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Approximate solutions in set-valued optimization problems with applications to maximal monotone operators

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Abstract

This paper is devoted to the study of efficient elements for set-valued maps. We propose two new notions of relative weak \(\epsilon \)-efficient element and strict relative weak \(\epsilon \)-efficient element of set-valued maps and provide new necessary optimality conditions for the proposed concepts. We provide existence results for efficient elements. The critical ingredients for the existence results for efficient elements are the well-known separation arguments and Fan’s lemma. As an application of the existence results, we derive relationships between the efficiency concepts and the local optimizers of certain optimization problems.

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Acknowledgements

The authors thank both the anonymous referee and the Editor-in-Chief for their valuable comments which led to an improved presentation of the results. They also express their gratitude to Boris Mordukhovich for his valuable remarks and several helpful suggestions which have considerably improved this work.

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Authors and Affiliations

  1. Department of Mathematics, University of Isfahan, Isfahan, 81745-163, Iran

    Malek Abbasi & Mahboubeh Rezaei

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  1. Malek Abbasi
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  2. Mahboubeh Rezaei
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Correspondence to Malek Abbasi.

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In this research, Mahboubeh Rezaei was partially supported by the Center of Excellence for Mathematics, University of Isfahan, Isfahan, Iran.

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Abbasi, M., Rezaei, M. Approximate solutions in set-valued optimization problems with applications to maximal monotone operators. Positivity 24, 779–797 (2020). https://doi.org/10.1007/s11117-019-00707-y

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