Algorithmic approaches to set optimization

集合优化问题在金融(动态多变量风险度量)和稳健优化(例如在考虑决策不确定性的情况下)等许多重要和及时的应用中得到了极大的关注。主要困难在于，目标函数的值现在是集合，而称为集合方法的实际相关最优化概念要求将这些集合作为一个整体进行比较。这也意味着，一般说来，存在无限多的最优解，其目的必须是找到该集合的表示。虽然集合优化的研究在不断增长，但它主要集中在理论上的见解，以及与其他最优概念或最优条件的导数概念的关系。到目前为止，关于求解集合优化问题的算法的研究非常有限。通过这个项目，我们打算通过提供新的理论见解，为集合优化方法的发展带来重大贡献，这些新的理论见解将用于新的求解算法。我们将沿着一个全新的方向，建立合适的多目标优化问题，然后利用参数相关的标量值子问题来解决这些问题。我们将允许集合优化问题是非线性的，但将需要其他关于光滑性的强假设来允许数值求解所产生的标量值子问题。此外，我们还将通过研究质量标准等概念来为这一日益增长的研究领域带来新的想法，这些概念用于评估现在是集合并的最优解集的图像的表示。我们还致力于将经典概念从标量优化作为局部最优解和近似最优解扩展到集值问题，这对这一领域的任何算法发展都将是重要的。我们的基本思想是利用极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，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依赖子问题来解决。并对这些参数进行自适应调整，以高质量和高数值效率找到集合优化问题的最优解集的表示。集合优化问题的难点(如非凸性)直接转化为子问题的难点。这限制了实际上可以解决的问题的类型。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研究提供一个全新的方向，如关于最优性条件的研究。

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.