Solving polynomial variational inequality problems via Lagrange multiplier expressions and Moment-SOS relaxations: Solving polynomial variational inequality problems...: J. Nie et al.
Saved in:
| Title: | Solving polynomial variational inequality problems via Lagrange multiplier expressions and Moment-SOS relaxations: Solving polynomial variational inequality problems...: J. Nie et al. |
|---|---|
| Authors: | Nie, Jiawang1 (AUTHOR) njw@math.ucsd.edu, Sun, Defeng2 (AUTHOR) defeng.sun@polyu.edu.hk, Tang, Xindong3 (AUTHOR) xdtang@hkbu.edu.hk, Zhang, Min4 (AUTHOR) zhangmin1206@gzhu.edu.cn |
| Source: | Computational Optimization & Applications. Mar2025, Vol. 90 Issue 2, p361-394. 34p. |
| Subjects: | Lagrange problem, Intersection theory, Polynomials, Lagrange multiplier, Algorithms |
| Abstract: | This paper focuses on the development of numerical methods for solving variational inequality problems (VIPs) with involved mappings and feasible sets characterized by polynomial functions. We propose a numerical algorithm for computing solutions to polynomial VIPs based on Lagrange multiplier expressions and the Moment-SOS hierarchy of semidefinite relaxations. Building upon this algorithm, we also extend to finding more or even all solutios to polynomial VIPs. This algorithm can find solutions to polynomial VIPs or determine their nonexistence within a finite number of steps, under some general assumptions. Moreover, it is demonstrated that if the VIP is represented by generic polynomial functions, a finite number of Karush–Kuhn–Tucker (KKT) points exist, and all solutions to the polynomial VIP are KKT points. The paper establishes that in such cases, the method is guaranteed to terminate within a finite number of iterations, with an upper bound for the number of KKT points determined using intersection theory. Finally, even when algorithms lack finite convergence, the paper demonstrates asymptotic convergence under specific continuity assumptions. Numerical experiments are conducted to illustrate the efficiency of the proposed methods. [ABSTRACT FROM AUTHOR] |
| Copyright of Computational Optimization & Applications is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
|
Full text is not displayed to guests.
Login for full access.
|
|
| FullText | Links: – Type: pdflink Text: Availability: 1 |
|---|---|
| Header | DbId: egs DbLabel: Engineering Source An: 183372641 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
| IllustrationInfo | |
| Items | – Name: Title Label: Title Group: Ti Data: Solving polynomial variational inequality problems via Lagrange multiplier expressions and Moment-SOS relaxations: Solving polynomial variational inequality problems...: J. Nie et al. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Nie%2C+Jiawang%22">Nie, Jiawang</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> njw@math.ucsd.edu</i><br /><searchLink fieldCode="AR" term="%22Sun%2C+Defeng%22">Sun, Defeng</searchLink><relatesTo>2</relatesTo> (AUTHOR)<i> defeng.sun@polyu.edu.hk</i><br /><searchLink fieldCode="AR" term="%22Tang%2C+Xindong%22">Tang, Xindong</searchLink><relatesTo>3</relatesTo> (AUTHOR)<i> xdtang@hkbu.edu.hk</i><br /><searchLink fieldCode="AR" term="%22Zhang%2C+Min%22">Zhang, Min</searchLink><relatesTo>4</relatesTo> (AUTHOR)<i> zhangmin1206@gzhu.edu.cn</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Computational+Optimization+%26+Applications%22">Computational Optimization & Applications</searchLink>. Mar2025, Vol. 90 Issue 2, p361-394. 34p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Lagrange+problem%22">Lagrange problem</searchLink><br /><searchLink fieldCode="DE" term="%22Intersection+theory%22">Intersection theory</searchLink><br /><searchLink fieldCode="DE" term="%22Polynomials%22">Polynomials</searchLink><br /><searchLink fieldCode="DE" term="%22Lagrange+multiplier%22">Lagrange multiplier</searchLink><br /><searchLink fieldCode="DE" term="%22Algorithms%22">Algorithms</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: This paper focuses on the development of numerical methods for solving variational inequality problems (VIPs) with involved mappings and feasible sets characterized by polynomial functions. We propose a numerical algorithm for computing solutions to polynomial VIPs based on Lagrange multiplier expressions and the Moment-SOS hierarchy of semidefinite relaxations. Building upon this algorithm, we also extend to finding more or even all solutios to polynomial VIPs. This algorithm can find solutions to polynomial VIPs or determine their nonexistence within a finite number of steps, under some general assumptions. Moreover, it is demonstrated that if the VIP is represented by generic polynomial functions, a finite number of Karush–Kuhn–Tucker (KKT) points exist, and all solutions to the polynomial VIP are KKT points. The paper establishes that in such cases, the method is guaranteed to terminate within a finite number of iterations, with an upper bound for the number of KKT points determined using intersection theory. Finally, even when algorithms lack finite convergence, the paper demonstrates asymptotic convergence under specific continuity assumptions. Numerical experiments are conducted to illustrate the efficiency of the proposed methods. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Computational Optimization & Applications is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
| PLink | https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=egs&AN=183372641 |
| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1007/s10589-024-00635-y Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 34 StartPage: 361 Subjects: – SubjectFull: Lagrange problem Type: general – SubjectFull: Intersection theory Type: general – SubjectFull: Polynomials Type: general – SubjectFull: Lagrange multiplier Type: general – SubjectFull: Algorithms Type: general Titles: – TitleFull: Solving polynomial variational inequality problems via Lagrange multiplier expressions and Moment-SOS relaxations: Solving polynomial variational inequality problems...: J. Nie et al. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Nie, Jiawang – PersonEntity: Name: NameFull: Sun, Defeng – PersonEntity: Name: NameFull: Tang, Xindong – PersonEntity: Name: NameFull: Zhang, Min IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 03 Text: Mar2025 Type: published Y: 2025 Identifiers: – Type: issn-print Value: 09266003 Numbering: – Type: volume Value: 90 – Type: issue Value: 2 Titles: – TitleFull: Computational Optimization & Applications Type: main |
| ResultId | 1 |
