A stochastic Lagrangian-based method for nonconvex optimization with nonlinear constraints

A stochastic Lagrangian-based method for nonconvex optimization with nonlinear constraints | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article A stochastic Lagrangian-based method for nonconvex optimization with nonlinear constraints Dimitri Papadimitriou, Bang Vu This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3302999/v2 This work is licensed under a CC BY 4.0 License Status: Posted Version 2 posted You are reading this latest preprint version Show more versions Abstract The Augmented Lagrangian Method (ALM) is one of the most common approaches for solving linear and nonlinear constrained problems. However, for non-convex objectives, handling non-linear inequality constraints remains challenging. In this paper, we propose a stochastic ALM with Backtracking Line Search that performs on a subset (mini-batch) of randomly selected points for the solving of nonconvex problems. The considered class of problems include both nonlinear equality and inequality constraints. Together with the formal proof of the convergence properties (in expectation) of the proposed algorithm and its computational complexity, the performance of the proposed algorithm are then numerically compared against both exact and inexact state-of-the-art ALM methods. Further, we apply the proposed stochastic ALM method to solve a multi-constrained network design problem. We perform extensive numerical executions on a set of instances extracted from SNDlib to study its behavior and performance, as well as potential improvement of this method. Then analysis and comparison of the results against those obtained by extending the method developed in [Contardo2021] to nonlinear constraints are provided for the approximation of separable nonconvex optimization programs. Mathematics Subject Classification (2020) 65K05 · 68Q25 · 90C46 · 90C30 · 90C25 Nonlinear optimization Constrained optimization Augmented Lagrangian Nonconvex Convex relaxation Network design Full Text Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 2 posted You are reading this latest preprint version Show more versions Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-3302999","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":230242582,"identity":"4a520f68-ea1b-458b-9cb3-7800e10cf407","order_by":0,"name":"Dimitri Papadimitriou","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA/UlEQVRIiWNgGAWjYDCCAyCCTYKBDcK2sWNgB7OJ0cIGZqclMzATpwWKGRgOMzYQ0sJ3u/nZgx9lFtF88r0PD3xsO89szsz87AFDxT27BhxaJO8cMzfsOSeR28bGbnBwZtttPstmNnMDhjPFybi0GNxIMJPgbQNpYWM4zNt2m9ngMIOZBGNbQjIuhxncSP8m+Reh5RzjhsPs3whoyTGTRrLlAFALD9gWO1xaJO+cKTeWAfsljeHgjHPJyQaHecoNEs4kJODSwne7fdvDN2V1ufObjzF/+FBmZ2dwvH3bgw8VCfa4tDBIYI0FoBWJDaRpAQLctoyCUTAKRsFIAwAJ6Vdh5ZazKwAAAABJRU5ErkJggg==","orcid":"","institution":"MCO Institute \u0026 Network Optimization Lab., Belgium Research Center (BeRC), Belgium","correspondingAuthor":true,"prefix":"","firstName":"Dimitri","middleName":"","lastName":"Papadimitriou","suffix":""},{"id":230242583,"identity":"d9043ba1-cfdb-45c5-bae3-c0d2d000cce1","order_by":1,"name":"Bang Vu","email":"","orcid":"","institution":"Network Optimization Lab., Belgium Research Center (BeRC)","correspondingAuthor":false,"prefix":"","firstName":"Bang","middleName":"","lastName":"Vu","suffix":""}],"badges":[],"createdAt":"2023-08-28 10:44:23","currentVersionCode":2,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-3302999/v2","doiUrl":"https://doi.org/10.21203/rs.3.rs-3302999/v2","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":78550965,"identity":"200abd7e-e053-46bf-949b-7c304f4a700e","added_by":"auto","created_at":"2025-03-14 19:34:03","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":739242,"visible":true,"origin":"","legend":"","description":"","filename":"jamcver01.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3302999/v2_covered_949461a6-d187-4d35-8797-95ebd87a0bff.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eA stochastic Lagrangian-based method for nonconvex optimization with nonlinear constraints\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Nonlinear optimization, Constrained optimization, Augmented Lagrangian, Nonconvex, Convex relaxation, Network design","lastPublishedDoi":"10.21203/rs.3.rs-3302999/v2","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3302999/v2","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThe Augmented Lagrangian Method (ALM) is one of the most common approaches for solving linear and nonlinear constrained problems. 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