A Many-Objective Optimization Algorithm Integrating Convergence and Diversity Metrics

A Many-Objective Optimization Algorithm Integrating Convergence and Diversity Metrics | 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 Many-Objective Optimization Algorithm Integrating Convergence and Diversity Metrics Yingjie Song, Xinjian Zhou This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8695531/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 13 You are reading this latest preprint version Abstract Currently, many critical fields in science, society, and engineering involve Many-objective Optimization Problems (MaOPs) composed of numerous decision variables. A key challenge with such problems is the difficulty in simultaneously maintaining good diversity and convergence during the search process. To address this challenge, this paper proposes a dual-indicator-based multi-objective optimization framework (TDC-MOEA), which transforms the multi-objective space into a bi-objective space based on convergence and diversity metrics. Firstly, the population is clustered into multiple sub-populations according to reference points, shifting the focus of subsequent operations from individuals to sub-populations, and the convergence and diversity metrics for each sub-population are calculated. Secondly, to further enhance convergence and diversity, a selection process is applied to each sub-population, aiming to improve both the local convergence within sub-populations and the overall diversity. This algorithmic framework incorporates polynomial crossover and binomial mutation as core evolutionary operators, ultimately constructing the TDC-MOEA algorithm. Experimental results demonstrate that the TDC-MOEA algorithm can obtain Pareto solutions with good convergence and a wide distribution, while also acquiring multiple Pareto solution sets for the original multi-objective optimization problem. Comparative results against seven state-of-the-art MaOP algorithms on 39 test instances show that the TDC-MOEA algorithm possesses strong competitiveness and superior overall performance. In a practical application, the algorithm effectively optimized five objectives—power output, ecological function, thermal efficiency, power density, and efficient power—for a simple endoreversible closed Brayton cycle, achieving favorable results. Many-Objective Optimization Problem Multi-Objective Framework Convergence Diversity Endoreversible Closed Brayton Cycle Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 20 Feb, 2026 Reviews received at journal 18 Feb, 2026 Reviewers agreed at journal 18 Feb, 2026 Reviewers agreed at journal 15 Feb, 2026 Reviewers agreed at journal 14 Feb, 2026 Reviewers agreed at journal 12 Feb, 2026 Reviews received at journal 12 Feb, 2026 Reviewers agreed at journal 12 Feb, 2026 Reviewers invited by journal 11 Feb, 2026 Editor invited by journal 03 Feb, 2026 Editor assigned by journal 30 Jan, 2026 Submission checks completed at journal 30 Jan, 2026 First submitted to journal 25 Jan, 2026 You are reading this latest preprint version 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-8695531","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":591999037,"identity":"8d931a9e-0baf-4292-9efe-b5930a9c3e5e","order_by":0,"name":"Yingjie Song","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA1UlEQVRIiWNgGAWjYNACHhsGhgM8IBYz0VrSGBiOkaaF4TAJWuTbew+/5pE5b893v/eYBEOFdWID+9kDeLUYnDmXZs3Dcztx5jG+NAmGM+mJDTx5Cfi1SOSYGQO1JBgc4zGTYGw7nNggwWOA32EzwFrO2UO0/CNCC8ONHOPHPDwHGDeAtTQQocXgzBkzxjk8yUC/5CVbJBxLN27jySHgsPYe4w9ve+zs+Q6fPXjjQ421bD/7GQIOY2Bgk+LtgTITQFxC6oGA+eOPH0QoGwWjYBSMgpELAHaVQnVugJ/yAAAAAElFTkSuQmCC","orcid":"","institution":"Shandong University of Business and Technology","correspondingAuthor":true,"prefix":"","firstName":"Yingjie","middleName":"","lastName":"Song","suffix":""},{"id":591999038,"identity":"7ee32a43-0c7d-43cf-b0d9-b43217e1ca72","order_by":1,"name":"Xinjian Zhou","email":"","orcid":"","institution":"Shandong University of Business and Technology","correspondingAuthor":false,"prefix":"","firstName":"Xinjian","middleName":"","lastName":"Zhou","suffix":""}],"badges":[],"createdAt":"2026-01-26 01:53:23","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8695531/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8695531/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":103049387,"identity":"234fa49c-f504-4cce-9b90-4c7d768ad95e","added_by":"auto","created_at":"2026-02-20 07:40:37","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":507952,"visible":true,"origin":"","legend":"","description":"","filename":"TDCMOEA.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8695531/v1_covered_7f33c245-177c-4b7d-8797-fec1dd001f38.