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Duality of Nonscalarized Multiobjective Linear Programs: Dual Balance, Level Sets, and Dual Clusters of Optimal Vectors

Author

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  • E. Galperin

    (Université du Québec à Montréal)

  • P. Jimenez Guerra

    (Universidad Nacional de Educación a Distancia)

Abstract

A new concept of duality is proposed for multiobjective linear programs. It is based on a set expansion process for the computation of optimal solutions without scalarization. The duality gap qualifications are investigated; the primal–dual balance set and level set equations are derived. It is demonstrated that the nonscalarized dual problem presents a cluster of optimal dual vectors that corresponds to a unique optimal primal vector. Comparisons are made with linear utility, minmax and minmin scalarizations. Connections to Pareto optimality are studied and relations to sensitivity and parametric programming are discussed. The ideas are illustrated by examples.

Suggested Citation

  • E. Galperin & P. Jimenez Guerra, 2001. "Duality of Nonscalarized Multiobjective Linear Programs: Dual Balance, Level Sets, and Dual Clusters of Optimal Vectors," Journal of Optimization Theory and Applications, Springer, vol. 108(1), pages 109-137, January.
    • Handle: RePEc:spr:joptap:v:108:y:2001:i:1:d:10.1023_a:1026465906067
    DOI: 10.1023/A:1026465906067
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    References listed on IDEAS

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    1. E. A. Galperin, 1997. "Pareto Analysis vis-à-vis Balance Space Approach in Multiobjective Global Optimization," Journal of Optimization Theory and Applications, Springer, vol. 93(3), pages 533-545, June.
    2. Ben-Israel, A & Ben-Tal, A & Charnes, A, 1977. "Necessary and Sufficient Conditions for a Pareto Optimum in Convex Programming," Econometrica, Econometric Society, vol. 45(4), pages 811-820, May.
    3. M. Ehrgott & H. W. Hamacher & K. Klamroth & S. Nickel & A. Schöbel & M. M. Wiecek, 1997. "Equivalence of Balance Points and Pareto Solutions in Multiple-Objective Programming," Journal of Optimization Theory and Applications, Springer, vol. 92(1), pages 209-212, January.
    4. Balbas, Alejandro & Heras, Antonio, 1993. "Duality theory for infinite-dimensional multiobjective linear programming," European Journal of Operational Research, Elsevier, vol. 68(3), pages 379-388, August.
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    Cited by:

    1. Luc, Dinh The, 2011. "On duality in multiple objective linear programming," European Journal of Operational Research, Elsevier, vol. 210(2), pages 158-168, April.
    2. A. Balbás & E. Galperin & P. Jiménez-Guerra, 2002. "Radial Solutions and Orthogonal Trajectories in Multiobjective Global Optimization," Journal of Optimization Theory and Applications, Springer, vol. 115(2), pages 315-344, November.
    3. N. Mahdavi-Amiri & F. Salehi Sadaghiani, 2017. "Strictly feasible solutions and strict complementarity in multiple objective linear optimization," 4OR, Springer, vol. 15(3), pages 303-326, September.

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