Solving set-valued optimization problems using a multiobjective approach
Solving set-valued optimization problems using a multiobjective approach
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DOI：
10.1080/02331934.2021.1988596
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发表时间：
2021-10
期刊：
影响因子：
2.2
通讯作者：
Gabriele Eichfelder;Stefan Rocktäschel
中科院分区：
文献类型：
--
作者：
Gabriele Eichfelder;Stefan Rocktäschel
摘要基于集合方法的集值优化问题是一个非常有意义的研究课题，因为它具有很强的实用性和与其他优化领域的许多相互依赖关系。然而，即使在具体的情况下，要解决这些优化问题也是一项非常困难的任务。在本文中，我们研究了集值优化问题，并发展了一个与之密切相关的多目标优化问题。证明了该子问题的弱极小解集与集值优化问题的弱极小元集密切相关，并且在一定意义下这些集可以任意接近。随后，我们引入了集值优化问题解集的逼近概念。我们在像空间中定义了一种可用于比较这类有限近似的质量度量，并概述了增强给定近似的过程。最后，我们给出了一些数值算例。
ABSTRACT Set-valued optimization using the set approach is a research topic of high interest due to its practical relevance and numerous interdependencies to other fields of optimization. However, it is a very difficult task to solve these optimization problems even for specific cases. In this paper, we study set-valued optimization problems and develop a multiobjective optimization problem that is strongly related to it. We prove that the set of weakly minimal solutions of this subproblem is closely related to the set of weakly minimal elements of the set-valued optimization problem and that these sets can get arbitrarily close in a certain sense. Subsequently, we introduce a concept of approximations of the solution set of the set-valued optimization problem. We define a quality measure in the image space that can be used to compare finite approximations of this kind and outline a procedure to enhance a given approximation. We conclude the paper with some numerical examples.