An inertial proximal splitting algorithm for hierarchical bilevel equilibria in Hilbert spaces
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915176823914496 |
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| author | Balhag, Aicha Mazgouri, Zakaria Riahi, Hassan Théra, Michel |
| author_facet | Balhag, Aicha Mazgouri, Zakaria Riahi, Hassan Théra, Michel |
| contents | In this article, we aim to approximate a solution to the bilevel equilibrium problem $\mathbf{(BEP})$ for short: find $\bar{x} \in \mathbf{S}_f$ such that $
g(\bar{x}, y) \geq 0, \,\, \forall y \in \mathbf{S}_f, $
where $
\mathbf{S}_f = \{ u \in \mathbf{K} : f(u, z) \geq 0, \forall z \in \mathbf{K} \}. $
Here, $\mathbf{K}$ is a closed convex subset of a real Hilbert space $\mathcal{H}$, and $f$ and $g$ are two real-valued bifunctions defined on $\mathbf{K} \times \mathbf{K}$. We propose an inertial version of the proximal splitting algorithm introduced by Z. Chbani and H. Riahi: \textit{Weak and strong convergence of prox-penalization and splitting algorithms for bilevel equilibrium problems}. \textit{Numer. Algebra Control Optim.}, 3 (2013), pp. 353-366. Under suitable conditions, we establish the weak and strong convergence of the sequence generated by the proposed iterative method. We also report a numerical example illustrating our theoretical result. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_20999 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An inertial proximal splitting algorithm for hierarchical bilevel equilibria in Hilbert spaces Balhag, Aicha Mazgouri, Zakaria Riahi, Hassan Théra, Michel Optimization and Control 90C33, 49J40, 46N10, 65K15, 65K10 In this article, we aim to approximate a solution to the bilevel equilibrium problem $\mathbf{(BEP})$ for short: find $\bar{x} \in \mathbf{S}_f$ such that $ g(\bar{x}, y) \geq 0, \,\, \forall y \in \mathbf{S}_f, $ where $ \mathbf{S}_f = \{ u \in \mathbf{K} : f(u, z) \geq 0, \forall z \in \mathbf{K} \}. $ Here, $\mathbf{K}$ is a closed convex subset of a real Hilbert space $\mathcal{H}$, and $f$ and $g$ are two real-valued bifunctions defined on $\mathbf{K} \times \mathbf{K}$. We propose an inertial version of the proximal splitting algorithm introduced by Z. Chbani and H. Riahi: \textit{Weak and strong convergence of prox-penalization and splitting algorithms for bilevel equilibrium problems}. \textit{Numer. Algebra Control Optim.}, 3 (2013), pp. 353-366. Under suitable conditions, we establish the weak and strong convergence of the sequence generated by the proposed iterative method. We also report a numerical example illustrating our theoretical result. |
| title | An inertial proximal splitting algorithm for hierarchical bilevel equilibria in Hilbert spaces |
| topic | Optimization and Control 90C33, 49J40, 46N10, 65K15, 65K10 |
| url | https://arxiv.org/abs/2502.20999 |