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Scalarization and Stability in Vector Optimization

Author

Listed:
  • E. Miglierina

    (Università dell'Insubria)

  • E. Molho

    (Università di Pavia)

Abstract

A class of scalarizations of vector optimization problems is studied in order to characterize weakly efficient, efficient, and properly efficient points of a nonconvex vector problem. A parallelism is established between the different solutions of the scalarized problem and the various efficient frontiers. In particular, properly efficient points correspond to stable solutions with respect to suitable perturbations of the feasible set.

Suggested Citation

  • E. Miglierina & E. Molho, 2002. "Scalarization and Stability in Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 114(3), pages 657-670, September.
    • Handle: RePEc:spr:joptap:v:114:y:2002:i:3:d:10.1023_a:1016031214488
    DOI: 10.1023/A:1016031214488
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    Cited by:

    1. Onetti Alberto & Verma Sameer, 2008. "Licensing and Business Models," Economics and Quantitative Methods qf0806, Department of Economics, University of Insubria.
    2. P. Oppezzi & A. M. Rossi, 2006. "Existence and Convergence of Pareto Minima," Journal of Optimization Theory and Applications, Springer, vol. 128(3), pages 653-664, March.
    3. S. J. Li & Y. D. Xu & S. K. Zhu, 2012. "Nonlinear Separation Approach to Constrained Extremum Problems," Journal of Optimization Theory and Applications, Springer, vol. 154(3), pages 842-856, September.
    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. Ginchev Ivan & Guerraggio Angelo & Rocca Matteo, 2003. "From scalar to vector optimization," Economics and Quantitative Methods qf0305, Department of Economics, University of Insubria.
    6. Radu Boţ & Sorin-Mihai Grad & Gert Wanka, 2007. "A general approach for studying duality in multiobjective optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 65(3), pages 417-444, June.
    7. Khushboo & C. S. Lalitha, 2018. "Scalarizations for a unified vector optimization problem based on order representing and order preserving properties," Journal of Global Optimization, Springer, vol. 70(4), pages 903-916, April.
    8. Shiva Kapoor & C. S. Lalitha, 2019. "Stability in unified semi-infinite vector optimization," Journal of Global Optimization, Springer, vol. 74(2), pages 383-399, June.
    9. Rocca Matteo & Papalia Melania, 2008. "Well-posedness in vector optimization and scalarization results," Economics and Quantitative Methods qf0807, Department of Economics, University of Insubria.
    10. Y. D. Xu & S. J. Li, 2013. "Optimality Conditions for Generalized Ky Fan Quasi-Inequalities with Applications," Journal of Optimization Theory and Applications, Springer, vol. 157(3), pages 663-684, June.
    11. Shiva Kapoor & C. S. Lalitha, 2019. "Stability and Scalarization for a Unified Vector Optimization Problem," Journal of Optimization Theory and Applications, Springer, vol. 182(3), pages 1050-1067, September.
    12. Xu, Y.D. & Li, S.J. & Teo, K.L., 2012. "Vector network equilibrium problems with capacity constraints of arcs," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 48(3), pages 567-577.
    13. 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.
    14. Ginchev Ivan & Guerraggio Angelo & Rocca Matteo, 2004. "Isolated minimizers, proper efficiency and stability for C0,1 constrained vector optimization problems," Economics and Quantitative Methods qf0404, Department of Economics, University of Insubria.
    15. B. Jiménez & V. Novo & A. Vílchez, 2020. "Characterization of set relations through extensions of the oriented distance," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 91(1), pages 89-115, February.
    16. S. W. Xiang & W. S. Yin, 2007. "Stability Results for Efficient Solutions of Vector Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 134(3), pages 385-398, September.
    17. 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.
    18. Y. D. Xu & S. J. Li, 2016. "On the solution continuity of parametric set optimization problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 84(1), pages 223-237, August.
    19. Crespi Giovanni P. & Ginchev Ivan & Rocca Matteo, 2004. "Two approaches toward constrained vector optimization and identity of the solutions," Economics and Quantitative Methods qf04013, Department of Economics, University of Insubria.
    20. C. Gutiérrez & L. Huerga & E. Köbis & C. Tammer, 2021. "A scalarization scheme for binary relations with applications to set-valued and robust optimization," Journal of Global Optimization, Springer, vol. 79(1), pages 233-256, January.
    21. Miglierina Enrico & Molho Elena, 2003. "Convergence of the minimal sets under convexity in vector optimization," Economics and Quantitative Methods qf0302, Department of Economics, University of Insubria.
    22. Miglierina Enrico & Molho Elena & Rocca Matteo, 2004. "Well-posedness and scalarization in vector optimization," Economics and Quantitative Methods qf0403, Department of Economics, University of Insubria.

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