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Existence and Uniqueness of Maximal Elements for Preference Relations: Variational Approach

Author

Listed:
  • Orestes Bueno

    (Universidad del Pacífico)

  • John Cotrina

    (Universidad del Pacífico)

  • Yboon García

    (Universidad del Pacífico)

Abstract

In this work, we reformulate the problem of existence of maximal elements for preference relations as a variational inequality problem in the sense of Stampacchia. Similarly, we establish the uniqueness of maximal elements using a variational inequality problem in the sense of Minty. In both of these approaches, we use the normal cone operator to find existence and uniqueness results, under mild assumptions. In addition, we provide an algorithm for finding such maximal elements, which is inspired by the steepest descent method for minimization. Under certain conditions, we prove that the sequence generated by this algorithm converges to a maximal element.

Suggested Citation

  • Orestes Bueno & John Cotrina & Yboon García, 2023. "Existence and Uniqueness of Maximal Elements for Preference Relations: Variational Approach," Journal of Optimization Theory and Applications, Springer, vol. 198(3), pages 1246-1263, September.
    • Handle: RePEc:spr:joptap:v:198:y:2023:i:3:d:10.1007_s10957-023-02251-y
    DOI: 10.1007/s10957-023-02251-y
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    References listed on IDEAS

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    2. Monica Milasi & Domenico Scopelliti, 2021. "A Variational Approach to the Maximization of Preferences Without Numerical Representation," Journal of Optimization Theory and Applications, Springer, vol. 190(3), pages 879-893, September.
    3. Rader, Trout, 1978. "Induced Preferences on Trades When Preferences May Be Intransitive and Incomplete," Econometrica, Econometric Society, vol. 46(1), pages 137-146, January.
    4. Campbell, Donald E. & Walker, Mark, 1990. "Maximal elements of weakly continuous relations," Journal of Economic Theory, Elsevier, vol. 50(2), pages 459-464, April.
    5. Mas-Colell, Andreu & Whinston, Michael D. & Green, Jerry R., 1995. "Microeconomic Theory," OUP Catalogue, Oxford University Press, number 9780195102680.
    6. Shafer, Wayne J, 1974. "The Nontransitive Consumer," Econometrica, Econometric Society, vol. 42(5), pages 913-919, September.
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