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Then several time discretizations of the continuous model are considered and analyzed via a unified discrete Lyapunov function. Moreover, two families of accelerated primal-dual methods are obtained, and nonergodic optimal mixed-type convergence rates shall be proved for the primal objective residual, the feasibility violation and the Lagrangian gap. Finally, numerical experiments are provided to validate the practical performances of the proposed methods.<\/p>","DOI":"10.1090\/mcom\/4063","type":"journal-article","created":{"date-parts":[[2024,12,18]],"date-time":"2024-12-18T10:19:34Z","timestamp":1734517174000},"page":"3009-3041","source":"Crossref","is-referenced-by-count":5,"title":["A unified differential equation solver approach for separable convex optimization: Splitting, acceleration and nonergodic rate"],"prefix":"10.1090","volume":"94","author":[{"given":"Hao","family":"Luo","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zihang","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2025,1,27]]},"reference":[{"key":"1","doi-asserted-by":"publisher","first-page":"Paper No. 2, 34","DOI":"10.1051\/cocv\/2017083","article-title":"Rate of convergence of the Nesterov accelerated gradient method in the subcritical case \ud835\udefc\u22643","volume":"25","author":"Attouch, Hedy","year":"2019","journal-title":"ESAIM Control Optim. 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