkowski距离的特例，其阶数为 $ {P = 2} $ 或者 $ {P = 1} $ ， 分别。因此，采用Minkowski距离进行收敛估计是很自然的，以覆盖具有不同凹凸度的各种类型的Pareto前沿（PF）。在本文中，提出了一种基于Minkowski距离的EA来解决MaOP。在所提出的算法中，首先，近似PF的凹凸度由的值表示 $ {P} $ ，是动态估算的。随后，明可夫斯基距离的阶数 $ {P} $ 用于估计每个解决方案的收敛性。最后，基于收敛性和多样性，通过综合方法选择最优解。在实验中，在一些广泛使用的基准问题上，将所提出的算法与五个最新的MaOEA进行了比较。此外，针对两个比较算法的修改版本，与建议的集成 $ {P} $ 估计方法和Minkowski距离也进行了设计和分析。实验结果表明，所提出的算法在解决MaOP方面与其他MaOEA相比具有很大的竞争力，并且两种经过比较的改进算法通常比其前身更有效。