The multiobjective evolutionary optimization algorithm (MOEA) is a powerful approach for tackling multiobjective optimization problems (MOPs), which can find a finite set of approximate Pareto solutions in a single run. However, under mild regularity conditions, the Pareto optimal set of a continuous MOP could be a low dimensional continuous manifold that contains infinite solutions. In addition, structure constraints on the whole optimal solution set, which characterize the patterns shared among all solutions, could be required in many real-life applications. It is very challenging for existing finite population based MOEAs to handle these structure constraints properly. In this work, we propose the first model-based algorithmic framework to learn the whole solution set with structure constraints for multiobjective optimization. In our approach, the Pareto optimality can be traded off with a preferred structure among the whole solution set, which could be crucial for many real-world problems. We also develop an efficient evolutionary learning method to train the set model with structure constraints. Experimental studies on benchmark test suites and real-world application problems demonstrate the promising performance of our proposed framework.

翻译：多目标进化优化算法（MOEA）是解决多目标优化问题（MOP）的强大方法，能够在单次运行中找到有限个近似帕累托解集。然而，在温和正则性条件下，连续MOP的帕累托最优集可能是包含无限解的低维连续流形。此外，在许多实际应用中，可能需要对整个最优解集施加结构约束，这些约束刻画了所有解之间共享的模式。现有基于有限种群的MOEA很难妥善处理这些结构约束。本文提出了首个基于模型的算法框架，用于学习具有结构约束的多目标优化完整解集。在该方法中，帕累托最优性可与整个解集中偏好的结构进行权衡，这对其许多实际问题至关重要。我们还开发了一种高效的进化学习方法，用于训练具有结构约束的集合模型。在基准测试套件和实际应用问题上的实验研究表明，我们提出的框架具有令人期待的性能。