Algorithms for Multi-objective Optimization Problems in Geodesy

在大地测量学中，多目标优化问题是普遍存在的。项目C1的一个例子是地图学中的聚类和聚合问题，其中必须将区域聚类并将其聚合到更大的区域中，并且有多个目标需要优化。项目C2的另一个例子是卫星雷达海面测高，其中需要从卫星数据估计海面高度(SSH)和有效波高(SWH)，这导致了一个多目标最短路径问题。处理多目标优化问题的常用方法是通过加权和将不同的准则组合成一个单目标。这种权重通常是人为的，可能会导致信息丢失。我们将充分利用多目标优化的潜力，特别是研究帕累托最优解集。我们将开发算法来有效地计算或近似这个集合，以解决上述问题和其他由大地测量学应用引起的问题。我们将在多目标聚类和聚合问题上与A2、B1、B2和C1合作，在多目标三角剖分问题上与C1和C2合作，在多目标最短路径问题上与B2、C1和C2合作。我们的算法将进行理论分析，但也将与项目C1、C2以及B2合作实施和测试。这将在实践中产生更好的解决方案，并产生新的理论结果。

In geodesy, multi-objective optimization problems are ubiquitous. An example from project C1 are clustering and aggregation problems in cartography where one has to cluster areas and aggregate them into larger regions and there are multiple objectives one wants to optimize. Another example from project C2 comes from satellite radar altimetry of the sea surface where one wants to estimate both the sea surface height (SSH) and the significant wave height (SWH) from satellite data, which leads to a multi-objective shortest path problem.A common approach to handle multi-objective optimization problems is to combine the different criteria into a single objective by taking a weighted sum. This weighting is often artificial and can result in a loss of information. We will make use of the full potential of multi-objective optimization and study in particular the set of Pareto-optimal solutions. We will develop algorithms for efficiently computing or approximating this set for the problems mentioned above and other problems motivated by applications from geodesy. We will collaborate with A2,B1, B2, and C1 on multi-objective clustering and aggregation problems, with C1 and C2 on multi-objective triangulations, and with B2, C1, and C2 on multi-objective shortest path problems. Our algorithms will be analyzed theoretically but also implemented and tested in cooperation with projects C1 and C2 as well as B2. This will lead to both improved solutions in practice and novel theoretical results.