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Constrained Extremum Problems and Image Space Analysis—Part III: Generalized Systems

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Abstract

In Part I, sufficient and necessary optimality conditions and the image regularity conditions of constrained scalar and vector extremum problems are reviewed for Image Space Analysis. Part II presents the main feature of the duality and penalization of constrained scalar and vector extremum problems by virtue of Image Space Analysis. In the light, as said in Part I and Part II, to describe the state of Image Space Analysis for constrained optimization, and to stress that it allows us to unify and generalize the several topics of Optimization, in this Part III, we continue to give an exhaustive literature review on separation functions, gap functions and error bounds for generalized systems. Part III also throws light on some research gaps and concludes with the scope of future research in this area.

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Acknowledgements

The authors would like to thank Professor Franco Giannessi for valuable comments and suggestions, which helped to improve the survey paper. This research was partially supported by the Natural Science Foundation of China (Grants Nos. 11571055, 11526165 and 11601437). The forth author is grateful for the kind hospitality of the institution when part of this work was carried out during a stay in the Department of Mathematics, University of Pisa.

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Authors and Affiliations

  1. College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China

    Shengjie Li & Manxue You

  2. College of Science, Chongqing University of Posts and Telecommunications, Chongqing, 400065, China

    Yangdong Xu

  3. Department of Economics and Mathematics, Southwestern University of Finance and Economics, Chengdu, 611130, China

    Shengkun Zhu

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  1. Shengjie Li
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  2. Yangdong Xu
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  3. Manxue You
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  4. Shengkun Zhu
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Correspondence to Shengjie Li.

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Li, S., Xu, Y., You, M. et al. Constrained Extremum Problems and Image Space Analysis—Part III: Generalized Systems. J Optim Theory Appl 177, 660–678 (2018). https://doi.org/10.1007/s10957-018-1249-x

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  • DOI: https://doi.org/10.1007/s10957-018-1249-x

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