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Constrained Extremum Problems and Image Space Analysis–Part I: Optimality Conditions

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

Image space analysis is a new tool for studying scalar and vector constrained extremum problems as well as generalized systems. In the last decades, the introduction of image space analysis has shown that the image space associated with the given problem provides a natural environment for the Lagrange theory of multipliers and that separation arguments turn out to be a fundamental mathematical tool for explaining, developing and improving such a theory. This work, with its 3 parts, aims at contributing to describe the state-of-the-art 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 1st part, after a short introduction of such an analysis, necessary and sufficient optimality conditions are treated. Duality and penalization are the contents of the 2nd part. The 3rd part deals with generalized systems, in particular, variational inequalities and Ky Fan inequalities. Some further developments are discussed in all the parts.

Suggested Citation

  • Shengjie Li & Yangdong Xu & Manxue You & Shengkun Zhu, 2018. "Constrained Extremum Problems and Image Space Analysis–Part I: Optimality Conditions," Journal of Optimization Theory and Applications, Springer, vol. 177(3), pages 609-636, June.
    • Handle: RePEc:spr:joptap:v:177:y:2018:i:3:d:10.1007_s10957-018-1247-z
    DOI: 10.1007/s10957-018-1247-z
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    References listed on IDEAS

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    Cited by:

    1. 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.
    2. 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.
    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. 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.

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