{"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/block-simultaneous-direction-method-of","title":"Block-Simultaneous Direction Method of Multipliers: A proximal primal-dual splitting algorithm for nonconvex problems with multiple constraints","arxiv_id":"1708.09066","date":"2017-08-30","proceeding":null,"authors":["Fred Moolekamp","Peter Melchior"],"abstract":"We introduce a generalization of the linearized Alternating Direction Method\nof Multipliers to optimize a real-valued function $f$ of multiple arguments\nwith potentially multiple constraints $g_\\circ$ on each of them. The function\n$f$ may be nonconvex as long as it is convex in every argument, while the\nconstraints $g_\\circ$ need to be convex but not smooth. If $f$ is smooth, the\nproposed Block-Simultaneous Direction Method of Multipliers (bSDMM) can be\ninterpreted as a proximal analog to inexact coordinate descent methods under\nconstraints. Unlike alternative approaches for joint solvers of\nmultiple-constraint problems, we do not require linear operators $L$ of a\nconstraint function $g(L\\ \\cdot)$ to be invertible or linked between each\nother. bSDMM is well-suited for a range of optimization problems, in particular\nfor data analysis, where $f$ is the likelihood function of a model and $L$\ncould be a transformation matrix describing e.g. finite differences or basis\ntransforms. We apply bSDMM to the Non-negative Matrix Factorization task of a\nhyperspectral unmixing problem and demonstrate convergence and effectiveness of\nmultiple constraints on both matrix factors. The algorithms are implemented in\npython and released as an open-source package.","url_abs":"http://arxiv.org/abs/1708.09066v1","url_pdf":"http://arxiv.org/pdf/1708.09066v1.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":"block-simultaneous-direction-method-of","repo_url":"https://github.com/pmelchior/proxmin","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"block-simultaneous-direction-method-of","repo_url":"https://github.com/fred3m/scarlet","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"MIT"}},{"paper_slug":"block-simultaneous-direction-method-of","repo_url":"https://github.com/gcmshadow/proxmin","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[{"task_slug":"hyperspectral-unmixing","task_name":"Hyperspectral Unmixing"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}