Algorithmic approaches to set optimization

集合优化问题最近受到了广泛的关注，因为它们出现在许多重要和及时的应用中，如金融（动态多元风险度量）和鲁棒优化（例如在考虑决策不确定性的情况下）。主要的困难是目标函数的值现在是集合，一个实用的相关的最优性概念，被称为集合方法，要求这些集合必须作为一个整体进行比较。这也意味着通常有无限个最优解，我们的目标是找到这个集合的表示。虽然集优化的研究在不断增长，但主要集中在与其他最优性概念或最优性条件的衍生概念的关系的理论见解上。到目前为止，对集优化问题求解算法的研究非常有限。在这个项目中，我们打算通过提供新的理论见解，将用于新的解决算法，为集合优化的发展做出重大贡献。我们将遵循一个全新的方向，通过制定合适的多目标优化问题，然后可以借助参数相关的标量值子问题来解决。我们将允许集合优化问题是非线性的，但需要其他强有力的假设，如平滑性，以允许在数值上解决出现的标量值子问题。此外，我们还将通过研究最优解集图像表示评估的质量标准等概念，为这一不断发展的研究领域带来新的想法，这些概念现在是集的并集。我们还致力于将经典概念从标量优化（作为局部和近似最优解）扩展到集值问题，这对该领域的任何算法发展都很重要。我们的基本思想是用最小值函数作为集合优化问题最优解的充分条件。在此基础上构造新的多目标优化问题，并在算法中自适应地选择这些问题的目标。由此产生的多目标问题将由参数相关子问题来解决。这些参数的导向也将自适应地完成，以寻找具有高质量和数值效率的集合优化问题的最优解集的表示。集合优化问题（如非凸性）的困难直接转移到子问题的困难上。这就限制了实际可以解决的问题类型。然而，理论结果将适用于更广泛的集合优化问题，并将为进一步的理论检验提供一个全新的方向，如最优性条件。

Set optimization problems have recently gained a lot of attraction as they appear in many important and timely applications such as finance (dynamic multivariate risk measures) and robust optimization (for instance in case decision uncertainty is taken into account). The main difficulty is that the values of the objective function are now sets and that a practical relevant optimality notion, known as set approach, requires that these sets have to be compared as a whole. This also implies that there is in general an infinite number of optimal solutions and it has to be the aim to find a representation of this set. While there is a continuously growing research in set optimization, it mostly concentrates on theoretical insights as on relations to other optimality concepts or derivative concepts for optimality conditions. So far there is only very limited research on algorithms for solving set optimization problems. With this project we intend to bring a significant contribution to the development of set optimization with the set approach by providing new theoretical insights which will be used for a new solution algorithm. We will follow a completely new direction by formulating suitable multiobjective optimization problems which can then be solved with the help of parameter dependent scalar-valued subproblems. We will allow the set optimization problems to be nonlinear, but other strong assumptions as on the smoothness will be required to allow to solve the arising scalar-valued subproblems numerically. Moreover, we will also bring new ideas to this growing research area by studying concepts like quality criteria for the evaluation of representations of the image of the optimal solution sets which are now union of sets. We also aim to contribute to the extension of classical concepts from scalar-valued optimization, as local and approximate optimal solutions, to set-valued problems which will be important for any algorithmic developments in this field. Our basic idea is to use minimal value functions for sufficient conditions for optimal solutions of the set optimization problem. Based on them new multiobjective optimization problems will be constructed where the objectives of these problems will be selected adaptively within the algorithm. The arising multiobjective problems will be solved by parameter dependent subproblems. The steering of these parameters will also be done adaptively to find representations of the optimal solution sets of the set optimization problem with a high quality and numerically efficient. The difficulties from the set optimization problem (as non-convexity) directly transfer to the difficulties of the subproblems. This limits the type of problems which can practically be solved. The theoretical results will nevertheless apply to wider classes of set optimization problems and will give a completely new direction also for further theoretical examinations as on optimality conditions.