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Nonlinear Separation Approach to Constrained Extremum Problems

Author

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  • S. J. Li

    (Chongqing University
    Chongqing University)

  • Y. D. Xu

    (Chongqing University)

  • S. K. Zhu

    (Chongqing University)

Abstract

In this paper, by virtue of a nonlinear scalarization function, two nonlinear weak separation functions, a nonlinear regular weak separation function, and a nonlinear strong separation function are first introduced, respectively. Then, by the image space analysis, a global saddle-point condition for a nonlinear function is investigated. It is shown that the existence of a saddle point is equivalent to a nonlinear separation of two suitable subsets of the image space. Finally, some necessary and sufficient optimality conditions are obtained for constrained extremum problems.

Suggested Citation

  • S. J. Li & Y. D. Xu & S. K. Zhu, 2012. "Nonlinear Separation Approach to Constrained Extremum Problems," Journal of Optimization Theory and Applications, Springer, vol. 154(3), pages 842-856, September.
    • Handle: RePEc:spr:joptap:v:154:y:2012:i:3:d:10.1007_s10957-012-0027-4
    DOI: 10.1007/s10957-012-0027-4
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    References listed on IDEAS

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    6. G. Mastroeni, 2010. "Some applications of the image space analysis to the duality theory for constrained extremum problems," Journal of Global Optimization, Springer, vol. 46(4), pages 603-614, April.
    7. A. Moldovan & L. Pellegrini, 2009. "On Regularity for Constrained Extremum Problems. Part 2: Necessary Optimality Conditions," Journal of Optimization Theory and Applications, Springer, vol. 142(1), pages 165-183, July.
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    Citations

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    Cited by:

    1. Shengkun Zhu, 2018. "Image Space Analysis to Lagrange-Type Duality for Constrained Vector Optimization Problems with Applications," Journal of Optimization Theory and Applications, Springer, vol. 177(3), pages 743-769, June.
    2. Hezhi Luo & Huixian Wu & Jianzhen Liu, 2015. "On Saddle Points in Semidefinite Optimization via Separation Scheme," Journal of Optimization Theory and Applications, Springer, vol. 165(1), pages 113-150, April.
    3. 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.
    4. Y. D. Xu & S. J. Li, 2013. "Optimality Conditions for Generalized Ky Fan Quasi-Inequalities with Applications," Journal of Optimization Theory and Applications, Springer, vol. 157(3), pages 663-684, June.
    5. Hong-Zhi Wei & Chun-Rong Chen & Sheng-Jie Li, 2018. "Characterizations for Optimality Conditions of General Robust Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 177(3), pages 835-856, June.
    6. Jiawei Chen & La Huang & Shengjie Li, 2018. "Separations and Optimality of Constrained Multiobjective Optimization via Improvement Sets," Journal of Optimization Theory and Applications, Springer, vol. 178(3), pages 794-823, September.
    7. Zhiang Zhou & Wang Chen & Xinmin Yang, 2019. "Scalarizations and Optimality of Constrained Set-Valued Optimization Using Improvement Sets and Image Space Analysis," Journal of Optimization Theory and Applications, Springer, vol. 183(3), pages 944-962, December.
    8. Hong-Zhi Wei & Chun-Rong Chen & Sheng-Jie Li, 2018. "A Unified Characterization of Multiobjective Robustness via Separation," Journal of Optimization Theory and Applications, Springer, vol. 179(1), pages 86-102, October.
    9. Jiawei Chen & Elisabeth Köbis & Markus Köbis & Jen-Chih Yao, 2018. "Image Space Analysis for Constrained Inverse Vector Variational Inequalities via Multiobjective Optimization," Journal of Optimization Theory and Applications, Springer, vol. 177(3), pages 816-834, June.
    10. Madhusudan Das & Chandal Nahak & Mahendra Prasad Biswal, 2024. "Treatment of Set-Valued Robustness via Separation and Scalarization," Journal of Optimization Theory and Applications, Springer, vol. 201(2), pages 843-865, May.
    11. S. K. Zhu & S. J. Li, 2014. "Unified Duality Theory for Constrained Extremum Problems. Part I: Image Space Analysis," Journal of Optimization Theory and Applications, Springer, vol. 161(3), pages 738-762, June.
    12. Qamrul Hasan Ansari & Elisabeth Köbis & Pradeep Kumar Sharma, 2019. "Characterizations of Multiobjective Robustness via Oriented Distance Function and Image Space Analysis," Journal of Optimization Theory and Applications, Springer, vol. 181(3), pages 817-839, June.
    13. Elisabeth Köbis & Markus A. Köbis & Xiaolong Qin, 2019. "Nonlinear Separation Approach to Inverse Variational Inequalities in Real Linear Spaces," Journal of Optimization Theory and Applications, Springer, vol. 183(1), pages 105-121, October.
    14. S. K. Zhu & S. J. Li, 2014. "Unified Duality Theory for Constrained Extremum Problems. Part II: Special Duality Schemes," Journal of Optimization Theory and Applications, Springer, vol. 161(3), pages 763-782, June.
    15. Manxue You & Shengjie Li, 2017. "Separation Functions and Optimality Conditions in Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 175(2), pages 527-544, November.

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