Fast Reflected Forward-Backward algorithm: achieving fast convergence rates for convex optimization with linear cone constraints

In this paper, we derive a Fast Reflected Forward-Backward (Fast RFB) algorithm to solve the problem of finding a zero of the sum of a maximally monotone operator and a monotone and Lipschitz continuous operator in a real Hilbert space. Our approach extends the class of reflected forward-backward me...

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Veröffentlicht in:Journal of scientific computing Jg. 105; H. 3; S. 73
Hauptverfasser: Boţ, Radu Ioan, Nguyen, Dang-Khoa, Zong, Chunxiang
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York Springer US 01.12.2025
Springer Nature B.V
Schlagworte:
Algorithms
Computational Mathematics and Numerical Analysis
Constraints
Convergence
Convex analysis
Convexity
Distance learning
Hilbert space
Lipschitz condition
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Methods
Numerical analysis
Operators (mathematics)
Optimization
Theoretical
saddle point problem
68Q25
Lyapunov analysis
65K05
fast primal-dual algorithm
reflected forward-backward splitting algorithm
90C25
90C47
Nesterov momentum
49M29
convergence of the iterates
monotone inclusion
fast convergence rates
ISSN:0885-7474, 1573-7691
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Zusammenfassung:In this paper, we derive a Fast Reflected Forward-Backward (Fast RFB) algorithm to solve the problem of finding a zero of the sum of a maximally monotone operator and a monotone and Lipschitz continuous operator in a real Hilbert space. Our approach extends the class of reflected forward-backward methods by introducing a Nesterov momentum term and a correction term, resulting in enhanced convergence performance. The iterative sequence of the proposed algorithm is proven to converge weakly, and the Fast RFB algorithm demonstrates impressive convergence rates, achieving o 1 k as k → + ∞ for both the discrete velocity and the tangent residual at the last-iterate . When applied to minimax problems with a smooth coupling term and nonsmooth convex regularizers, the resulting algorithm demonstrates significantly improved convergence properties compared to the current state of the art in the literature. For convex optimization problems with linear cone constraints, our approach yields a fully splitting primal-dual algorithm that ensures not only the convergence of iterates to a primal-dual solution, but also a last-iterate convergence rate of o 1 k as k → + ∞ for the objective function value, feasibility measure, and complementarity condition. This represents the most competitive theoretical result currently known for algorithms addressing this class of optimization problems. Numerical experiments are performed to illustrate the convergence behavior of Fast RFB.
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ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-025-03103-9