Necessary and Sufficient Second-Order Optimality Conditions on Hadamard Manifolds

Gabriel Ruiz-Garzón, Jaime Ruiz-Zapatero, Rafaela Osuna-Gómez and Antonio Rufián-Lizana
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Gabriel Ruiz-Garzón: Instituto de Desarrollo Social y Sostenible (INDESS), Universidad de Cádiz,11003 Cádiz, Spain
Jaime Ruiz-Zapatero: Department of Physics and Astronomy, University College London, London WC1E 6BT, UK
Rafaela Osuna-Gómez: Departamento de Estadística e I.O., Universidad de Sevilla, 41004 Sevilla, Spain
Antonio Rufián-Lizana: Departamento de Estadística e I.O., Universidad de Sevilla, 41004 Sevilla, Spain

Mathematics, 2020, vol. 8, issue 7, 1-12

Abstract: 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 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 derivatives, 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 “Higgs Boson like” potentials, among others.

Keywords: Hadamard manifold; second-order optimality conditions; generalized convexity (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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