Existence and uniqueness of maximal elements for preference relations: Variational approach

Orestes Bueno; John Cotrina; Yboon García

doi:10.1007/s10957-023-02251-y

Existence and uniqueness of maximal elements for preference relations: Variational approach

Research output: Contribution to journal › Article in a journal › peer-review

3 Scopus citations

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.

Original language	English
Pages (from-to)	1246-1263
Number of pages	18
Journal	Journal of Optimization Theory and Applications
Volume	198
Issue number	3
Early online date	27 Jul 2023
DOIs	https://doi.org/10.1007/s10957-023-02251-y
State	Published - Sep 2023

Bibliographical note

Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

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Keywords

Maximal elements
Minty variational inequality
Stampacchia variational inequality
Steepest descent method

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10.1007/s10957-023-02251-y

Cite this

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title = "Existence and uniqueness of maximal elements for preference relations: Variational approach",

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.",

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author = "Orestes Bueno and John Cotrina and Yboon Garc{\'i}a",

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TY - JOUR

T1 - Existence and uniqueness of maximal elements for preference relations

T2 - Variational approach

AU - Bueno, Orestes

AU - Cotrina, John

AU - García, Yboon

PY - 2023/9

Y1 - 2023/9

N2 - 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.

AB - 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.

KW - Maximal elements

KW - Minty variational inequality

KW - Stampacchia variational inequality

KW - Steepest descent method

KW - Elementos máximos

KW - Desigualdad variacional de Minty

KW - Desigualdad variacional de Stampacchia

KW - Método de descenso más pronunciado

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UR - https://www.mendeley.com/catalogue/b5046b6a-8039-3f56-a7dc-b78bf39c95d8/

U2 - 10.1007/s10957-023-02251-y

DO - 10.1007/s10957-023-02251-y

M3 - Article in a journal

AN - SCOPUS:85163314105

SN - 0022-3239

VL - 198

SP - 1246

EP - 1263

JO - Journal of Optimization Theory and Applications

JF - Journal of Optimization Theory and Applications

IS - 3

ER -

Existence and uniqueness of maximal elements for preference relations: Variational approach

Abstract

Bibliographical note

UN SDGs

Keywords

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