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"A Many-Objective Optimization Algorithm Integrating Convergence and Diversity Metrics","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"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":"discover-applied-sciences","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"","sideBox":"Learn more about [Discover Applied Sciences](https://link.springer.com/journal/42452)","snPcode":"42452","submissionUrl":"https://submission.springernature.com/new-submission/42452/3","title":"Discover Applied Sciences","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Discover Series","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"Many-Objective Optimization Problem, Multi-Objective Framework, Convergence, Diversity, Endoreversible Closed Brayton Cycle","lastPublishedDoi":"10.21203/rs.3.rs-8695531/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8695531/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eCurrently, many critical fields in science, society, and engineering involve Many-objective Optimization Problems (MaOPs) composed of numerous decision variables. A key challenge with such problems is the difficulty in simultaneously maintaining good diversity and convergence during the search process. To address this challenge, this paper proposes a dual-indicator-based multi-objective optimization framework (TDC-MOEA), which transforms the multi-objective space into a bi-objective space based on convergence and diversity metrics. Firstly, the population is clustered into multiple sub-populations according to reference points, shifting the focus of subsequent operations from individuals to sub-populations, and the convergence and diversity metrics for each sub-population are calculated. Secondly, to further enhance convergence and diversity, a selection process is applied to each sub-population, aiming to improve both the local convergence within sub-populations and the overall diversity. This algorithmic framework incorporates polynomial crossover and binomial mutation as core evolutionary operators, ultimately constructing the TDC-MOEA algorithm. Experimental results demonstrate that the TDC-MOEA algorithm can obtain Pareto solutions with good convergence and a wide distribution, while also acquiring multiple Pareto solution sets for the original multi-objective optimization problem. Comparative results against seven state-of-the-art MaOP algorithms on 39 test instances show that the TDC-MOEA algorithm possesses strong competitiveness and superior overall performance. In a practical application, the algorithm effectively optimized five objectives\u0026mdash;power output, ecological function, thermal efficiency, power density, and efficient power\u0026mdash;for a simple endoreversible closed Brayton cycle, achieving favorable results.\u003c/p\u003e","manuscriptTitle":"A Many-Objective Optimization Algorithm Integrating Convergence and Diversity Metrics","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-02-17 06:19:02","doi":"10.21203/rs.3.rs-8695531/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2026-02-20T06:18:23+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-02-18T23:02:48+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"243305069260192605864304397676655523234","date":"2026-02-18T22:49:09+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"246361082840786840883625713380773364387","date":"2026-02-15T06:57:36+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"250013241012426799280797066065860901569","date":"2026-02-15T03:32:49+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"230140451949632999769601959421183597215","date":"2026-02-12T10:42:19+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-02-12T07:17:16+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"118773584233695218557378955181494807656","date":"2026-02-12T05:51:40+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2026-02-12T04:44:40+00:00","index":"","fulltext":""},{"type":"editorInvited","content":"","date":"2026-02-04T04:28:42+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2026-01-30T12:22:27+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2026-01-30T12:19:44+00:00","index":"","fulltext":""},{"type":"submitted","content":"Discover Applied Sciences","date":"2026-01-26T01:51:00+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"discover-applied-sciences","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"","sideBox":"Learn more about [Discover Applied Sciences](https://link.springer.com/journal/42452)","snPcode":"42452","submissionUrl":"https://submission.springernature.com/new-submission/42452/3","title":"Discover Applied Sciences","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Discover Series","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"99db8a96-cdaf-4855-92ef-d1b4733eea5d","owner":[],"postedDate":"February 17th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[],"tags":[],"updatedAt":"2026-03-03T05:53:12+00:00","versionOfRecord":[],"versionCreatedAt":"2026-02-17 06:19:02","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8695531","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8695531","identity":"rs-8695531","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}