Necessary and Sufficient Second-Order Optimality Conditions on Hadamard Manifolds
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 b...
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MDPI AG
2020-07-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/8/7/1152 |
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| author | Gabriel Ruiz-Garzón Jaime Ruiz-Zapatero Rafaela Osuna-Gómez Antonio Rufián-Lizana |
| author_facet | Gabriel Ruiz-Garzón Jaime Ruiz-Zapatero Rafaela Osuna-Gómez Antonio Rufián-Lizana |
| author_sort | Gabriel Ruiz-Garzón |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-10T18:29:17Z |
| format | Article |
| id | doaj.art-0c9ba23dd157488ebbf8feeeba3f7696 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T18:29:17Z |
| publishDate | 2020-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-0c9ba23dd157488ebbf8feeeba3f76962023-11-20T06:42:50ZengMDPI AGMathematics2227-73902020-07-0187115210.3390/math8071152Necessary and Sufficient Second-Order Optimality Conditions on Hadamard ManifoldsGabriel Ruiz-Garzón0Jaime Ruiz-Zapatero1Rafaela Osuna-Gómez2Antonio Rufián-Lizana3Instituto de Desarrollo Social y Sostenible (INDESS), Universidad de Cádiz,11003 Cádiz, SpainDepartment of Physics and Astronomy, University College London, London WC1E 6BT, UKDepartamento de Estadística e I.O., Universidad de Sevilla, 41004 Sevilla, SpainDepartamento de Estadística e I.O., Universidad de Sevilla, 41004 Sevilla, SpainThis 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.https://www.mdpi.com/2227-7390/8/7/1152Hadamard manifoldsecond-order optimality conditionsgeneralized convexity |
| spellingShingle | Gabriel Ruiz-Garzón Jaime Ruiz-Zapatero Rafaela Osuna-Gómez Antonio Rufián-Lizana Necessary and Sufficient Second-Order Optimality Conditions on Hadamard Manifolds Mathematics Hadamard manifold second-order optimality conditions generalized convexity |
| title | Necessary and Sufficient Second-Order Optimality Conditions on Hadamard Manifolds |
| title_full | Necessary and Sufficient Second-Order Optimality Conditions on Hadamard Manifolds |
| title_fullStr | Necessary and Sufficient Second-Order Optimality Conditions on Hadamard Manifolds |
| title_full_unstemmed | Necessary and Sufficient Second-Order Optimality Conditions on Hadamard Manifolds |
| title_short | Necessary and Sufficient Second-Order Optimality Conditions on Hadamard Manifolds |
| title_sort | necessary and sufficient second order optimality conditions on hadamard manifolds |
| topic | Hadamard manifold second-order optimality conditions generalized convexity |
| url | https://www.mdpi.com/2227-7390/8/7/1152 |
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