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Connectedness of the Efficient Set for Strictly Quasiconcave Sets

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  • J. Benoist

    (Limoges University)

Abstract

Given a closed subset X in % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaW% baaSqabeaacaWGUbaaaaaa!387D! , we show the connectedness of its efficient points or nondominated points when X is sequentially strictly quasiconcave. In the particular case of a maximization problem with n continuous and strictly quasiconcave objective functions on a compact convex feasible region of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaW% baaSqabeaacaWGWbaaaaaa!387F! , we deduce the connectedness of the efficient frontier of the problem. This work solves the open problem of the efficient frontier for strictly quasiconcave vector maximization problems.

Suggested Citation

  • J. Benoist, 1998. "Connectedness of the Efficient Set for Strictly Quasiconcave Sets," Journal of Optimization Theory and Applications, Springer, vol. 96(3), pages 627-654, March.
    • Handle: RePEc:spr:joptap:v:96:y:1998:i:3:d:10.1023_a:1022616612527
    DOI: 10.1023/A:1022616612527
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    References listed on IDEAS

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    1. Bonnisseau, Jean-Marc & Cornet, Bernard, 1988. "Existence of equilibria when firms follow bounded losses pricing rules," Journal of Mathematical Economics, Elsevier, vol. 17(2-3), pages 119-147, April.
    2. E. U. Choo & D. R. Atkins, 1983. "Connectedness in Multiple Linear Fractional Programming," Management Science, INFORMS, vol. 29(2), pages 250-255, February.
    3. A. Daniilidis & N. Hadjisavvas & S. Schaible, 1997. "Connectedness of the Efficient Set for Three-Objective Quasiconcave Maximization Problems," Journal of Optimization Theory and Applications, Springer, vol. 93(3), pages 517-524, June.
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    Cited by:

    1. E. Miglierina & E. Molho & F. Patrone & S. Tijs, 2008. "Axiomatic approach to approximate solutions in multiobjective optimization," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 31(2), pages 95-115, November.
    2. N. T. T. Huong & J.-C. Yao & N. D. Yen, 2020. "Geoffrion’s proper efficiency in linear fractional vector optimization with unbounded constraint sets," Journal of Global Optimization, Springer, vol. 78(3), pages 545-562, November.
    3. J. Benoist & N. Popovici, 2001. "Contractibility of the Efficient Frontier of Three-Dimensional Simply-Shaded Sets," Journal of Optimization Theory and Applications, Springer, vol. 111(1), pages 81-116, October.
    4. S.T. Hackman & U. Passy, 2002. "Maximizing a Linear Fractional Function on a Pareto Efficient Frontier," Journal of Optimization Theory and Applications, Springer, vol. 113(1), pages 83-103, April.
    5. A. Daniilidis & N. Hadjisavvas, 1999. "Characterization of Nonsmooth Semistrictly Quasiconvex and Strictly Quasiconvex Functions," Journal of Optimization Theory and Applications, Springer, vol. 102(3), pages 525-536, September.
    6. N. Q. Huy & N. D. Yen, 2005. "Contractibility of the Solution Sets in Strictly Quasiconcave Vector Maximization on Noncompact Domains," Journal of Optimization Theory and Applications, Springer, vol. 124(3), pages 615-635, March.
    7. Y. D. Hu & C. Ling, 2000. "Connectedness of Cone Superefficient Point Sets in Locally Convex Topological Vector Spaces," Journal of Optimization Theory and Applications, Springer, vol. 107(2), pages 433-446, November.
    8. J. Benoist, 2001. "Contractibility of the Efficient Set in Strictly Quasiconcave Vector Maximization," Journal of Optimization Theory and Applications, Springer, vol. 110(2), pages 325-336, August.

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