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

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Optimization, Variational Analysis and Applications (IFSOVAA 2020)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 355))

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.

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Acknowledgements

In this research, the first author was supported by DST-SERB Project No. MTR/2017/000135 and the second author was supported by UGC-Dr. D.S. Kothari Post Doctoral Fellowship (DSKPDF) [F.4-2/2006 (BSR)/MA/19-20/0040]. All the authors acknowledge the constructive comments of the unknown referees which helped in bringing the chapter in the present form.

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  1. Department of Mathematics, Aligarh Muslim University, Aligarh, 202002, India

    Qamrul Hasan Ansari

  2. Department of Mathematics, University of Delhi South Campus, New Delhi, 110021, India

    Pradeep Kumar Sharma

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  1. Qamrul Hasan Ansari
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Correspondence to Qamrul Hasan Ansari .

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  1. Institute of Science, Banaras Hindu University, Varanasi, Uttar Pradesh, India

    Vivek Laha

  2. Institut de Mathématiques de Toulouse, Paul Sabatier University, Toulouse, France

    Pierre Maréchal

  3. Institute of Science, Banaras Hindu University, Varanasi, Uttar Pradesh, India

    S. K. Mishra

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Ansari, Q.H., Sharma, P.K. (2021). Set Order Relations, Set Optimization, and Ekeland’s Variational Principle. In: Laha, V., Maréchal, P., Mishra, S.K. (eds) Optimization, Variational Analysis and Applications. IFSOVAA 2020. Springer Proceedings in Mathematics & Statistics, vol 355. Springer, Singapore. https://doi.org/10.1007/978-981-16-1819-2_6

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