Multi-objective optimization problems (MOPs) naturally arise in many applications. Since for such problems one can expect an entire set of optimal solutions, a common task in set based multi-objective optimization is to compute N solutions along the Pareto set/front of a given MOP. In this work, we propose and discuss the set based Newton methods for the performance indicators Generational Distance (GD), Inverted Generational Distance (IGD), and the averaged Hausdorff distance ${\Delta }_p$ for reference set problems for unconstrained MOPs. The methods hence directly utilize the set based scalarization problems that are induced by these indicators and manipulate all N candidate solutions in each iteration. We demonstrate the applicability of the methods on several benchmark problems, and also show how the reference set approach can be used in a bootstrap manner to compute Pareto front approximations in certain cases.

中文翻译：

---

多目标参考集问题的平均Hausdorff距离的基于集的牛顿方法

在许多应用程序中自然会产生多目标优化问题（MOP）。由于对于此类问题，人们可以期望获得一整套最佳解，因此基于集合的多目标优化中的一项常见任务是沿着给定MOP的Pareto集/前沿计算N个解。在这项工作中，我们针对性能指标世代距离（GD），逆世代距离（IGD）和平均Hausdorff距离提出并讨论基于集合的牛顿法${\Delta }_p$供无约束的MOP使用的参考集问题。因此，这些方法直接利用由这些指标引起的基于集合的标量问题，并在每次迭代中操纵所有N个候选解。我们演示了该方法在几个基准问题上的适用性，还展示了如何在某些情况下以自举方式使用参考集方法来计算Pareto前沿逼近。