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Definition and Characterization of Geoffrion Proper Efficiency for Real Vector Optimization with Infinitely Many Criteria

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  • Alexander Engau

    (University of Colorado Denver)

Abstract

The concept and characterization of proper efficiency is of significant theoretical and computational interest, in multiobjective optimization and decision-making, to prevent solutions with unbounded marginal rates of substitution. In this paper, we propose a slight modification to the original definition in the sense of Geoffrion, which maintains the common characterizations of properly efficient points as solutions to weighted sums or series and augmented or modified weighted Tchebycheff norms, also if the number of objective functions is countably infinite. We give new proofs and counterexamples which demonstrate that such results become invalid for infinitely many criteria with respect to the original definition, in general, and we address the motivation and practical relevance of our findings for possible applications in stochastic optimization and decision-making under uncertainty.

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  • Alexander Engau, 2015. "Definition and Characterization of Geoffrion Proper Efficiency for Real Vector Optimization with Infinitely Many Criteria," Journal of Optimization Theory and Applications, Springer, vol. 165(2), pages 439-457, May.
    • Handle: RePEc:spr:joptap:v:165:y:2015:i:2:d:10.1007_s10957-014-0608-5
    DOI: 10.1007/s10957-014-0608-5
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    References listed on IDEAS

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    1. H. Isermann, 1974. "Technical Note—Proper Efficiency and the Linear Vector Maximum Problem," Operations Research, INFORMS, vol. 22(1), pages 189-191, February.
    2. E. U. Choo & D. R. Atkins, 1983. "Proper Efficiency in Nonconvex Multicriteria Programming," Mathematics of Operations Research, INFORMS, vol. 8(3), pages 467-470, August.
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    Cited by:

    1. Engau, Alexander & Sigler, Devon, 2020. "Pareto solutions in multicriteria optimization under uncertainty," European Journal of Operational Research, Elsevier, vol. 281(2), pages 357-368.
    2. Fereshteh Akbari & Mehrdad Ghaznavi & Esmaile Khorram, 2018. "A Revised Pascoletti–Serafini Scalarization Method for Multiobjective Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 178(2), pages 560-590, August.
    3. Alexander Engau, 2017. "Proper Efficiency and Tradeoffs in Multiple Criteria and Stochastic Optimization," Mathematics of Operations Research, INFORMS, vol. 42(1), pages 119-134, January.

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