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Program."],"published-print":{"date-parts":[[2023,6]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    This work aims to minimize a continuously differentiable convex function with Lipschitz continuous gradient under linear equality constraints. The proposed inertial algorithm results from the discretization of the second-order primal-dual dynamical system with asymptotically vanishing damping term addressed by Bo\u0163 and Nguyen (J. Differential Equations 303:369\u2013406, 2021), and it is formulated in terms of the Augmented Lagrangian associated with the minimization problem. The general setting we consider for the inertial parameters covers the three classical rules by Nesterov, Chambolle\u2013Dossal and Attouch\u2013Cabot used in the literature to formulate fast gradient methods. For these rules, we obtain in the convex regime convergence rates of order\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$${\\mathcal {O}}\\left( 1\/k^{2} \\right) $$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>O<\/mml:mi>\n                            <mml:mfenced>\n                              <mml:mn>1<\/mml:mn>\n                              <mml:mo>\/<\/mml:mo>\n                              <mml:msup>\n                                <mml:mi>k<\/mml:mi>\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:msup>\n                            <\/mml:mfenced>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    for the primal-dual gap, the feasibility measure, and the objective function value. In addition, we prove that the generated sequence of primal-dual iterates converges to a primal-dual solution in a general setting that covers the two latter rules. This is the first result which provides the convergence of the sequence of iterates generated by a fast algorithm for linearly constrained convex optimization problems without additional assumptions such as strong convexity. We also emphasize that all convergence results of this paper are compatible with the ones obtained in Bo\u0163 and Nguyen (J. Differential Equations 303:369\u2013406, 2021) in the continuous setting.\n                  <\/jats:p>","DOI":"10.1007\/s10107-022-01879-4","type":"journal-article","created":{"date-parts":[[2022,8,16]],"date-time":"2022-08-16T10:05:04Z","timestamp":1660644304000},"page":"147-197","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":34,"title":["Fast Augmented Lagrangian Method in the convex regime with convergence guarantees for the iterates"],"prefix":"10.1007","volume":"200","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4469-314X","authenticated-orcid":false,"given":"Radu Ioan","family":"Bo\u0163","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ern\u00f6 Robert","family":"Csetnek","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dang-Khoa","family":"Nguyen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2022,8,16]]},"reference":[{"issue":"1","key":"1879_CR1","doi-asserted-by":"publisher","first-page":"849","DOI":"10.1137\/17M1114739","volume":"28","author":"H Attouch","year":"2018","unstructured":"Attouch, H., Cabot, A.: Convergence rates of inertial forward-backward algorithms. 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