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Symmetry
Volume 12
Issue 5
10.3390/sym12050804
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Article

Solutions of Optimization Problems on Hadamard Manifolds with Lipschitz Functions

by
Gabriel Ruiz-Garzón
1,*,†,‡,
Rafaela Osuna-Gómez
2,‡ and
Antonio Rufián-Lizana
2,‡
1
Instituto de Desarrollo Social y Sostenible (INDESS), Universidad de Cádiz, 11406 Cádiz, Spain
2
Departamento de Estadística e I.O., Universidad de Sevilla, 41012 Sevilla, Spain
*
Author to whom correspondence should be addressed.
†
Current address: Instituto de Desarrollo Social y Sostenible (INDESS), Universidad de Cádiz, Campus de Jerez de la Frontera, Avda. de la Universidad s/n, 11405 Jerez de la Frontera, Spain.
‡
These authors contributed equally to this work.
Symmetry 2020, 12(5), 804; https://doi.org/10.3390/sym12050804
Submission received: 13 April 2020 / Revised: 1 May 2020 / Accepted: 3 May 2020 / Published: 12 May 2020
Versions Notes

Abstract

The aims of this paper are twofold. First, it is shown, for the first time, which types of nonsmooth functions are characterized by all vector critical points as being efficient or weakly efficient solutions of vector optimization problems in constrained and unconstrained scenarios on Hadamard manifolds. This implies the need to extend different concepts, such as the Karush–Kuhn–Tucker vector critical points and generalized invexity functions, to Hadamard manifolds. The relationships between these quantities are clarified through a great number of explanatory examples. Second, we present an economic application proving that Nash’s critical and equilibrium points coincide in the case of invex payoff functions. This is done on Hadamard manifolds, a particular case of noncompact Riemannian symmetric spaces.
Keywords: generalized convexity; Hadamard manifold; efficient solution; vector critical point; Nash equilibrium point generalized convexity; Hadamard manifold; efficient solution; vector critical point; Nash equilibrium point

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MDPI and ACS Style

Ruiz-Garzón, G.; Osuna-Gómez, R.; Rufián-Lizana, A. Solutions of Optimization Problems on Hadamard Manifolds with Lipschitz Functions. Symmetry 2020, 12, 804. https://doi.org/10.3390/sym12050804

AMA Style

Ruiz-Garzón G, Osuna-Gómez R, Rufián-Lizana A. Solutions of Optimization Problems on Hadamard Manifolds with Lipschitz Functions. Symmetry. 2020; 12(5):804. https://doi.org/10.3390/sym12050804

Chicago/Turabian Style

Ruiz-Garzón, Gabriel, Rafaela Osuna-Gómez, and Antonio Rufián-Lizana. 2020. "Solutions of Optimization Problems on Hadamard Manifolds with Lipschitz Functions" Symmetry 12, no. 5: 804. https://doi.org/10.3390/sym12050804

APA Style

Ruiz-Garzón, G., Osuna-Gómez, R., & Rufián-Lizana, A. (2020). Solutions of Optimization Problems on Hadamard Manifolds with Lipschitz Functions. Symmetry, 12(5), 804. https://doi.org/10.3390/sym12050804

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