Optimality conditions for set optimization using a directional derivative based on generalized Steiner sets
Optimality conditions for set optimization using a directional derivative based on generalized Steiner sets
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DOI：
10.1080/02331934.2020.1812605
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发表时间：
2019-11
期刊：
影响因子：
2.2
通讯作者：
Robert Baier;Gabriele Eichfelder;T. Gerlach
中科院分区：
文献类型：
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作者：
Robert Baier;Gabriele Eichfelder;T. Gerlach
摘要近年来，集优化问题引起了人们越来越多的关注，例如不确定多目标优化问题导致了具有集值目标函数的问题。因此，从实践的角度来看，最重要的是所谓的集合方法是有意义的。然而，这些问题的最优性条件，例如使用方向导数，仍然非常有限。一个有用的方向导数的关键方面是定义一个有用的集合差，用于差商中分子的求值。我们在这里提出了一个新的集差，它避免了使用凸船体，并适用于任意凸集，而不是严格凸集。新的集差是基于广义Steiner集的新概念。我们介绍了广义Steiner集的Banach空间以及在这个空间中使用Steiner点的凸集的嵌入。在这个Banach空间中，我们可以很容易地定义一个差和一个方向导数。我们使用后者的新的最优性条件集优化。数值例子说明了新的概念。
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.