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The proposed algorithm can be interpreted as an inexact inertial forward\u2013backward algorithm equipped with a prefixed number of inner primal\u2013dual iterations for the proximal evaluation and a \u201cwarm\u2013start\u201d strategy for starting the inner loop, and generalizes several nested primal\u2013dual algorithms already available in the literature. By appropriately choosing the inertial parameters, we prove the convergence of the iterates to a saddle point of the problem, and provide an<jats:italic>O<\/jats:italic>(1\/<jats:italic>n<\/jats:italic>) convergence rate on the primal\u2013dual gap evaluated at the corresponding ergodic sequences. Numerical experiments on some image restoration problems show that the combination of the \u201cwarm\u2013start\u201d strategy with an appropriate choice of the inertial parameters is strictly required in order to guarantee the convergence to the real minimum point of the objective function.<\/jats:p>","DOI":"10.1007\/s10589-022-00410-x","type":"journal-article","created":{"date-parts":[[2022,8,26]],"date-time":"2022-08-26T17:27:50Z","timestamp":1661534870000},"page":"85-123","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["A nested primal\u2013dual FISTA-like scheme for composite convex optimization problems"],"prefix":"10.1007","volume":"84","author":[{"given":"S.","family":"Bonettini","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"M.","family":"Prato","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1232-5481","authenticated-orcid":false,"given":"S.","family":"Rebegoldi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2022,8,26]]},"reference":[{"issue":"6","key":"410_CR1","doi-asserted-by":"publisher","first-page":"1217","DOI":"10.1088\/0266-5611\/10\/6\/003","volume":"10","author":"R Acar","year":"1994","unstructured":"Acar, R., Vogel, C.R.: Analysis of bounded variation penalty methods for ill-posed problems. 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