Approximate Efficient Solutions of the Vector Optimization Problem on Hadamard Manifolds via Vector Variational Inequalities

Ruiz-Garzón, Gabriel; Osuna-Gómez, Rafaela; Rufián-Lizana, Antonio; Hernández-Jiménez, Beatriz

doi:10.3390/math8122196

Open AccessArticle

Approximate Efficient Solutions of the Vector Optimization Problem on Hadamard Manifolds via Vector Variational Inequalities

by

Gabriel Ruiz-Garzón

^1,*,†,‡

,

Rafaela Osuna-Gómez

^2,‡

,

Antonio Rufián-Lizana

^2,‡

and

Beatriz Hernández-Jiménez

^3,‡

¹

Instituto de Desarrollo Social y Sostenible (INDESS), Universidad de Cádiz, 11003 Cádiz, Spain

²

Departamento de Estadística e I.O., Universidad de Sevilla, 41004 Sevilla, Spain

³

Departamento de Economía, Métodos Cuantitativos e Historia Económica, Universidad Pablo de Olavide, 41013 Sevilla, Spain

^*

Author to whom correspondence should be addressed.

^†

Instituto de Desarrollo Social y Sostenible (INDESS), Universidad de Cádiz, Campus de Jerez de la Frontera, Avda. de la Universidad s/n, Jerez de la Frontera, Cádiz 11405, Spain.

^‡

These authors contributed equally to this work.

Mathematics 2020, 8(12), 2196; https://doi.org/10.3390/math8122196

Submission received: 12 November 2020 / Revised: 5 December 2020 / Accepted: 6 December 2020 / Published: 10 December 2020

(This article belongs to the Special Issue Multiobjective Optimization: Methodology, Computational Implementation and Real Models)

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Abstract

This article has two objectives. Firstly, we use the vector variational-like inequalities problems to achieve local approximate (weakly) efficient solutions of the vector optimization problem within the novel field of the Hadamard manifolds. Previously, we introduced the concepts of generalized approximate geodesic convex functions and illustrated them with examples. We see the minimum requirements under which critical points, solutions of Stampacchia, and Minty weak variational-like inequalities and local approximate weakly efficient solutions can be identified, extending previous results from the literature for linear Euclidean spaces. Secondly, we show an economical application, again using solutions of the variational problems to identify Stackelberg equilibrium points on Hadamard manifolds and under geodesic convexity assumptions.

Keywords: generalized convexity; Hadamard manifold; approximate efficient solution; Stackelberg equilibrium point

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MDPI and ACS Style

Ruiz-Garzón, G.; Osuna-Gómez, R.; Rufián-Lizana, A.; Hernández-Jiménez, B. Approximate Efficient Solutions of the Vector Optimization Problem on Hadamard Manifolds via Vector Variational Inequalities. Mathematics 2020, 8, 2196. https://doi.org/10.3390/math8122196

AMA Style

Ruiz-Garzón G, Osuna-Gómez R, Rufián-Lizana A, Hernández-Jiménez B. Approximate Efficient Solutions of the Vector Optimization Problem on Hadamard Manifolds via Vector Variational Inequalities. Mathematics. 2020; 8(12):2196. https://doi.org/10.3390/math8122196

Chicago/Turabian Style

Ruiz-Garzón, Gabriel, Rafaela Osuna-Gómez, Antonio Rufián-Lizana, and Beatriz Hernández-Jiménez. 2020. "Approximate Efficient Solutions of the Vector Optimization Problem on Hadamard Manifolds via Vector Variational Inequalities" Mathematics 8, no. 12: 2196. https://doi.org/10.3390/math8122196

APA Style

Ruiz-Garzón, G., Osuna-Gómez, R., Rufián-Lizana, A., & Hernández-Jiménez, B. (2020). Approximate Efficient Solutions of the Vector Optimization Problem on Hadamard Manifolds via Vector Variational Inequalities. Mathematics, 8(12), 2196. https://doi.org/10.3390/math8122196

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Approximate Efficient Solutions of the Vector Optimization Problem on Hadamard Manifolds via Vector Variational Inequalities

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