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Optim."],"published-print":{"date-parts":[[2018,1]]},"abstract":"<jats:p>In a Hilbert space $\\mathcal H$, assuming $(\\alpha_k)$ a general sequence of nonnegative numbers, we analyze the convergence properties of the inertial forward-backward algorithm $(IFB)\\{\\begin{array}{l} y_k=x_k+\\alpha_k(x_k-x_{k-1}), x_{k+1}={\\rm prox}_{s\\Psi}(y_k-s\\nabla \\Phi(y_k)) \\end{array},$ where $\\Psi: \\mathcal H \\to \\mathbb R \\cup \\lbrace + \\infty \\rbrace $ is a proper lower-semicontinuous convex function, and $\\Phi: \\mathcal H \\to \\mathbb R$ is a differentiable convex function, whose gradient is Lipschitz continuous. Various options for the sequence $(\\alpha_k)$ are considered in the literature. Among them, the Nesterov choice leads to the FISTA algorithm and accelerates convergence from $\\mathcal{O}(1\/k)$ to $\\mathcal{O}(1\/k^2)$ for the values. Several variants are used to guarantee the convergence of the iterates or to improve the rate of convergence for the values. For the design of fast optimization methods, the tuning of the sequence $(\\alpha_k)$ is a subtle issue, which we deal with in this paper in general. We show that the convergence rate of the algorithm can be obtained simply by analyzing the sequence of positive real numbers $(\\alpha_k)$. In addition to the case $\\alpha_k= 1 -\\frac{\\alpha}{k} $ with $\\alpha\\geq 3$, our results apply equally well to $\\alpha_k = 1- \\frac{\\alpha}{k^r}$, with an exponent $0&lt;r&lt;1$, and to Polyak's heavy ball method. Thus, we unify most of the existing results based on the accelerated gradient method of Nesterov. In the process, we improve some of them and discover new ones.<\/jats:p>","DOI":"10.1137\/17m1114739","type":"journal-article","created":{"date-parts":[[2018,3,15]],"date-time":"2018-03-15T15:21:59Z","timestamp":1521127319000},"page":"849-874","source":"Crossref","is-referenced-by-count":100,"title":["Convergence Rates of Inertial Forward-Backward Algorithms"],"prefix":"10.1137","volume":"28","author":[{"given":"Hedy","family":"Attouch","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Alexandre","family":"Cabot","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2018,3,15]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012998335802"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1023\/A:1011253113155"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2017.06.024"},{"key":"atypb4","doi-asserted-by":"crossref","unstructured":"H. Attouch, Z. Chbani, J. Peypouquet, and P. Redont,\n                      Fast convergence of inertial dynamics and algorithms with asymptotic vanishing damping\n                      , Math. Program. (2016),doi:10.1007\/s10107-016-0992-8.","DOI":"10.1007\/s10107-016-0992-8"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1137\/15M1046095"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1137\/130910294"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1137\/140994964"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1016\/0022-1236(78)90093-9"},{"key":"atypb9","doi-asserted-by":"crossref","unstructured":"H. Bauschke and P. Combettes,\n                      Convex Analysis and Monotone Operator Theory in Hilbert spaces\n                      , CMS Books in Math., Springer, New York, 2011.","DOI":"10.1007\/978-1-4419-9467-7"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1137\/080716542"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2015.04.016"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1137\/15M1012657"},{"key":"atypb13","unstructured":"H. Bre\u0301zis,\n                      Ope\u0301rateurs maximaux monotones dans les espaces de Hilbert et e\u0301quations d'e\u0301volution\n                      , Lecture Notes 5, North-Holland, Amsterdam, 1972."},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1007\/BF02761171"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1016\/0022-1236(75)90027-0"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-09-04785-0"},{"key":"atypb17","first-page":"33","volume":"17","author":"Cabot A.","year":"2009","journal-title":"Electron. J. Differential Equations"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1007\/s11228-010-0140-1"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1007\/s10957-015-0746-4"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1137\/050626090"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1137\/15M1029485"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1137\/0329022"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1080\/00036811.2014.903569"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-015-0949-3"},{"key":"atypb25","first-page":"1970","author":"Liang J.","year":"2014","journal-title":"Advances in Neural Information Processing Systems"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1007\/s10851-014-0523-2"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.3906\/mat-1512-28"},{"key":"atypb28","first-page":"372","volume":"27","author":"Nesterov Y.","year":"1983","journal-title":"Soviet Math. Dokl."},{"key":"atypb29","doi-asserted-by":"crossref","unstructured":"Y. Nesterov,\n                      Introductory Lectures on Convex Optimization: A Basic Course\n                      , Appl. Optim. 87, Kluwer, Boston, MA, 2004.","DOI":"10.1007\/978-1-4419-8853-9"},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1007\/s10208-013-9150-3"},{"key":"atypb31","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1967-11761-0"},{"key":"atypb32","first-page":"123","volume":"1","author":"Parikh N.","year":"2013","journal-title":"Found. Trends Optim."},{"key":"atypb33","doi-asserted-by":"publisher","DOI":"10.1016\/0041-5553(64)90137-5"},{"key":"atypb34","unstructured":"B. T. Polyak,\n                      Introduction to Optimization\n                      , Optimization Software, New York, 1987."},{"key":"atypb35","first-page":"1","volume":"17","author":"Su W.","year":"2016","journal-title":"J. Mach. Learn. 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