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Stability of Optimal Points with Respect to Improvement Sets

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Abstract

The aim of this paper is to study the stability of optimal point sets based on the improvement set E by using the scalarization method and the density results. Under the convergence of a sequence of sets in the sense of Wijsman, we derive the convergence of the sets of E-optimal points, weak E-optimal points, E-quasi-optimal points, E-Benson proper optimal points, E-super optimal points and E-strictly optimal points in the sense of Wijsman. Moreover, we obtain the semicontinuity of E-optimal point mapping, weak E-optimal point mapping, E-quasi-optimal point mapping, E-Benson proper optimal point mapping, E-super optimal point mapping and E-strictly optimal point mapping. Finally, we make a new attempt to establish Lipschitz continuity of these E-optimal point mappings under some suitable conditions.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China [Grant No. 11801257], [Grant No. 11991024], [Grant No. 12171063], the Natural Science Foundation of Jiangxi Province [Grant No. 20232BAB211012], the Project for Creative Research Groups at Institutions of Higher Education in Chongqing [Grant No. CXQT20014] and the Natural Science Foundation of Chongqing [Grant No. cstc2022ycjh], [Grant No. bgzxm0114].

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Authors and Affiliations

  1. School of Statistics and Data Science, Jiangxi University of Finance and Economics, Nanchang, 330013, Jiangxi, China

    Yu Han

  2. School of Mathematical Sciences, Chongqing Normal University, Chongqing, 400047, China

    Ke Quan Zhao

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  1. Yu Han
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  2. Ke Quan Zhao
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Correspondence to Yu Han.

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Communicated by Mathias Ehrgott.

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Han, Y., Zhao, K.Q. Stability of Optimal Points with Respect to Improvement Sets. J Optim Theory Appl 199, 904–930 (2023). https://doi.org/10.1007/s10957-023-02308-y

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