[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-120587-en":3,"doc-seo-120587-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},120587,1099514068365,"Aurelia","https://ap-avatar.wpscdn.com/avatar/10000253d8d9f28188e?_k=1776742907772140068",8,"Research & Report","Parallel and distributed Machine Learning on Augmented Lagrangian Algorithms - Research","Constrained optimization drives large-scale machine learning, especially when data and computation are split across multiple processors or agents. This paper studies augmented Lagrangian–based algorithms for parallel and distributed settings, covering Lagrangian relaxation, the method of multipliers, ADMM, Bertsekas’ algorithm, Tatjewski’s method, and SALA. A unified theoretical framework analyzes convergence and decomposition strategies, while experiments on regularized linear systems and K-means clustering compare speed, quality, scalability, and computational efficiency under different numerical conditions.","INTL JOURNAL OF ELECTRONICS AND TELECOMMUNICATIONS, 2025, VOL. 71, NO. 4, PP. 1–10  \nManuscript received September 4, 2025; revised October 2025 . doi: 10.24425/ijet.2025.155472  \nParallel and distributed Machine Learning on Augmented Lagrangian Algorithms  \nAnthony Nwachukwu, and Andrzej Karbowski  \nAbstract—Constrained optimization is central to large-scale machine learning, particularly in parallel and distributed environments. This paper presents a comprehensive study of augmented Lagrangian–based algorithms for such problems, including classical Lagrangian relaxation, the method of multipliers, the Alternating Direction Method of Multipliers (ADMM), Bertsekas’algorithm, Tatjewski’s method, and the Separable Augmented Lagrangian Algorithm (SALA). We develop a unified theoretical framework, analyze convergence properties and decomposition strategies, and evaluate these methods on two representative classes of tasks: regularized linear systems and K-means clustering. Numerical experiments on synthetic and real-world datasets show that Bertsekas’ method consistently achieves the best balance of convergence speed and solution quality, while ADMM offers practical scalability under decomposition but struggles in high-dimensional or ill-conditioned settings. Tatjewski’s method benefits significantly from partitioning, whereas the classical Augmented Lagrangian approach proves computationally inefficient for large-scale problems. These findings clarify the trade-offs among augmented Lagrangian algorithms, highlighting Bertsekas’ method as the most effective for distributed optimization and providing guidance for algorithm selection in large-scale machine learning applications.  \nKeywords—Augmented Lagrangian, Optimization, Machine Learning, Alternating Direction Method of Multipliers, Parallel Computing, Convex and Non-convex Optimization, ADMM, Distributed Computing, Clustering, Support Vector Machine, Regression  \nI. INTRODUCTION  \nMODERN machine learning (ML) tasks frequently in  \nvolve optimizing high-dimensional models under explicit constraints, such as parameter bounds, resource limitations, or fairness criteria. Examples include risk minimization with regularization, network flow, and structured prediction. To meet the demands of scale and privacy, data and computations are often distributed across multiple processors or agents, requiring parallel algorithms for constrained optimization. In such distributed settings, classical single-machine solvers are typically inadequate due to communication bottlenecks.  \nAugmented Lagrangian techniques address these challenges by combining Lagrange multipliers with quadratic penalties. This formulation mitigates duality gaps while preserving separability, making it well-suited for parallel implementations.  \nA. Nwachukwu and A. Karbowski are with Faculty of Electronics and Information Technology, Warsaw University of Technology, Warsaw, Poland (e-mail: [anthonychukwuemeka.nwachukwu@gmail.com](anthonychukwuemeka.nwachukwu@gmail.com), an[drzej.karbowski@pw.edu.pl](drzej.karbowski@pw.edu.pl)).  \nA prominent example is the Alternating Direction Method of Multipliers (ADMM), which integrates dual decomposition with augmented penalties to enable independent updates across machines, followed by consistency enforcement. ADMM has become popular in distributed ML for its simplicity and effectiveness.  \nBeyond ADMM, other algorithms extend this approach for distributed optimization. Ordinary Lagrangian relaxation (dual decomposition) suffers from duality gaps and slow convergence. The classic method of multipliers improves convergence but requires coupled updates, limiting scalability. Bertsekas’ algorithm introduces damped multiplier updates to improve separability and convergence speed. Tatjewski’s method similarly refines decomposition with scaled multipliers. The Separable Augmented Lagrangian Algorithm (SALA) further exploits primal reformulation and resource-based splitting for parallel ef","cbCaikCUXCpxliBt","https://ap.wps.com/l/cbCaikCUXCpxliBt","pdf",1085386,1,10,"English","en",105,"# Introduction\n## Constrained optimization in large-scale ML\n## Augmented Lagrangian techniques and parallel separability\n# Review of Augmented Lagrangian–Based Algorithms\n## Classical Lagrangian formulation","[{\"question\":\"Which augmented Lagrangian algorithms are compared in the paper?\",\"answer\":\"The paper compares ordinary Lagrangian relaxation, the classical augmented Lagrangian (multiplier) method, ADMM, Bertsekas’ algorithm, Tatjewski’s method, and the SALA scheme.\"},{\"question\":\"What types of tasks are used to evaluate the methods?\",\"answer\":\"Evaluation focuses on regularized linear systems and K-means clustering, using numerical experiments on synthetic and real-world datasets.\"},{\"question\":\"How do the methods differ in practical performance and scalability?\",\"answer\":\"Results indicate Bertsekas’ method achieves the best convergence speed/solution-quality balance, ADMM scales well under decomposition but can struggle in high-dimensional or ill-conditioned cases, and partitioning helps Tatjewski’s method significantly.\"}]","Parallel and distributed Machine Learning on Augmented Lagrangian Algorithms - Research | PDF",1785730776,25,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"parallel-and-distributed-machine-learning-on-augmented-lagrangian-algorithms-research","",{"@graph":36,"@context":85},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/parallel-and-distributed-machine-learning-on-augmented-lagrangian-algorithms-research/120587/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-03",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"Which augmented Lagrangian algorithms are compared in the paper?","Question",{"text":75,"@type":76},"The paper compares ordinary Lagrangian relaxation, the classical augmented Lagrangian (multiplier) method, ADMM, Bertsekas’ algorithm, Tatjewski’s method, and the SALA scheme.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What types of tasks are used to evaluate the methods?",{"text":80,"@type":76},"Evaluation focuses on regularized linear systems and K-means clustering, using numerical experiments on synthetic and real-world datasets.",{"name":82,"@type":73,"acceptedAnswer":83},"How do the methods differ in practical performance and scalability?",{"text":84,"@type":76},"Results indicate Bertsekas’ method achieves the best convergence speed/solution-quality balance, ADMM scales well under decomposition but can struggle in high-dimensional or ill-conditioned cases, and partitioning helps Tatjewski’s method significantly.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,134],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":21,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":21,"slug":133},"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]