Backward–forward algorithms for structured monotone inclusions in Hilbert spaces
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 ba...
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| Vydané v: | Journal of mathematical analysis and applications Ročník 457; číslo 2; s. 1095 - 1117 |
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| Jazyk: | English |
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Elsevier Inc
15.01.2018
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| ISSN: | 0022-247X, 1096-0813 |
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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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| AbstractList | 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. |
| Author | Peypouquet, Juan Redont, Patrick Attouch, Hédy |
| Author_xml | – sequence: 1 givenname: Hédy surname: Attouch fullname: Attouch, Hédy email: hedy.attouch@univ-montp2.fr organization: Institut Montpelliérain Alexander Grothendieck, IMAG UMR 5149 CNRS, Université Montpellier 2, place Eugène Bataillon, 34095 Montpellier Cedex 5, France – sequence: 2 givenname: Juan orcidid: 0000-0002-8551-0522 surname: Peypouquet fullname: Peypouquet, Juan email: juan.peypouquet@usm.cl organization: Departamento de Matemática, Universidad Técnica Federico Santa María, Avenida España 1680, Valparaíso, Chile – sequence: 3 givenname: Patrick surname: Redont fullname: Redont, Patrick email: patrick.redont@univ-montp2.fr organization: Institut Montpelliérain Alexander Grothendieck, IMAG UMR 5149 CNRS, Université Montpellier 2, place Eugène Bataillon, 34095 Montpellier Cedex 5, France |
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| Title | Backward–forward algorithms for structured monotone inclusions in Hilbert spaces |
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