This paper presents a hybrid approach that combines an evolutionary algorithm with a classical multi-objective optimization technique to incorporate the preferences of the decision maker into the search process. The preferences are given as a vector of goals, which represent the desirable values for each objective. The proposed approach enhances the goal-constraint technique in such a way that, instead of use the provided \(\varepsilon \) values to compute the upper bounds of the restated problem, it uses only the information of the vector of goals to generate the constraints. The bounds of the region of interest are obtained using an efficient constrained evolutionary optimization algorithm. Then, an interpolation method is placed in charge of populating such a region. It is worth noting that although goal-constraint is able to obtain the bounds of problems regardless of their number objectives, the interpolation method adopted in this paper is restricted to bi-objective problems. The proposed approach was validated using problems from the ZDT, DTLZ, and WFG benchmarks. In addition, it was compared with two well-known algorithms that use the g-dominance approach to incorporate the preferences of the decision maker. The results corroborate that the incorporation of a priori preferences into the proposed approach is useful to direct the search efforts towards the decision’s maker region of interest.

中文翻译：

---

使用基于目标约束的方法来发现多目标问题中的关注区域

本文提出了一种混合方法，该方法将进化算法与经典的多目标优化技术相结合，将决策者的偏好纳入搜索过程。偏好是作为目标的向量给出的，代表了每个目标的期望值。所提出的方法以一种方式增强了目标约束技术，而不是使用提供的\（\ varepsilon \）值以计算重述问题的上限，它仅使用目标向量的信息来生成约束。使用有效的约束进化优化算法获得感兴趣区域的边界。然后，放置插值方法来负责填充这样的区域。值得注意的是，尽管目标约束能够获得问题的边界而不管其目标是什么，但本文采用的插值方法仅限于双目标问题。使用ZDT，DTLZ和WFG基准测试中的问题验证了所提出的方法。此外，还与使用g的两种著名算法进行了比较。-主导方法，以纳入决策者的偏好。结果证实，将先验偏好合并到所提议的方法中有助于将搜索工作引向决策者感兴趣的区域。