Convergence of Adaptive Algorithms for Equilibrium Problems in Hadamard Spaces
In this paper, we consider the equilibrium problems under the setting of Hadamard spaces. For an approximate solution of equilibrium problems, iterative adaptive two-stage proximal algorithms are proposed and studied. At each step of the algorithms, the sequential minimization of two special strongl...
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| Format: | Article |
| Language: | English |
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Oles Honchar Dnipro National University
2025-03-01
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| Series: | Journal of Optimization, Differential Equations and Their Applications |
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| Online Access: | https://model-dnu.dp.ua/index.php/SM/article/view/209 |
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| author | Serhii V. Denysov Oleksandra Yu. Kovalenko Vladimir V. Semenov |
| author_facet | Serhii V. Denysov Oleksandra Yu. Kovalenko Vladimir V. Semenov |
| author_sort | Serhii V. Denysov |
| collection | DOAJ |
| description | In this paper, we consider the equilibrium problems under the setting of Hadamard spaces. For an approximate solution of equilibrium problems, iterative adaptive two-stage proximal algorithms are proposed and studied. At each step of the algorithms, the sequential minimization of two special strongly convex functions should be done. Our self-adaptive algorithms do not calculate bifunction values at additional points and do not require knowledge of the bifunction’s Lipschitz constants. For pseudomonotone bifunctions of the Lipschitz type that are weakly upper semicontinuous in the first variable and convex and lower semicontinuous in the second variable, we prove convergence theorems about sequences generated by proposed algorithms. First, we show weak convergence of generated sequences to a solution of the equilibrium problem. Then we prove strong convergence of Halpern regularization of adaptive extraproximal algorithm. Also it is shown that proposed algorithms are applicable to variational inequalities with Lipschitz-continuous, sequentially weakly continuous and pseudo-monotone operators acting in Hilbert spaces. |
| format | Article |
| id | doaj-art-934d9e6b979d4db388b2cab680adf9a1 |
| institution | OA Journals |
| issn | 2617-0108 2663-6824 |
| language | English |
| publishDate | 2025-03-01 |
| publisher | Oles Honchar Dnipro National University |
| record_format | Article |
| series | Journal of Optimization, Differential Equations and Their Applications |
| spelling | doaj-art-934d9e6b979d4db388b2cab680adf9a12025-08-20T02:35:40ZengOles Honchar Dnipro National UniversityJournal of Optimization, Differential Equations and Their Applications2617-01082663-68242025-03-01331426710.15421/142503201Convergence of Adaptive Algorithms for Equilibrium Problems in Hadamard SpacesSerhii V. Denysov0Oleksandra Yu. Kovalenko1Vladimir V. Semenov2Harbour. Space Institute of TechnologyTaras Shevchenko National University of KyivTaras Shevchenko National University of KyivIn this paper, we consider the equilibrium problems under the setting of Hadamard spaces. For an approximate solution of equilibrium problems, iterative adaptive two-stage proximal algorithms are proposed and studied. At each step of the algorithms, the sequential minimization of two special strongly convex functions should be done. Our self-adaptive algorithms do not calculate bifunction values at additional points and do not require knowledge of the bifunction’s Lipschitz constants. For pseudomonotone bifunctions of the Lipschitz type that are weakly upper semicontinuous in the first variable and convex and lower semicontinuous in the second variable, we prove convergence theorems about sequences generated by proposed algorithms. First, we show weak convergence of generated sequences to a solution of the equilibrium problem. Then we prove strong convergence of Halpern regularization of adaptive extraproximal algorithm. Also it is shown that proposed algorithms are applicable to variational inequalities with Lipschitz-continuous, sequentially weakly continuous and pseudo-monotone operators acting in Hilbert spaces.https://model-dnu.dp.ua/index.php/SM/article/view/209hadamard spaceequilibrium problempseudo-monotonicityextraproximal algorithmadaptive algorithmregularizationconvergence |
| spellingShingle | Serhii V. Denysov Oleksandra Yu. Kovalenko Vladimir V. Semenov Convergence of Adaptive Algorithms for Equilibrium Problems in Hadamard Spaces Journal of Optimization, Differential Equations and Their Applications hadamard space equilibrium problem pseudo-monotonicity extraproximal algorithm adaptive algorithm regularization convergence |
| title | Convergence of Adaptive Algorithms for Equilibrium Problems in Hadamard Spaces |
| title_full | Convergence of Adaptive Algorithms for Equilibrium Problems in Hadamard Spaces |
| title_fullStr | Convergence of Adaptive Algorithms for Equilibrium Problems in Hadamard Spaces |
| title_full_unstemmed | Convergence of Adaptive Algorithms for Equilibrium Problems in Hadamard Spaces |
| title_short | Convergence of Adaptive Algorithms for Equilibrium Problems in Hadamard Spaces |
| title_sort | convergence of adaptive algorithms for equilibrium problems in hadamard spaces |
| topic | hadamard space equilibrium problem pseudo-monotonicity extraproximal algorithm adaptive algorithm regularization convergence |
| url | https://model-dnu.dp.ua/index.php/SM/article/view/209 |
| work_keys_str_mv | AT serhiivdenysov convergenceofadaptivealgorithmsforequilibriumproblemsinhadamardspaces AT oleksandrayukovalenko convergenceofadaptivealgorithmsforequilibriumproblemsinhadamardspaces AT vladimirvsemenov convergenceofadaptivealgorithmsforequilibriumproblemsinhadamardspaces |