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Duality Theory in Interval-Valued Linear Programming Problems

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  • Hsien-Chung Wu

    (National Kaohsiung Normal University)

Abstract

The weak and strong duality theorems in interval-valued linear programming problems are derived in this paper. The primal and dual interval-valued linear programming problems are formulated by proposing the concept of a scalar (inner) product of closed intervals. We introduce a solution concept that is essentially similar to the notion of nondominated solution in multiobjective programming problems by imposing a partial ordering on the set of all closed intervals. Under these settings, the weak and strong duality theorems for interval-valued linear programming problems are derived naturally.

Suggested Citation

  • Hsien-Chung Wu, 2011. "Duality Theory in Interval-Valued Linear Programming Problems," Journal of Optimization Theory and Applications, Springer, vol. 150(2), pages 298-316, August.
    • Handle: RePEc:spr:joptap:v:150:y:2011:i:2:d:10.1007_s10957-011-9842-2
    DOI: 10.1007/s10957-011-9842-2
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    References listed on IDEAS

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    1. A. L. Soyster, 1974. "Technical Note—A Duality Theory for Convex Programming with Set-Inclusive Constraints," Operations Research, INFORMS, vol. 22(4), pages 892-898, August.
    2. H. C. Wu, 2010. "Duality Theory for Optimization Problems with Interval-Valued Objective Functions," Journal of Optimization Theory and Applications, Springer, vol. 144(3), pages 615-628, March.
    3. Soyster, A. L., 1979. "Inexact linear programming with generalized resource sets," European Journal of Operational Research, Elsevier, vol. 3(4), pages 316-321, July.
    4. H. C. Wu, 2008. "Wolfe Duality for Interval-Valued Optimization," Journal of Optimization Theory and Applications, Springer, vol. 138(3), pages 497-509, September.
    5. David J. Thuente, 1980. "Technical Note—Duality Theory for Generalized Linear Programs with Computational Methods," Operations Research, INFORMS, vol. 28(4), pages 1005-1011, August.
    6. A. L. Soyster, 1973. "Technical Note—Convex Programming with Set-Inclusive Constraints and Applications to Inexact Linear Programming," Operations Research, INFORMS, vol. 21(5), pages 1154-1157, October.
    7. James E. Falk, 1976. "Technical Note—Exact Solutions of Inexact Linear Programs," Operations Research, INFORMS, vol. 24(4), pages 783-787, August.
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    Cited by:

    1. Hsien-Chung Wu, 2019. "Normed Interval Space and Its Topological Structure," Mathematics, MDPI, vol. 7(10), pages 1-22, October.
    2. A. K. Bhurjee & G. Panda, 2016. "Sufficient optimality conditions and duality theory for interval optimization problem," Annals of Operations Research, Springer, vol. 243(1), pages 335-348, August.
    3. Debdas Ghosh, 2016. "A Newton method for capturing efficient solutions of interval optimization problems," OPSEARCH, Springer;Operational Research Society of India, vol. 53(3), pages 648-665, September.
    4. B. Japamala Rani & Krishna Kummari, 2023. "Duality for fractional interval-valued optimization problem via convexificator," OPSEARCH, Springer;Operational Research Society of India, vol. 60(1), pages 481-500, March.

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