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Separation Functions and Optimality Conditions in Vector Optimization

Author

Listed:
  • Manxue You

    (Chongqing University)

  • Shengjie Li

    (Chongqing University)

Abstract

In this paper, we propose weak separation functions in the image space for general constrained vector optimization problems on strong and weak vector minimum points. Gerstewitz function is applied to construct a special class of nonlinear separation functions as well as the corresponding generalized Lagrangian functions. By virtue of such nonlinear separation functions, we derive Lagrangian-type sufficient optimality conditions in a general context. Especially for nonconvex problems, we establish Lagrangian-type necessary optimality conditions under suitable restriction conditions, and we further deduce Karush–Kuhn–Tucker necessary conditions in terms of Clarke subdifferentials.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:175:y:2017:i:2:d:10.1007_s10957-016-1029-4
    DOI: 10.1007/s10957-016-1029-4
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. A. Moldovan & L. Pellegrini, 2009. "On Regularity for Constrained Extremum Problems. Part 1: Sufficient Optimality Conditions," Journal of Optimization Theory and Applications, Springer, vol. 142(1), pages 147-163, July.
    4. Hezhi Luo & Huixian Wu & Jianzhen Liu, 2013. "Some Results on Augmented Lagrangians in Constrained Global Optimization via Image Space Analysis," Journal of Optimization Theory and Applications, Springer, vol. 159(2), pages 360-385, November.
    5. H. Z. Luo & G. Mastroeni & H. X. Wu, 2010. "Separation Approach for Augmented Lagrangians in Constrained Nonconvex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 144(2), pages 275-290, February.
    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. J. Li & S. Q. Feng & Z. Zhang, 2013. "A Unified Approach for Constrained Extremum Problems: Image Space Analysis," Journal of Optimization Theory and Applications, Springer, vol. 159(1), pages 69-92, October.
    8. 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.
    9. Regina S. Burachik & M. M. Rizvi, 2012. "On Weak and Strong Kuhn–Tucker Conditions for Smooth Multiobjective Optimization," Journal of Optimization Theory and Applications, Springer, vol. 155(2), pages 477-491, November.
    10. 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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    Cited by:

    1. 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.
    2. 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.
    3. 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.
    4. Elena Constantin, 2020. "Second-Order Optimality Conditions in Locally Lipschitz Inequality-Constrained Multiobjective Optimization," Journal of Optimization Theory and Applications, Springer, vol. 186(1), pages 50-67, July.

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