{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/a-new-primal-dual-algorithm-for-minimizing","title":"A new primal-dual algorithm for minimizing the sum of three functions with a linear operator","arxiv_id":"1611.09805","date":"2016-11-29","proceeding":null,"authors":["Ming Yan"],"abstract":"In this paper, we propose a new primal-dual algorithm for minimizing $f(x) +\ng(x) + h(Ax)$, where $f$, $g$, and $h$ are proper lower semi-continuous convex\nfunctions, $f$ is differentiable with a Lipschitz continuous gradient, and $A$\nis a bounded linear operator. The proposed algorithm has some famous\nprimal-dual algorithms for minimizing the sum of two functions as special\ncases. E.g., it reduces to the Chambolle-Pock algorithm when $f = 0$ and the\nproximal alternating predictor-corrector when $g = 0$. For the general convex\ncase, we prove the convergence of this new algorithm in terms of the distance\nto a fixed point by showing that the iteration is a nonexpansive operator. In\naddition, we prove the $O(1/k)$ ergodic convergence rate in the primal-dual\ngap. With additional assumptions, we derive the linear convergence rate in\nterms of the distance to the fixed point. Comparing to other primal-dual\nalgorithms for solving the same problem, this algorithm extends the range of\nacceptable parameters to ensure its convergence and has a smaller per-iteration\ncost. The numerical experiments show the efficiency of this algorithm.","url_abs":"http://arxiv.org/abs/1611.09805v4","url_pdf":"http://arxiv.org/pdf/1611.09805v4.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"a-new-primal-dual-algorithm-for-minimizing","repo_url":"https://github.com/mingyan08/PD3O","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":{"status":"ok"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1611.09805","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}