In this paper, we study a first-order solution method for a particular class of set optimization problems where the solution concept is given by the set approach. We consider the case in which the set-valued objective mapping is identified by a finite number of continuously differentiable selections. The corresponding set optimization problem is then equivalent to find optimistic solutions to vector optimization problems under uncertainty with a finite uncertainty set. We develop optimality conditions for these types of problems and introduce two concepts of critical points. Furthermore, we propose a descent method and provide a convergence result to points satisfying the optimality conditions previously derived. Some numerical examples illustrating the performance of the method are also discussed. This paper is a modified and polished version of Chapter 5 in the dissertation by Quintana (On set optimization with set relations: a scalarization approach to optimality conditions and algorithms, Martin-Luther-Universität Halle-Wittenberg, 2020).

在本文中，我们研究了特定类别的集合优化问题的一阶求解方法，其中求解概念由集合方法给出。我们考虑设置值目标映射由有限数量的连续可微选择标识的情况。那么相应的集合优化问题就等价于在一个有限的不确定集的不确定性下找到向量优化问题的乐观解。我们为这些类型的问题开发了最优条件，并引入了两个临界点概念。此外，我们提出了一种下降方法，并为满足先前导出的最优条件的点提供收敛结果。还讨论了一些说明该方法性能的数值例子。