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Our study is based on the non-autonomous version of the Polyak heavy ball method, which, at time t<\/jats:italic>, is associated with the strongly convex function obtained by adding to f<\/jats:italic> a Tikhonov regularization term with vanishing coefficient $$\\varepsilon (t)$$<\/jats:tex-math>\n \n \u03b5<\/mml:mi>\n (<\/mml:mo>\n t<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. In this dynamic, the damping coefficient is proportional to the square root of the Tikhonov regularization parameter $$\\varepsilon (t)$$<\/jats:tex-math>\n \n \u03b5<\/mml:mi>\n (<\/mml:mo>\n t<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. By adjusting the speed of convergence of $$\\varepsilon (t)$$<\/jats:tex-math>\n \n \u03b5<\/mml:mi>\n (<\/mml:mo>\n t<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> towards zero, we will obtain both rapid convergence towards the infimal value of f<\/jats:italic>, and the strong convergence of the trajectories towards the element of minimum norm of the set of minimizers of f<\/jats:italic>. In particular, we obtain an improved version of the dynamic of Su-Boyd-Cand\u00e8s for the accelerated gradient method of Nesterov. This study naturally leads to corresponding first-order algorithms obtained by temporal discretization. In the case of a proper lower semicontinuous and convex function f<\/jats:italic>, we study the proximal algorithms in detail, and show that they benefit from similar properties.\n<\/jats:p>","DOI":"10.1007\/s00186-024-00867-y","type":"journal-article","created":{"date-parts":[[2024,6,27]],"date-time":"2024-06-27T18:15:14Z","timestamp":1719512114000},"page":"307-347","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["Convex optimization via inertial algorithms with vanishing Tikhonov regularization: fast convergence to the minimum norm solution"],"prefix":"10.1007","volume":"99","author":[{"given":"Hedy","family":"Attouch","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5140-1144","authenticated-orcid":false,"given":"Szil\u00e1rd Csaba","family":"L\u00e1szl\u00f3","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2024,6,27]]},"reference":[{"key":"867_CR1","doi-asserted-by":"publisher","first-page":"539","DOI":"10.1051\/cocv:2001100","volume":"6","author":"F Alvarez","year":"2001","unstructured":"Alvarez F, Attouch H (2001) Convergence and asymptotic stabilization for some damped hyperbolic equations with non-isolated equilibria. 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