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Quasi \(\epsilon \)-solutions in a semi-infinite programming problem with locally Lipschitz data

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Abstract

Under the fulfilment of the limiting constraint qualification, a necessary condition for a quasi \(\epsilon \)-solution to a semi-infinite programming problem (SIP) by means of employing some advanced tools of variational analysis and generalized differential is established. Sufficient conditions for such a quasi \(\epsilon \)-solution to problem (SIP) are also investigated in light of generalized convex functions defined in terms of the limiting subdifferential of locally Lipschitz functions. Finally, a Wolfe type dual model in approximate form is formulated, and weak, strong and converse-like duality theorems are proposed. Besides, we give some simple examples to illustrate the obtained results.

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Acknowledgements

The authors would like to express their sincere thanks to anonymous referees for the valuable suggestions and comments for the paper.

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Authors and Affiliations

  1. School of Mathematical Sciences, Soochow University, Suzhou, 215006, Jiangsu Province, China

    Liguo Jiao & Yuying Zhou

  2. Department of Applied Mathematics, Pukyong National University, Busan, 48513, Republic of Korea

    Do Sang Kim

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  1. Liguo Jiao
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  2. Do Sang Kim
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  3. Yuying Zhou
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Correspondence to Do Sang Kim.

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Liguo Jiao was partially supported by National Center for Theoretical Sciences (NCTS) of Taiwan. Do Sang Kim was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korean Government (NRF-2019R1A2C1008672). Yuying Zhou was supported by the National Natural Sciences Foundation of China (11771319).

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Jiao, L., Kim, D.S. & Zhou, Y. Quasi \(\epsilon \)-solutions in a semi-infinite programming problem with locally Lipschitz data. Optim Lett 15, 1759–1772 (2021). https://doi.org/10.1007/s11590-019-01457-2

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  • DOI: https://doi.org/10.1007/s11590-019-01457-2

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