Set Order Relations, Set Optimization, and Ekeland’s Variational Principle

Qamrul Hasan Ansari; Pradeep Kumar Sharma

doi:10.1007/978-981-16-1819-2_6

Set Order Relations, Set Optimization, and Ekeland’s Variational Principle

Qamrul Hasan Ansari^*, Pradeep Kumar Sharma

^*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

25 Scopus citations

Abstract

This chapter provides a brief survey on different kinds of set order relations which are used to compare the objective values of set-valued maps and play a key role to study set optimization problems. The solution concepts of set optimization problems and their relationships with respect to different kinds of set order relations are provided. The nonlinear scalarization functions for vector-valued maps as well as for set-valued maps are very useful to study the optimality solutions of vector optimization/set optimization problems. A survey of such nonlinear scalarization functions for vector-valued maps/set-valued maps is given. We give some new results on the existence of optimal solutions of set optimization problems. In the end, we gather some recent results, namely, Ekeland’s variational principle and some equivalent variational principle for set-valued maps with respect to different kinds of set order relations.

Original language	English
Title of host publication	Optimization, Variational Analysis and Applications - IFSOVAA-2020
Editors	Vivek Laha, Pierre Maréchal, S. K. Mishra
Publisher	Springer
Pages	103-165
Number of pages	63
ISBN (Print)	9789811618185
DOIs	https://doi.org/10.1007/978-981-16-1819-2_6
State	Published - 2021
Externally published	Yes

Publication series

Name	Springer Proceedings in Mathematics and Statistics
Volume	355
ISSN (Print)	2194-1009
ISSN (Electronic)	2194-1017

Bibliographical note

Publisher Copyright:
© 2021, The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.

Keywords

49J53
58E30
90C29
90C30
90C46

ASJC Scopus subject areas

General Mathematics

Access to Document

10.1007/978-981-16-1819-2_6

Cite this

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title = "Set Order Relations, Set Optimization, and Ekeland{\textquoteright}s Variational Principle",

abstract = "This chapter provides a brief survey on different kinds of set order relations which are used to compare the objective values of set-valued maps and play a key role to study set optimization problems. The solution concepts of set optimization problems and their relationships with respect to different kinds of set order relations are provided. The nonlinear scalarization functions for vector-valued maps as well as for set-valued maps are very useful to study the optimality solutions of vector optimization/set optimization problems. A survey of such nonlinear scalarization functions for vector-valued maps/set-valued maps is given. We give some new results on the existence of optimal solutions of set optimization problems. In the end, we gather some recent results, namely, Ekeland{\textquoteright}s variational principle and some equivalent variational principle for set-valued maps with respect to different kinds of set order relations.",

keywords = "49J53, 58E30, 90C29, 90C30, 90C46",

author = "Ansari, \{Qamrul Hasan\} and Sharma, \{Pradeep Kumar\}",

note = "Publisher Copyright: {\textcopyright} 2021, The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.",

year = "2021",

doi = "10.1007/978-981-16-1819-2\_6",

language = "English",

isbn = "9789811618185",

series = "Springer Proceedings in Mathematics and Statistics",

publisher = "Springer",

pages = "103--165",

editor = "Vivek Laha and Pierre Mar{\'e}chal and Mishra, \{S. K.\}",

booktitle = "Optimization, Variational Analysis and Applications - IFSOVAA-2020",

address = "Germany",

}

Set Order Relations, Set Optimization, and Ekeland’s Variational Principle. / Ansari, Qamrul Hasan; Sharma, Pradeep Kumar.
Optimization, Variational Analysis and Applications - IFSOVAA-2020. ed. / Vivek Laha; Pierre Maréchal; S. K. Mishra. Springer, 2021. p. 103-165 (Springer Proceedings in Mathematics and Statistics; Vol. 355).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

TY - GEN

T1 - Set Order Relations, Set Optimization, and Ekeland’s Variational Principle

AU - Ansari, Qamrul Hasan

AU - Sharma, Pradeep Kumar

PY - 2021

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N2 - This chapter provides a brief survey on different kinds of set order relations which are used to compare the objective values of set-valued maps and play a key role to study set optimization problems. The solution concepts of set optimization problems and their relationships with respect to different kinds of set order relations are provided. The nonlinear scalarization functions for vector-valued maps as well as for set-valued maps are very useful to study the optimality solutions of vector optimization/set optimization problems. A survey of such nonlinear scalarization functions for vector-valued maps/set-valued maps is given. We give some new results on the existence of optimal solutions of set optimization problems. In the end, we gather some recent results, namely, Ekeland’s variational principle and some equivalent variational principle for set-valued maps with respect to different kinds of set order relations.

AB - This chapter provides a brief survey on different kinds of set order relations which are used to compare the objective values of set-valued maps and play a key role to study set optimization problems. The solution concepts of set optimization problems and their relationships with respect to different kinds of set order relations are provided. The nonlinear scalarization functions for vector-valued maps as well as for set-valued maps are very useful to study the optimality solutions of vector optimization/set optimization problems. A survey of such nonlinear scalarization functions for vector-valued maps/set-valued maps is given. We give some new results on the existence of optimal solutions of set optimization problems. In the end, we gather some recent results, namely, Ekeland’s variational principle and some equivalent variational principle for set-valued maps with respect to different kinds of set order relations.

KW - 49J53

KW - 58E30

KW - 90C29

KW - 90C30

KW - 90C46

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AN - SCOPUS:85113309029

SN - 9789811618185

T3 - Springer Proceedings in Mathematics and Statistics

SP - 103

EP - 165

BT - Optimization, Variational Analysis and Applications - IFSOVAA-2020

A2 - Laha, Vivek

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PB - Springer

ER -

Set Order Relations, Set Optimization, and Ekeland’s Variational Principle

Abstract

Publication series

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

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Cite this