ABSTRACT

Set-optimization has attracted increasing interest in the last years, as for instance uncertain multiobjective optimization problems lead to such problems with a set-valued objective function. Thereby, from a practical point of view, most of all the so-called set approach is of interest. However, optimality conditions for these problems, for instance using directional derivatives, are still very limited. The key aspect for a useful directional derivative is the definition of a useful set difference for the evaluation of the numerator in the difference quotient. We present here a new set difference which avoids the use of a convex hull and which applies to arbitrary convex sets, and not to strictly convex sets only. The new set difference is based on the new concept of generalized Steiner sets. We introduce the Banach space of generalized Steiner sets as well as an embedding of convex sets in this space using Steiner points. In this Banach space we can easily define a difference and a directional derivative. We use the latter for new optimality conditions for set optimization. Numerical examples illustrate the new concepts.

摘要

集合优化在过去几年中引起了越来越多的兴趣，例如不确定的多目标优化问题会导致具有集合值目标函数的此类问题。因此，从实践的角度来看，最重要的是所谓的集合方法。然而，这些问题的最优条件，例如使用方向导数，仍然非常有限。有用的方向导数的关键方面是定义有用的集差，用于评估差商中的分子。我们在这里提出了一个新的集合差异，它避免使用凸包并且适用于任意凸集，而不是仅适用于严格凸集。新集差异基于广义施泰纳集的新概念。我们介绍了广义施泰纳集的巴拿赫空间，以及使用施泰纳点在该空间中嵌入凸集。在这个 Banach 空间中，我们可以很容易地定义一个差分和一个方向导数。我们将后者用于集合优化的新最优条件。数值示例说明了新概念。