Multi-objective evolutionary algorithms (MOEAs) are a widely accepted choice for the numerical treatment of multi-objective optimization problems (MOPs). For constrained problems, however, these methods still have room for improvement to compute satisfactory approximations of the solution sets. A possible remedy is the hybridization of MOEAs with specialized local search mechanisms; which is not a simple task due to their high cost. In this work, we consider the information of the constraints when performing the local search, and propose a new and effective way to compute descent directions for constrained bi-objective optimization problems. Since the directions are computed via neighborhood sampling, the method is perfectly suited for the use within MOEAs or any other population based algorithm as the samples can be taken precisely from the populations. The new method can be used as local search engine within, in principle, any MOEA. As demonstrator, we will consider two particular hybrids. Numerical results on some benchmark problems support the benefits of the novel approach.

Though this work focuses on the bi-objective case, this represents an important step to formalize gradient-free multi-objective descent directions and its efficient interleaving into MOEAs.

对于多目标优化问题（MOP）的数值处理，多目标进化算法（MOEA）是一种广为接受的选择。但是，对于受约束的问题，这些方法仍然有改进的余地，可以计算出令人满意的解决方案集近似值。一种可能的补救方法是将MOEA与专门的本地搜索机制进行混合；由于成本高，这不是一件容易的事。在这项工作中，我们在执行局部搜索时考虑约束的信息，并提出了一种新的有效方法来计算约束双目标优化问题的下降方向。由于这些方向是通过邻域采样计算得出的，该方法非常适合在MOEA或任何其他基于总体的算法中使用，因为可以从总体中精确地获取样本。原则上，该新方法可用作任何MOEA内的本地搜索引擎。作为演示者，我们将考虑两个特殊的混合动力车。关于某些基准问题的数值结果支持了该新方法的优势。

尽管这项工作着眼于双目标情况，但这代表了迈向正式化无梯度多目标下降方向及其有效插入MOEA的重要一步。