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Augmented Lagrangian (AL) methods are a well known class of algorithms for solving constrained optimization problems. They have been extended to the solution of saddle-point systems of linear equations. We study an AL (SPAL) algorithm for unsymmetric saddle-point systems and derive convergence and semi-convergence properties, even when the system is singular. At each step, our SPAL requires the exact solution of a linear system of the same size but with an SPD (2,2) block. To improve efficiency, we introduce an inexact SPAL algorithm. We establish its convergence properties under reasonable assumptions. Specifically, we use a gradient method, known as the Barzilai-Borwein (BB) method, to solve the linear system at each iteration. We call the result the augmented Lagrangian BB (SPALBB) algorithm and study its convergence. Numerical experiments on test problems from Navier-Stokes equations and coupled Stokes-Darcy flow show that SPALBB is more robust and efficient than BICGSTAB and GMRES. SPALBB often requires the least CPU time, especially on large systems.

Submission history

From: Dominique Orban [view email]
[v1] Tue, 23 Apr 2024 00:24:38 UTC (12,704 KB)

||||I|||| Skip to main content We gratefully acknowledge support from the Simons Foundation, member institutions , and all contributors. Donate > math > arXiv:2404.14636 Help | Advanced Search All fields Title Author Abstract Comments Journal reference ACM classification MSC classification Report number arXiv identifier DOI ORCID arXiv author ID Help pages Full text Search GO quick links * Login * Help Pages * About Mathematics > Numerical Analysis arXiv:2404.14636 (math) [Submitted on 23 Apr 2024] Title: An inexact augmented Lagrangian algorithm for unsymmetric saddle-point systems Authors: N. Huang, Y.-H. Dai, D. Orban, M. A. Saunders View a PDF of the paper titled An inexact augmented Lagrangian algorithm for unsymmetric saddle-point systems, by N. Huang and 3 other authors View PDF Abstract: Augmented Lagrangian (AL) methods are a well known class of algorithms for solving constrained optimization problems. They have been extended to the solution of saddle-point systems of linear equations. We study an AL (SPAL) algorithm for unsymmetric saddle-point systems and derive convergence and semi-convergence properties, even when the system is singular. At each step, our SPAL requires the exact solution of a linear system of the same size but with an SPD (2,2) block. To improve efficiency, we introduce an inexact SPAL algorithm. We establish its convergence properties under reasonable assumptions. Specifically, we use a gradient method, known as the Barzilai-Borwein (BB) method, to solve the linear system at each iteration. We call the result the augmented Lagrangian BB (SPALBB) algorithm and study its convergence. Numerical experiments on test problems from Navier-Stokes equations and coupled Stokes-Darcy flow show that SPALBB is more robust and efficient than BICGSTAB and GMRES. SPALBB often requires the least CPU time, especially on large systems. Subjects: Numerical Analysis (math.NA) ; Data Structures and Algorithms (cs.DS) MSC classes: 65F10, 65F50 Report number: G-2024-30 Cite as: arXiv:2404.14636 [math.NA] (or arXiv:2404.14636v1 [math.NA] for this version) https://doi.org/10.13140/RG.2.2.17308.09602 Related DOI: Focus to learn more DOI(s) linking to related resources Submission history From: Dominique Orban [view email] [v1] Tue, 23 Apr 2024 00:24:38 UTC (12,704 KB) Full-text links: Access Paper: View a PDF of the paper titled An inexact augmented Lagrangian algorithm for unsymmetric saddle-point systems, by N. Huang and 3 other authors * View PDF * TeX Source * Other Formats view license Current browse context: math.NA < prev | next > new | recent | 2404 Change to browse by: cs cs.DS cs.NA math References & Citations * NASA ADS * Google Scholar * Semantic Scholar a export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Code, Data and Media Associated with this Article Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub Toggle DagsHub (What is DagsHub?) GotitPub Toggle Gotit.pub (What is GotitPub?) Links to Code Toggle Papers with Code (What is Papers with Code?) ScienceCast Toggle ScienceCast (What is ScienceCast?) Demos Demos Replicate Toggle Replicate (What is Replicate?) Spaces Toggle Hugging Face Spaces (What is Spaces?) Spaces Toggle TXYZ.AI (What is TXYZ.AI?) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower (What are Influence Flowers?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Core recommender toggle CORE Recommender (What is CORE?) About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs. Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?) * About * Help * Click here to contact arXiv Contact * Click here to subscribe Subscribe * Copyright * Privacy Policy * Web Accessibility Assistance * arXiv Operational Status Get status notifications via email or slack

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