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On vector equilibria, vector optimisation and vector variational inequalities

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On vector equilibria, vector optimisation and vector variational inequalities. / Raith, Andrea; Ehrgott, Matthias.
In: Journal of Multi-Criteria Decision Analysis, Vol. 18, No. 1-2, 01.2011, p. 39-54.

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Harvard

Raith, A & Ehrgott, M 2011, 'On vector equilibria, vector optimisation and vector variational inequalities', Journal of Multi-Criteria Decision Analysis, vol. 18, no. 1-2, pp. 39-54. https://doi.org/10.1002/mcda.477

APA

Raith, A., & Ehrgott, M. (2011). On vector equilibria, vector optimisation and vector variational inequalities. Journal of Multi-Criteria Decision Analysis, 18(1-2), 39-54. https://doi.org/10.1002/mcda.477

Vancouver

Raith A, Ehrgott M. On vector equilibria, vector optimisation and vector variational inequalities. Journal of Multi-Criteria Decision Analysis. 2011 Jan;18(1-2):39-54. doi: 10.1002/mcda.477

Author

Raith, Andrea ; Ehrgott, Matthias. / On vector equilibria, vector optimisation and vector variational inequalities. In: Journal of Multi-Criteria Decision Analysis. 2011 ; Vol. 18, No. 1-2. pp. 39-54.

Bibtex

@article{516909a096af4173b07ab739ad79ee1a,
title = "On vector equilibria, vector optimisation and vector variational inequalities",
abstract = "It is well-known that, under certain conditions, network equilibrium, optimization and variational inequality problems are equivalent. Hence, solution algorithms to solve any of the three problems can be used to solve the other problems. Vector network equilibrium problems lead to analogous definitions of vector optimization (VOP) and vector variational inequality (VVI) problems. Investigating whether a similar equivalence exists in the vector valued case suggests itself, in particular to derive solution algorithms for vector equilibrium problems (VEQ). Unfortunately, the three problems are no longer equivalent in the vector valued case. We show under which assumptions a solution of VOP solves VEQ. Even though a solution of VVI is a solution of VEQ, the converse is not true. We demonstrate structural properties of solutions of VEQ that prevent them from being solutions of VVI and show under which assumptions VVI and VEQ are equivalent. We also comment in more detail on some results within the literature related to concepts of vector equilibria.",
keywords = "vector equilibrium, vector variational inequality , vector optimization , traffic assignment",
author = "Andrea Raith and Matthias Ehrgott",
year = "2011",
month = jan,
doi = "10.1002/mcda.477",
language = "English",
volume = "18",
pages = "39--54",
journal = "Journal of Multi-Criteria Decision Analysis",
issn = "1057-9214",
publisher = "John Wiley and Sons Ltd",
number = "1-2",

}

RIS

TY - JOUR

T1 - On vector equilibria, vector optimisation and vector variational inequalities

AU - Raith, Andrea

AU - Ehrgott, Matthias

PY - 2011/1

Y1 - 2011/1

N2 - It is well-known that, under certain conditions, network equilibrium, optimization and variational inequality problems are equivalent. Hence, solution algorithms to solve any of the three problems can be used to solve the other problems. Vector network equilibrium problems lead to analogous definitions of vector optimization (VOP) and vector variational inequality (VVI) problems. Investigating whether a similar equivalence exists in the vector valued case suggests itself, in particular to derive solution algorithms for vector equilibrium problems (VEQ). Unfortunately, the three problems are no longer equivalent in the vector valued case. We show under which assumptions a solution of VOP solves VEQ. Even though a solution of VVI is a solution of VEQ, the converse is not true. We demonstrate structural properties of solutions of VEQ that prevent them from being solutions of VVI and show under which assumptions VVI and VEQ are equivalent. We also comment in more detail on some results within the literature related to concepts of vector equilibria.

AB - It is well-known that, under certain conditions, network equilibrium, optimization and variational inequality problems are equivalent. Hence, solution algorithms to solve any of the three problems can be used to solve the other problems. Vector network equilibrium problems lead to analogous definitions of vector optimization (VOP) and vector variational inequality (VVI) problems. Investigating whether a similar equivalence exists in the vector valued case suggests itself, in particular to derive solution algorithms for vector equilibrium problems (VEQ). Unfortunately, the three problems are no longer equivalent in the vector valued case. We show under which assumptions a solution of VOP solves VEQ. Even though a solution of VVI is a solution of VEQ, the converse is not true. We demonstrate structural properties of solutions of VEQ that prevent them from being solutions of VVI and show under which assumptions VVI and VEQ are equivalent. We also comment in more detail on some results within the literature related to concepts of vector equilibria.

KW - vector equilibrium

KW - vector variational inequality

KW - vector optimization

KW - traffic assignment

U2 - 10.1002/mcda.477

DO - 10.1002/mcda.477

M3 - Journal article

VL - 18

SP - 39

EP - 54

JO - Journal of Multi-Criteria Decision Analysis

JF - Journal of Multi-Criteria Decision Analysis

SN - 1057-9214

IS - 1-2

ER -