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Proper or Weak Efficiency via Saddle Point Conditions in Cone-Constrained Nonconvex Vector Optimization Problems

Author

Listed:
  • Fabián Flores-Bazán

    (Universidad de Concepción)

  • Giandomenico Mastroeni

    (University of Pisa)

  • Cristián Vera

    (Universidad Arturo Prat, Casilla 121)

Abstract

Motivated by many applications (for instance, some production models in finance require infinity-dimensional commodity spaces, and the preference is defined in terms of an ordering cone having possibly empty interior), this paper deals with a unified model, which involves preference relations that are not necessarily transitive or reflexive. Our study is carried out by means of saddle point conditions for the generalized Lagrangian associated with a cone-constrained nonconvex vector optimization problem. We establish a necessary and sufficient condition for the existence of a saddle point in case the multiplier vector related to the objective function belongs to the quasi-interior of the polar of the ordering set. Moreover, exploiting suitable Slater-type constraints qualifications involving the notion of quasi-relative interior, we obtain several results concerning the existence of a saddle point, which serve to get efficiency, weak efficiency and proper efficiency. Such results generalize, to the nonconvex vector case, existing conditions in the literature. As a by-product, we propose a notion of properly efficient solution for a vector optimization problem with explicit constraints. Applications to optimality conditions for vector optimization problems are provided with particular attention to bicriteria problems, where optimality conditions for efficiency, proper efficiency and weak efficiency are stated, both in a geometric form and by means of the level sets of the objective functions.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:181:y:2019:i:3:d:10.1007_s10957-019-01486-y
    DOI: 10.1007/s10957-019-01486-y
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. 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.
    4. Fabián Flores-Bazán & William Echegaray & Fernando Flores-Bazán & Eladio Ocaña, 2017. "Primal or dual strong-duality in nonconvex optimization and a class of quasiconvex problems having zero duality gap," Journal of Global Optimization, Springer, vol. 69(4), pages 823-845, December.
    5. R. Zeng & R. J. Caron, 2006. "Generalized Motzkin Theorems of the Alternative and Vector Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 131(2), pages 281-299, November.
    6. 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.
    7. 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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    Cited by:

    1. Min Feng & Shengjie Li & Jie Wang, 2022. "On Tucker-Type Alternative Theorems and Necessary Optimality Conditions for Nonsmooth Multiobjective Optimization," Journal of Optimization Theory and Applications, Springer, vol. 195(2), pages 480-503, November.
    2. 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.

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