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

Garzón, Gabriel Ruiz; Osuna-Gómez, R.; Ruiz-Zapatero, Jaime

doi:10.3390/sym11081037

Cited by 20 publications

(16 citation statements)

References 26 publications

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“…Once introduced the concept of invexity, we can know discuss the existence of critical points. For this purpose, we will adopt the definition given by Ruiz-Garzón et al [13]: This result is a generalization to Riemannian manifolds of similar result to the one achieved by Craven and Glover [16].…”

Section: Unconstrained Casementioning

confidence: 99%

“…In Ruiz-Garzón et al [13] we obtained first-order optimality conditions for the scalar and vector optimization problem on Riemannian manifolds, but not second order conditions, it will be discussed in this article. Also, in Ruiz-Garzón et al [14] we prove the existence of optimality conditions from KKT to constrained vector optimization problems on Hadamard manifolds as a particular case of equilibrium vector problems with constraints.…”

Section: Introductionmentioning

confidence: 99%

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Second-Order Optimality Conditions: An extension to Hadamard Manifolds

Garzón¹,

Ruiz-Zapatero²,

Osuna-Gómez³

et al. 2020

Preprint

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This work is intended to lead a study of necessary and sufficient optimality conditions for scalar optimization problems on Hadamard manifolds. In the context of this geometry, we obtain and present new function types characterized by the property of having all their second-order stationary points to be global minimums. In order to do so, we extend the concept convexity in Euclidean space to a more general notion of invexity on Hadamard manifolds. This is done employing notions of second-order directional derivative, second-order pseudoinvexity functions and the second-order Karush-Kuhn-Tucker-pseudoinvexity problem. Thus, we prove that every second-order stationary point is a global minimum if and only if the problem is either second-order pseudoinvex or second-order KKT-pseudoinvex depending on whether the problem regards unconstrained or constrained scalar optimization respectively. This result has not been presented in the literature before. Finally, examples of these new characterizations are provided in the context of \textit{"Higgs Boson like"} potentials among others.

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Section: Unconstrained Casementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Second-Order Optimality Conditions: An extension to Hadamard Manifolds

Garzón¹,

Ruiz-Zapatero²,

Osuna-Gómez³

et al. 2020

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Two years later, Ruiz-Garzón et al [26] extended these properties on Riemannian manifolds in the smooth case. In 2019, Ruiz-Garzón et al [27] showed the existence of KKT optimality conditions for weakly efficient Pareto solutions for vector equilibrium problems, with particular focus on the Nash equilibrium problem, but only in the differential case.…”

Section: Introductionmentioning

confidence: 99%

Solutions of Optimization Problems on Hadamard Manifolds with Lipschitz Functions

2020

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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.

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“…More recently, in 2019, in Ruiz-Garzón et al [25], we studied the constrained vector optimization problem as a particular case of the equilibrium vector with constraints problem on Hadamard manifolds.…”

Section: Introductionmentioning

confidence: 99%

Vectorial Variational Inequalities on Hadamard Manifolds: A Tool to Achieve Efficient Approximate Solutions

Garzón¹,

Osuna-Gómez²,

Rufián-Lizana³

et al. 2020

Preprint

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This article has two objectives. Firstly, we will use the vector variational-like inequalities problems to achieve local approximate (weakly) efficient solutions of Vector Optimization Problem within the novel field of the Hadamard manifolds. Previously, we will introduce the concepts of generalized approximate geodesic convex functions and illustrate them with examples. We will 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 will show an economical application, using again solutions of the variational problems to identify with Stackelberg equilibrium points on Hadamard manifolds and under geodesic convexity assumptions.

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

Cited by 20 publications

References 26 publications

Second-Order Optimality Conditions: An extension to Hadamard Manifolds

Second-Order Optimality Conditions: An extension to Hadamard Manifolds

Solutions of Optimization Problems on Hadamard Manifolds with Lipschitz Functions

Vectorial Variational Inequalities on Hadamard Manifolds: A Tool to Achieve Efficient Approximate Solutions

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