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Set-Valued Systems with Infinite-Dimensional Image and Applications

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Listed:
  • J. Li

    (China West Normal University)

  • L. Yang

    (China West Normal University)

Abstract

In infinite-dimensional spaces, we investigate a set-valued system from the image perspective. By exploiting the quasi-interior and the quasi-relative interior, we give some new equivalent characterizations of (proper, regular) linear separation and establish some new sufficient and necessary conditions for the impossibility of the system under new assumptions, which do not require the set to have nonempty interior. We also present under mild assumptions the equivalence between (proper, regular) linear separation and saddle points of Lagrangian functions for the system. These results are applied to obtain some new saddle point sufficient and necessary optimality conditions of vector optimization problems.

Suggested Citation

  • J. Li & L. Yang, 2018. "Set-Valued Systems with Infinite-Dimensional Image and Applications," Journal of Optimization Theory and Applications, Springer, vol. 179(3), pages 868-895, December.
    • Handle: RePEc:spr:joptap:v:179:y:2018:i:3:d:10.1007_s10957-016-1041-8
    DOI: 10.1007/s10957-016-1041-8
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. G. Mastroeni & T. Rapcsák, 2000. "On Convex Generalized Systems," Journal of Optimization Theory and Applications, Springer, vol. 104(3), pages 605-627, March.
    4. R. I. Boţ & E. R. Csetnek & A. Moldovan, 2008. "Revisiting Some Duality Theorems via the Quasirelative Interior in Convex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 139(1), pages 67-84, October.
    5. Fabián Flores-Bazán & Fernando Flores-Bazán & Cristián Vera, 2012. "A complete characterization of strong duality in nonconvex optimization with a single constraint," Journal of Global Optimization, Springer, vol. 53(2), pages 185-201, June.
    6. F. Cammaroto & B. Di Bella, 2005. "Separation Theorem Based on the Quasirelative Interior and Application to Duality Theory," Journal of Optimization Theory and Applications, Springer, vol. 125(1), pages 223-229, 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. 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.
    10. A. Maugeri & F. Raciti, 2010. "Remarks on infinite dimensional duality," Journal of Global Optimization, Springer, vol. 46(4), pages 581-588, April.
    11. Z. A. Zhou & X. M. Yang, 2011. "Optimality Conditions of Generalized Subconvexlike Set-Valued Optimization Problems Based on the Quasi-Relative Interior," Journal of Optimization Theory and Applications, Springer, vol. 150(2), pages 327-340, August.
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