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Stability and Scalarization in Vector Optimization Using Improvement Sets

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  • C. S. Lalitha

    (University of Delhi South Campus)

  • Prashanto Chatterjee

    (University of Delhi)

Abstract

The aim of this paper is to study certain aspects of stability and scalarization of a nonconvex vector optimization problem through improvement sets. This paper attempts to investigate an open problem on stability posed by Chicco et al. The notion of stability is studied through Painlevé–Kuratowski set-convergence, where we establish sufficiency conditions for the lower and upper set-convergences of optimal solution sets of a family of perturbed vector problems, both in the given space and its image space. The perturbations are performed both on the objective function and the feasible set. Further, by using a nonlinear scalarization function defined in terms of an improvement set, we establish lower and upper Painlevé–Kuratowski set-convergences of sequences of approximate solution sets of certain scalarized problems. We then link these set-convergences with the set-convergences of optimal solution sets of the perturbed problems. Finally, we investigate the stability and scalarization of a linear vector optimization problem in finite dimensional spaces.

Suggested Citation

  • C. S. Lalitha & Prashanto Chatterjee, 2015. "Stability and Scalarization in Vector Optimization Using Improvement Sets," Journal of Optimization Theory and Applications, Springer, vol. 166(3), pages 825-843, September.
    • Handle: RePEc:spr:joptap:v:166:y:2015:i:3:d:10.1007_s10957-014-0686-4
    DOI: 10.1007/s10957-014-0686-4
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    References listed on IDEAS

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    1. SALINETTI, Gabriella & WETS, Roger J.-B., 1979. "On the convergence of sequences of convex sets in finite dimensions," LIDAM Reprints CORE 352, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. M. Chicco & F. Mignanego & L. Pusillo & S. Tijs, 2011. "Vector Optimization Problems via Improvement Sets," Journal of Optimization Theory and Applications, Springer, vol. 150(3), pages 516-529, September.
    3. Gutiérrez, C. & Jiménez, B. & Novo, V., 2012. "Improvement sets and vector optimization," European Journal of Operational Research, Elsevier, vol. 223(2), pages 304-311.
    4. C. S. Lalitha & Prashanto Chatterjee, 2012. "Stability for Properly Quasiconvex Vector Optimization Problem," Journal of Optimization Theory and Applications, Springer, vol. 155(2), pages 492-506, November.
    5. Gutiérrez, C. & Jiménez, B. & Novo, V., 2010. "Optimality conditions via scalarization for a new [epsilon]-efficiency concept in vector optimization problems," European Journal of Operational Research, Elsevier, vol. 201(1), pages 11-22, February.
    6. C. S. Lalitha & Prashanto Chatterjee, 2012. "Stability and Scalarization of Weak Efficient, Efficient and Henig Proper Efficient Sets Using Generalized Quasiconvexities," Journal of Optimization Theory and Applications, Springer, vol. 155(3), pages 941-961, December.
    7. J. H. Qiu & Y. Hao, 2010. "Scalarization of Henig Properly Efficient Points in Locally Convex Spaces," Journal of Optimization Theory and Applications, Springer, vol. 147(1), pages 71-92, October.
    8. E. Miglierina & E. Molho & M. Rocca, 2005. "Well-Posedness and Scalarization in Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 126(2), pages 391-409, August.
    9. C. Gutiérrez & B. Jiménez & V. Novo, 2011. "A generic approach to approximate efficiency and applications to vector optimization with set-valued maps," Journal of Global Optimization, Springer, vol. 49(2), pages 313-342, February.
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    Cited by:

    1. C. Gutiérrez & L. Huerga & B. Jiménez & V. Novo, 2020. "Optimality conditions for approximate proper solutions in multiobjective optimization with polyhedral cones," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 28(2), pages 526-544, July.
    2. 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.
    3. C. Gutiérrez & L. Huerga & B. Jiménez & V. Novo, 2018. "Approximate solutions of vector optimization problems via improvement sets in real linear spaces," Journal of Global Optimization, Springer, vol. 70(4), pages 875-901, April.

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