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Convergence of Solutions to Set Optimization Problems with the Set Less Order Relation

Authors:
Lam Quoc Anh at Can Tho University
Lam Quoc Anh
Duy Tran at FPT University
Duy Tran
Dinh Vinh Hien at Ho Chi Minh City University of Food Industry
Dinh Vinh Hien
Daishi Kuroiwa at Shimane University
Daishi Kuroiwa
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This article investigates stability conditions for set optimization problems with the set less order relation in the senses of Panilevé–Kuratowski and Hausdorff convergence. Properties of various kinds of convergences for elements in the image space are discussed. Taking such properties into account, formulations of internal and external stability of the solutions are studied in the image space in terms of the convergence of a solution sets sequence of perturbed set optimization problems to a solution set of the given problem.
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Journal of Optimization Theory and Applications (2020) 185:416–432
https://doi.org/10.1007/s10957-020-01657-2
Convergence of Solutions to Set Optimization Problems
with the Set Less Order Relation
Lam Quoc Anh1·Tran Quoc Duy2,3 ·Dinh Vinh Hien4,5 ·Daishi Kuroiwa6·
Narin Petrot7
Received: 26 March 2019 / Accepted: 12 March 2020 / Published online: 30 March 2020
© Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract
This article investigates stability conditions for set optimization problems with the set
less order relation in the senses of Panilevé–Kuratowski and Hausdorff convergence.
Properties of various kinds of convergences for elements in the image space are dis-
cussed. Taking such properties into account, formulations of internal and external
stability of the solutions are studied in the image space in terms of the convergence
of a solution sets sequence of perturbed set optimization problems to a solution set of
the given problem.
Keywords Set optimization ·Set less order relation ·Internal stability ·External
stability
Mathematics Subject Classification 49M37 ·90C30 ·65K05 ·47J20
1 Introduction
Set-valued optimization is an interesting and important branch of applied mathematics,
that is designed to solve optimization problems, in which either the objective mapping
or the constraint mappings are set-valued mappings acting between abstract spaces.
Concerning set-valued optimization, there are two main approaches, depending on
what the notion of minimality is considered. The classical one is vector approach,
where the definition of a minimizer considers only an efficient point of the union of
all images of the set-valued objective mapping and identifies the image set containing
this minimal element as the “best” available [1,2]. However, a serious disadvantage of
this approach is that, in general, only one element does not necessarily imply that the
Communicated by Anil Aswani.
BTran Quoc Duy
tranquocduy@tdtu.edu.vn
Extended author information available on the last page of the article
123
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
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