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

Author

Listed:
  • Shengjie Li

    (Chongqing University)

  • Yangdong Xu

    (Chongqing University of Posts and Telecommunications)

  • Manxue You

    (Chongqing University)

  • Shengkun Zhu

    (Southwestern University of Finance and Economics)

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.

Suggested Citation

  • Shengjie Li & Yangdong Xu & Manxue You & Shengkun Zhu, 2018. "Constrained Extremum Problems and Image Space Analysis—Part III: Generalized Systems," Journal of Optimization Theory and Applications, Springer, vol. 177(3), pages 660-678, June.
    • Handle: RePEc:spr:joptap:v:177:y:2018:i:3:d:10.1007_s10957-018-1249-x
    DOI: 10.1007/s10957-018-1249-x
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    References listed on IDEAS

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    1. S.-M. Guu & J. Li, 2014. "Vector quasi-equilibrium problems: separation, saddle points and error bounds for the solution set," Journal of Global Optimization, Springer, vol. 58(4), pages 751-767, April.
    2. J. Li & G. Mastroeni, 2016. "Image Convexity of Generalized Systems with Infinite-Dimensional Image and Applications," Journal of Optimization Theory and Applications, Springer, vol. 169(1), pages 91-115, April.
    3. Suhel Ahmad Khan & Jia-Wei Chen, 2015. "Gap Functions and Error Bounds for Generalized Mixed Vector Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 166(3), pages 767-776, September.
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    Cited by:

    1. Yang-Dong Xu & Cheng-Ling Zhou & Sheng-Kun Zhu, 2021. "Image Space Analysis for Set Optimization Problems with Applications," Journal of Optimization Theory and Applications, Springer, vol. 191(1), pages 311-343, October.
    2. Kenneth O. Kortanek & Guolin Yu & Qinghong Zhang, 2021. "Strong duality for standard convex programs," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 94(3), pages 413-436, December.
    3. Hong-Zhi Wei & Chun-Rong Chen & Sheng-Jie Li, 2020. "Robustness Characterizations for Uncertain Optimization Problems via Image Space Analysis," Journal of Optimization Theory and Applications, Springer, vol. 186(2), pages 459-479, August.
    4. Hong-Zhi Wei & Chun-Rong Chen & Sheng-Jie Li, 2020. "A Unified Approach Through Image Space Analysis to Robustness in Uncertain Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 184(2), pages 466-493, February.

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