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Necessary and Sufficient Optimality Conditions for Vector Equilibrium Problems on Hadamard Manifolds

1
Departamento de Estadística e I.O., Universidad de Cádiz, 11405 Cádiz, Spain
2
Departamento de Estadística e I.O., Universidad de Sevilla, 41012 Sevilla, Spain
3
Department of Physics and Astronomy, University College of London, London WC1E 6BT, UK
*
Author to whom correspondence should be addressed.
Current address: Departamento de Estadística e I.O., Universidad de Cádiz, Campus de Jerez de la Frontera, Avda. de la Universidad s/n, 11405, Jerez de la Frontera, Cádiz, Spain.
These authors contributed equally to this work.
Symmetry 2019, 11(8), 1037; https://doi.org/10.3390/sym11081037
Received: 18 July 2019 / Revised: 5 August 2019 / Accepted: 8 August 2019 / Published: 12 August 2019
(This article belongs to the Special Issue Geometry of Submanifolds and Homogeneous Spaces)
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PDF [249 KB, uploaded 12 August 2019]

Abstract

The aim of this paper is to show the existence and attainability of Karush–Kuhn–Tucker optimality conditions for weakly efficient Pareto points for vector equilibrium problems with the addition of constraints in the novel context of Hadamard manifolds, as opposed to the classical examples of Banach, normed or Hausdorff spaces. More specifically, classical necessary and sufficient conditions for weakly efficient Pareto points to the constrained vector optimization problem are presented. The results described in this article generalize results obtained by Gong (2008) and Wei and Gong (2010) and Feng and Qiu (2014) from Hausdorff topological vector spaces, real normed spaces, and real Banach spaces to Hadamard manifolds, respectively. This is done using a notion of Riemannian symmetric spaces of a noncompact type as special Hadarmard manifolds. View Full-Text
Keywords: vector equilibrium problem; generalized convexity; hadamard manifolds; weakly efficient pareto points vector equilibrium problem; generalized convexity; hadamard manifolds; weakly efficient pareto points
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Ruiz-Garzón, G.; Osuna-Gómez, R.; Ruiz-Zapatero, J. Necessary and Sufficient Optimality Conditions for Vector Equilibrium Problems on Hadamard Manifolds. Symmetry 2019, 11, 1037.

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