Backward–forward algorithms for structured monotone inclusions in Hilbert spaces |
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Authors: | Hédy Attouch Juan Peypouquet Patrick Redont |
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Institution: | 1. Institut Montpelliérain Alexander Grothendieck, IMAG UMR 5149 CNRS, Université Montpellier 2, place Eugène Bataillon, 34095 Montpellier Cedex 5, France;2. Departamento de Matemática, Universidad Técnica Federico Santa María, Avenida España 1680, Valparaíso, Chile |
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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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Keywords: | Monotone inclusion Forward–backward algorithm Proximal-gradient method |
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