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Keywords = reflected–forward–backward splitting method

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13 pages, 306 KB  
Article
A Reflected–Forward–Backward Splitting Method for Monotone Inclusions Involving Lipschitz Operators in Banach Spaces
by Changchi Huang, Jigen Peng, Liqian Qin and Yuchao Tang
Mathematics 2026, 14(2), 245; https://doi.org/10.3390/math14020245 - 8 Jan 2026
Abstract
The reflected–forward–backward splitting (RFBS) method is well-established for solving monotone inclusion problems involving Lipschitz continuous operators in Hilbert spaces, where it converges weakly under mild assumptions. Extending this method to Banach spaces presents significant challenges, primarily due to the nonlinearity of the duality [...] Read more.
The reflected–forward–backward splitting (RFBS) method is well-established for solving monotone inclusion problems involving Lipschitz continuous operators in Hilbert spaces, where it converges weakly under mild assumptions. Extending this method to Banach spaces presents significant challenges, primarily due to the nonlinearity of the duality mapping. In this paper, we propose and analyze an RFBS algorithm in the setting of real Banach spaces that are 2-uniformly convex and uniformly smooth. To the best of our knowledge, this work presents the first strong (R-linear) convergence result for the RFBS method in such Banach spaces, achieved under a newly adapted notion of strong monotonicity. Our results thus establish a foundational theoretical guarantee for RFBS in Banach spaces under strengthened monotonicity conditions, while highlighting the open problem of proving weak convergence for the general monotone case. Full article
(This article belongs to the Special Issue Nonlinear Functional Analysis: Theory, Methods, and Applications)
16 pages, 15923 KB  
Article
Aggregative Game for Distributed Charging Strategy of PEVs in a Smart Charging Station
by Ti Kang, Huaqing Li and Lifeng Zheng
Axioms 2023, 12(2), 186; https://doi.org/10.3390/axioms12020186 - 10 Feb 2023
Viewed by 2019
Abstract
This paper proposes a charging strategy for plug-in electric vehicles (PEVs) in a smart charging station (SCS) that considers load constraints and time anxieties. Due to the rapidly growing load demand of PEVs and the load capacity investments in infrastructure, PEV charging needs [...] Read more.
This paper proposes a charging strategy for plug-in electric vehicles (PEVs) in a smart charging station (SCS) that considers load constraints and time anxieties. Due to the rapidly growing load demand of PEVs and the load capacity investments in infrastructure, PEV charging needs to be subject to overload limits, beyond which failures can occur. The time anxiety is presented to address some of the uncertainties that may arise while charging PEVs. Under an aggregative game framework, this paper constructs a price-driven charging model to minimize costs by choosing the optimal charging strategy. Meanwhile, since the driver information is an aggregated item in the PEV cost function, the drivers’ privacy can be protected. Then, a distributed reflected forward–backward (RFB) splitting method is developed to search for the generalized Nash equilibria (GNE) of the game. The convergence of the proposed algorithm and the effectiveness of the charging strategy are verified by the detailed simulation and results. Full article
(This article belongs to the Special Issue Applied Mathematics in Energy and Mechanical Engineering)
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16 pages, 301 KB  
Article
A Symmetric FBF Method for Solving Monotone Inclusions
by Aviv Gibali and Yekini Shehu
Symmetry 2020, 12(9), 1456; https://doi.org/10.3390/sym12091456 - 4 Sep 2020
Cited by 4 | Viewed by 4330
Abstract
The forward–backward–forward (FBF) splitting method is a popular iterative procedure for finding zeros of the sum of maximal monotone and Lipschitz continuous monotone operators. In this paper, we introduce a forward–backward–forward splitting method with reflection steps (symmetric) in real Hilbert spaces. Weak and [...] Read more.
The forward–backward–forward (FBF) splitting method is a popular iterative procedure for finding zeros of the sum of maximal monotone and Lipschitz continuous monotone operators. In this paper, we introduce a forward–backward–forward splitting method with reflection steps (symmetric) in real Hilbert spaces. Weak and strong convergence analyses of the proposed method are established under suitable assumptions. Moreover, a linear convergence rate of an inertial modified forward–backward–forward splitting method is also presented. Full article
(This article belongs to the Special Issue Symmetry in Optimization and Control with Real World Applications)
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