Relaxed and Inertial Nonlinear Forward–Backward with Momentum

Fernando Roldán () and Cristian Vega ()
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Fernando Roldán: Universidad de Concepción
Cristian Vega: Universidad de Tarapacá

Journal of Optimization Theory and Applications, 2025, vol. 206, issue 2, No 2, 30 pages

Abstract: Abstract In this article, we study inertial algorithms for numerically solving monotone inclusions involving the sum of a maximally monotone and a cocoercive operator. In particular, we analyze the convergence of inertial and relaxed versions of the nonlinear forward–backward with momentum (NFBM). We propose an inertial version of NFBM including a relaxation step and a second version considering a double-inertial step with additional momentum. By applying NFBM to specific monotone inclusions, we derive inertial and relaxed versions of algorithms such as forward–backward, forward-half-reflected-backward (FHRB), Chambolle–Pock, Condat–Vũ, among others, thereby recovering and extending previous results from the literature for solving monotone inclusions involving maximally monotone, cocoercive, monotone and Lipschitz, and linear bounded operators. We also present numerical experiments on image restoration, comparing the proposed inertial and relaxed algorithms. In particular, we compare the inertial and relaxed FHRB with its non-inertial and momentum versions. Additionally, we compare the numerical convergence for larger step-sizes versus relaxation parameters and introduce a restart strategy that incorporates larger step-sizes and inertial steps to further enhance numerical convergence.

Keywords: Operator splitting; Monotone operators; Monotone inclusion; Inertial methods; Convex optimization; 47H05; 65K05; 65K15; 90C25 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10957-025-02694-5

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