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On the Existence of Weak Efficient Solutions of Nonconvex Vector Optimization Problems

Author

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  • César Gutiérrez

    (University of Valladolid)

  • Rubén López

    (Universidad de Tarapacá)

Abstract

We study vector optimization problems with solid non-polyhedral convex ordering cones, without assuming any convexity or quasiconvexity assumption. We state a Weierstrass-type theorem and existence results for weak efficient solutions for coercive and noncoercive problems. Our approach is based on a new coercivity notion for vector-valued functions, two realizations of the Gerstewitz scalarization function, asymptotic analysis and a regularization of the objective function. We define new boundedness and lower semicontinuity properties for vector-valued functions and study their properties. These new tools rely heavily on the solidness of the ordering cone through the notion of colevel and level sets. As a consequence of this approach, we improve various existence results from the literature, since weaker assumptions are required.

Suggested Citation

  • César Gutiérrez & Rubén López, 2020. "On the Existence of Weak Efficient Solutions of Nonconvex Vector Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 185(3), pages 880-902, June.
    • Handle: RePEc:spr:joptap:v:185:y:2020:i:3:d:10.1007_s10957-020-01667-0
    DOI: 10.1007/s10957-020-01667-0
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    References listed on IDEAS

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    1. S. Deng, 1998. "On Efficient Solutions in Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 96(1), pages 201-209, January.
    2. X. X. Huang & X. Q. Yang & K. L. Teo, 2004. "Characterizing Nonemptiness and Compactness of the Solution Set of a Convex Vector Optimization Problem with Cone Constraints and Applications," Journal of Optimization Theory and Applications, Springer, vol. 123(2), pages 391-407, November.
    3. ,, 2002. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 18(1), pages 193-194, February.
    4. ,, 2002. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 18(2), pages 541-545, April.
    5. Fabián Flores-Bazán & Giandomenico Mastroeni & Cristián Vera, 2019. "Proper or Weak Efficiency via Saddle Point Conditions in Cone-Constrained Nonconvex Vector Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 181(3), pages 787-816, June.
    6. ,, 2002. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 18(4), pages 1007-1017, August.
    7. ,, 2002. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 18(6), pages 1461-1465, December.
    8. César Gutiérrez & Rubén López & Vicente Novo, 2014. "Existence and Boundedness of Solutions in Infinite-Dimensional Vector Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 162(2), pages 515-547, August.
    9. S. Deng, 1998. "Characterizations of the Nonemptiness and Compactness of Solution Sets in Convex Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 96(1), pages 123-131, January.
    10. F. Flores-Bazán & C. Vera, 2006. "Characterization of the Nonemptiness and Compactness of Solution Sets in Convex and Nonconvex Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 130(2), pages 185-207, August.
    11. Kazufumi Ito & Karl Kunisch, 2014. "A Note on the Existence of Nonsmooth Nonconvex Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 163(3), pages 697-706, December.
    12. ,, 2002. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 18(5), pages 1273-1289, October.
    13. M. Bianchi & N. Hadjisavvas & S. Schaible, 1997. "Vector Equilibrium Problems with Generalized Monotone Bifunctions," Journal of Optimization Theory and Applications, Springer, vol. 92(3), pages 527-542, March.
    14. S. Deng, 2009. "Characterizations of the Nonemptiness and Boundedness of Weakly Efficient Solution Sets of Convex Vector Optimization Problems in Real Reflexive Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 140(1), pages 1-7, January.
    15. ,, 2002. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 18(3), pages 819-821, June.
    16. Charalambos D. Aliprantis & Kim C. Border, 2006. "Infinite Dimensional Analysis," Springer Books, Springer, edition 0, number 978-3-540-29587-7, September.
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