Convergence of a Relaxed Inertial Forward–Backward Algorithm for Structured Monotone Inclusions

Abstract : In this paper, we study the backward forward algorithm as a splitting method to solve structured monotone inclusions, and convex minimization problems in Hilbert spaces. It has a natural link with the forward backward algorithm and has the same computational complexity, since it involves the same basic blocks, but organized differently. Surprisingly enough, this kind of iteration arises when studying the time discretization of the regularized Newton method for maximally monotone operators. First, we show that these two methods enjoy remarkable involutive relations, which go far beyond the evident inversion of the order in which the forward and backward steps are applied. Next, we establish several convergence properties for both methods, some of which were unknown even for the forward backward algorithm. This brings further insight into this well-known scheme. Finally, we specialize our results to structured convex minimization problems, the gradient projection algorithms, and give a numerical illustration of theoretical interest.
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https://hal-univ-bourgogne.archives-ouvertes.fr/hal-02173692
Contributeur : Imb - Université de Bourgogne <>
Soumis le : jeudi 4 juillet 2019 - 16:11:55
Dernière modification le : vendredi 5 juillet 2019 - 01:29:22

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Hedy Attouch, Alexandre Cabot. Convergence of a Relaxed Inertial Forward–Backward Algorithm for Structured Monotone Inclusions. Applied Mathematics and Optimization, Springer Verlag (Germany), 2019, pp.1-52. ⟨https://link-springer-com.proxy-scd.u-bourgogne.fr/article/10.1007%2Fs00245-019-09584-z#citeas⟩. ⟨10.1007/s00245-019-09584-z⟩. ⟨hal-02173692⟩

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