Abstract
背景问题
Decomposition based evolutionary algorithms have been quite successful in solving optimization problems involving two and three objectives. Recently there have been some attempts to exploit the strengths of decomposition based approaches to deal with many objective optimization problems. Performance of such approaches are largely dependent on three key factors i.e., (a) means of reference point generation (b) schemes to simultaneously deal with convergence and diversity and finally (c) methods to associate solutions to reference directions.
基于分解的进化算法在解决涉及两个和三个目标的优化问题方面非常成功。 最近，已经有一些尝试利用基于分解的方法的优势来处理许多客观的优化问题。 这些方法的性能在很大程度上取决于三个关键因素，即（a）参考点生成方法（b）同时处理收敛和多样性的方案，以及（c）将解决方案与参考方向相关联的方法。
方法
In this paper, we introduce a decomposition based evolutionary algorithm, wherein, uniformly distributed reference points are generated via systematic sampling, balance between convergence and diversity is maintained using two independent distance measures and a simple preemptive distance comparison scheme is used for association. In order to deal with constraints, an adaptive epsilon formulation is used.
在本文中，我们引入了一种基于分解的进化算法，其中，通过系统采样生成均匀分布的参考点，使用两个独立的距离测量来维持收敛和多样性之间的平衡，并且使用简单的抢占距离比较方案来进行关联。 为了处理约束，使用自适应epsilon公式。
实验结果
The performance of the algorithm is evaluated using standard benchmark problems i.e., DTLZ1- DTLZ4 for 3, 5, 8, 10 and 15 objectives, WFG1-WFG9, the car side impact problem, the water resource management problem and the constrained ten-objective general aviation aircraft (GAA) design problem. Results of problems involving redundant objectives and disconnected Pareto fronts are also included in this study to illustrate the capability of the algorithm. The study clearly highlights that the proposed algorithm is better or at par with recent reference direction based approaches for many objective optimization.
使用标准基准问题评估算法的性能，即针对3,5,8,10和15目标的DTLZ1-DTLZ4，WFG1-WFG9，汽车侧面碰撞问题，水资源管理问题和约束十目标一般航空飞机（GAA）设计问题。 本研究还包括涉及冗余目标和断开的Pareto前沿的问题的结果，以说明算法的能力。 该研究清楚地强调了针对多目标优化，与基于参考方向的方法相比所提出的算法更好或持恒。
关键词
many-objective optimization, decomposition, evolutionary algorithm, adaptive epsilon constraint-handling
多目标优化，分解，进化算法，自适应epsilon约束处理
Conclusion
方法
In this paper, a decomposition based evolutionary algorithm is introduced and its performance is demonstrated using unconstrained and constrained many objective optimization problems. The approach utilizes reference directions to guide the search, wherein the reference directions are generated using a systematic sampling scheme as introduced by Das and Dennis [28]. The algorithm is designed using a steady state form. In an attempt to alleviate the problems associated with scalarization (commonly encountered in the context of reference direction based methods), the balance between diversity and convergence is maintained using a simple preemptive distance comparison scheme. Such a process also eliminates the need for a detailed association and niching operation as employed in NSGA-III. In order to deal with constraints, an epsilon level comparison is used which is known to be more effective than methods employing feasibility first principles.
本文介绍了一种基于分解的进化算法，并利用无约束和约束的多目标优化问题证明了其性能。 该方法利用参考方向来指导搜索，其中参考方向使用Das和Dennis [28]引入的系统采样方案生成。 该算法使用稳态形式设计。 为了减轻与标量化相关的问题（通常在基于参考方向的方法的上下文中遇到），使用简单的抢先距离比较方案来维持多样性和收敛之间的平衡。 这种方法还消除了对NSGA-III中使用的详细关联和小细节操作的需要。 为了处理约束，使用已知比使用可行性第一原理的方法更有效的ε水平比较。
实验证明及未来研究内容
The performance of the algorithm is evaluated using standard benchmark problems i.e., DTLZ1-DTLZ4 for 3, 5, 8, 10 an 15 objectives, WFG1-WFG9, the car side impact problem, the water resource management problem and the constrained ten-objective general aviation aircraft (GAA) design problem. Results of problems involving redundant objectives and disconnected Pareto fronts are also included in this study to illustrate the capability of the algorithm. The results indicate that the proposed algorithm is able to deal with unconstrained and constrained many-objective optimization problems better or at par with existing state of the art algorithms such as NSGA-III and MOEA/D-PBI. In the current form, the population size is set to be the same as the number of reference directions. It is clear that for problems with a large number of reference directions, evolving a large population is not practical and there is a potential to develop archive based schemes. Another direct use of this algorithm would be for robust multi-objective optimization problems, where one can formulate the problem as a many objective optimization problem and obtain solutions with various levels of robustness simultaneously. These directions are currently being explored by the authors.
使用标准基准问题评估算法的性能，即DTLZ1-DTLZ4用于3,5,8,10和15个目标，WFG1-WFG9，汽车侧面碰撞问题，水资源管理问题和约束十目标通用航空飞机（GAA）设计问题。本研究还包括涉及冗余目标和断开的Pareto前沿的问题的结果，以说明算法的能力。结果表明，该算法能够更好地处理无约束和约束的多目标优化问题，或者与现有的NSGA-III和MOEA / D-PBI等现有算法相媲美。在当前形式中，种群大小设置为与参考方向的数量相同。很明显，对于具有大量参考方向的问题，进化大量种群是不实际的，并且存在开发基于档案的方案的潜力。该算法的另一个直接用途是用于鲁棒的多目标优化问题，其中可以将问题表述为许多客观优化问题并且同时获得具有各种稳健性水平的解决方案。这些方向目前正由作者探讨。