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

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  • Gabriel Ruiz-Garzón

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

  • Rafaela Osuna-Gómez

    (Departamento de Estadística e I.O., Universidad de Sevilla, 41004 Sevilla, Spain
    These authors contributed equally to this work.)

  • Antonio Rufián-Lizana

    (Departamento de Estadística e I.O., Universidad de Sevilla, 41004 Sevilla, Spain
    These authors contributed equally to this work.)

  • Beatriz Hernández-Jiménez

    (Departamento de Economía, Métodos Cuantitativos e Historia Económica, Universidad Pablo de Olavide, 41013 Sevilla, Spain
    These authors contributed equally to this work.)

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.

Suggested Citation

  • Gabriel Ruiz-Garzón & Rafaela Osuna-Gómez & Antonio Rufián-Lizana & Beatriz Hernández-Jiménez, 2020. "Approximate Efficient Solutions of the Vector Optimization Problem on Hadamard Manifolds via Vector Variational Inequalities," Mathematics, MDPI, vol. 8(12), pages 1-19, December.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:12:p:2196-:d:459572
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    References listed on IDEAS

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    1. Sheng-lan Chen & Nan-Jing Huang & Donal O'Regan, 2014. "Geodesic B -Preinvex Functions and Multiobjective Optimization Problems on Riemannian Manifolds," Journal of Applied Mathematics, Hindawi, vol. 2014, pages 1-12, March.
    2. Szilárd Nagy, 2013. "Stackelberg equilibria via variational inequalities and projections," Journal of Global Optimization, Springer, vol. 57(3), pages 821-828, November.
    3. A. J. Novak & G. Feichtinger & G. Leitmann, 2010. "A Differential Game Related to Terrorism: Nash and Stackelberg Strategies," Journal of Optimization Theory and Applications, Springer, vol. 144(3), pages 533-555, March.
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