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Characterizing Nonemptiness and Compactness of the Solution Set of a Convex Vector Optimization Problem with Cone Constraints and Applications

Author

Listed:
  • X. X. Huang

    (Chongqing Normal University)

  • X. Q. Yang

    (Hong Kong Polytechnic University)

  • K. L. Teo

    (Hong Kong Polytechnic University)

Abstract

In this paper, we characterize the nonemptiness and compactness of the set of weakly efficient solutions of a convex vector optimization problem with cone constraints in terms of the level-boundedness of the component functions of the objective on the perturbed sets of the original constraint set. This characterization is then applied to carry out the asymptotic analysis of a class of penalization methods. More specifically, under the assumption of nonemptiness and compactness of the weakly efficient solution set, we prove the existence of a path of weakly efficient solutions to the penalty problem and its convergence to a weakly efficient solution of the original problem. Furthermore, for any efficient point of the original problem, there exists a path of efficient solutions to the penalty problem whose function values (with respect to the objective function of the original problem) converge to this efficient point.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:123:y:2004:i:2:d:10.1007_s10957-004-5155-z
    DOI: 10.1007/s10957-004-5155-z
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    References listed on IDEAS

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    1. A. Auslender & R. Cominetti & M. Haddou, 1997. "Asymptotic Analysis for Penalty and Barrier Methods in Convex and Linear Programming," Mathematics of Operations Research, INFORMS, vol. 22(1), pages 43-62, February.
    2. 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.
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    Cited by:

    1. H. Luo & X. Huang & J. Peng, 2012. "Generalized weak sharp minima in cone-constrained convex optimization with applications," Computational Optimization and Applications, Springer, vol. 53(3), pages 807-821, December.
    2. Yarui Duan & Liguo Jiao & Pengcheng Wu & Yuying Zhou, 2022. "Existence of Pareto Solutions for Vector Polynomial Optimization Problems with Constraints," Journal of Optimization Theory and Applications, Springer, vol. 195(1), pages 148-171, October.
    3. S. Deng, 2010. "Boundedness and Nonemptiness of the Efficient Solution Sets in Multiobjective Optimization," Journal of Optimization Theory and Applications, Springer, vol. 144(1), pages 29-42, January.
    4. 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.
    5. 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.
    6. Jiang-hua Fan & Yan Jing & Ren-you Zhong, 2015. "Nonemptiness and boundedness of solution sets for vector variational inequalities via topological method," Journal of Global Optimization, Springer, vol. 63(1), pages 181-193, September.
    7. 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.
    8. X. Huang & J. Yao, 2013. "Characterizations of the nonemptiness and compactness for solution sets of convex set-valued optimization problems," Journal of Global Optimization, Springer, vol. 55(3), pages 611-626, March.

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