When solving constrained multiobjective optimization problems (CMOPs), the most commonly used way of measuring constraint violation is to calculate the sum of all constraint violations of a solution as its distance to feasibility. However, this kind of constraint violation measure may not reflect the distance of an infeasible solution from feasibility for some problems, for example, when an infeasible solution closer to a feasible region does not have a smaller constraint violation than the one farther away from a feasible region. Unfortunately, no set of artificial benchmark problems focusing on this area exists. To remedy this issue, a set of CMOPs with deceptive constraints is introduced in this article. It is the first attempt to consider CMOPs with deceptive constraints (DCMOPs). Based on our previous work, which designed a set of directed weight vectors to solve CMOPs, this article proposes a cooperative framework with an improved version of directed weight vectors to solve DCMOPs. Specifically, the cooperative framework consists of two switchable phases. The first phase uses two subpopulations—one to explore feasible regions and the other to explore the entire space. The two subpopulations provide useful information about the optimal direction of objective improvement to each other. The second phase aims mainly at finding Pareto-optimal solutions. Then an infeasibility utilization strategy is used to improve the objective function values. The two phases are switchable based on the information found to date at any time in the evolutionary process. The experimental results show that this method significantly outperforms the algorithms with which it is compared on most of the DCMOPs, in terms of reliability and stability in finding a set of well-distributed optimal solutions.

中文翻译：

---

基于改进版有向权重向量的合作进化框架，用于具有欺骗性约束的约束多目标优化

在求解受约束的多目标优化问题 (CMOP) 时，最常用的测量约束违反的方法是计算解决方案的所有违反约束的总和，作为其与可行性的距离。然而，对于某些问题，这种违反约束的度量可能无法反映不可行解与可行性的距离，例如，当接近可行域的不可行解的约束违反并不比远离可行域的不可行解更小。地区。不幸的是，不存在针对该领域的一组人工基准问题。为了解决这个问题，本文介绍了一组具有欺骗性约束的 CMOP。这是考虑具有欺骗性约束 (DCMOP) 的 CMOP 的首次尝试。根据我们之前的工作，设计了一组有向权向量来解决 CMOP，本文提出了一个协作框架，其中包含改进版的有向权向量来解决 DCMOP。具体来说，合作框架由两个可切换的阶段组成。第一阶段使用两个亚群——一个探索可行区域，另一个探索整个空间。这两个亚群相互提供了关于目标改进的最佳方向的有用信息。第二阶段主要是寻找帕累托最优解。然后使用不可行性利用策略来提高目标函数值。这两个阶段可以根据迄今为止在进化过程中随时发现的信息进行切换。