Academic Journal
A Low-Rank Augmented Lagrangian Method for Doubly Nonnegative Relaxations of Mixed-Binary Quadratic Programs.
| Τίτλος: | A Low-Rank Augmented Lagrangian Method for Doubly Nonnegative Relaxations of Mixed-Binary Quadratic Programs. |
|---|---|
| Συγγραφείς: | Hou, Di1 (AUTHOR) dihou@u.nus.edu, Tang, Tianyun2 (AUTHOR) ttang@u.nus.edu, Toh, Kim-Chuan1 (AUTHOR) mattohkc@nus.edu.sg |
| Πηγή: | Operations Research. May/Jun2026, Vol. 74 Issue 3, p1626-1647. 22p. |
| Θεματικοί όροι: | *Mathematical optimization, Optimization algorithms, Matrix decomposition, Semidefinite programming, Convex programming, Quadratic programming |
| Περίληψη: | Low-Rank Approaches to Large-Scale DNN Optimization Doubly nonnegative (DNN) relaxations are a powerful tool for approximating large-scale mixed-binary quadratic programs, but their size—often involving millions of constraints—makes them difficult to solve. We propose RiNNAL, a Riemannian augmented Lagrangian method that exploits the low-rank structure often present in optimal solutions. By applying a low-rank decomposition, RiNNAL reformulates most quadratic constraints into simpler affine ones, reducing problem complexity and mitigating issues caused by violations of Slater's condition. A key innovation is showing that the required metric projection onto a certain algebraic variety, although nonconvex in form, can be solved as a convex optimization problem under mild regularity conditions, enforced through constraint reformulation. RiNNAL is versatile, handling not only DNN relaxations but also general semidefinite programs with polyhedral constraints. Extensive experiments on challenging benchmarks confirm its efficiency and robustness, making RiNNAL a promising solver for large-scale optimization problems. Doubly nonnegative (DNN) programming problems are challenging to solve because of their huge number of Ω(n2) constraints and Ω(n2) variables. In this work, we introduce RiNNAL, a method for solving DNN relaxations of large-scale mixed-binary quadratic programs by leveraging their solutions' possible low-rank property. RiNNAL is a globally convergent Riemannian augmented Lagrangian method (ALM) that penalizes the nonnegativity and complementarity constraints while preserving all other constraints as an algebraic variety. After applying the low-rank decomposition to the ALM subproblem, the resulting feasible region becomes an algebraic variety with favorable geometric properties. Our low-rank decomposition model improves upon the standard Burer-Monteiro (BM) model by equivalently reformulating most quadratic constraints after the BM decomposition into fewer, more manageable affine constraints, which also helps to alleviate Slater's condition violations in the primal DNN problem. Moreover, we make the crucial step to show that the metric projection onto the algebraic variety, although nonconvex, can be transformed into a solvable convex optimization problem under certain regularity conditions, which can be ensured by a constraint reformulation strategy. RiNNAL is able to handle general semidefinite programming (SDP) with additional polyhedral cone constraints, thus serving as a prototype algorithm for solving general DNN problems. Numerous numerical experiments are conducted to validate the efficiency of the proposed RiNNAL method. Funding: This research was supported by the Ministry of Education, Singapore, under its Academic Research Fund Tier 3 grant call [Grant MOE-2019-T3-1-010]. Supplemental Material: All supplemental materials, including the code, data, and files required to reproduce the results, are available at https://doi.org/10.1287/opre.2024.1137. [ABSTRACT FROM AUTHOR] |
| Copyright of Operations Research is the property of INFORMS: Institute for Operations Research & the Management Sciences 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.) | |
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| Items | – Name: Title Label: Title Group: Ti Data: A Low-Rank Augmented Lagrangian Method for Doubly Nonnegative Relaxations of Mixed-Binary Quadratic Programs. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Hou%2C+Di%22">Hou, Di</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> dihou@u.nus.edu</i><br /><searchLink fieldCode="AR" term="%22Tang%2C+Tianyun%22">Tang, Tianyun</searchLink><relatesTo>2</relatesTo> (AUTHOR)<i> ttang@u.nus.edu</i><br /><searchLink fieldCode="AR" term="%22Toh%2C+Kim-Chuan%22">Toh, Kim-Chuan</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> mattohkc@nus.edu.sg</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Operations+Research%22">Operations Research</searchLink>. May/Jun2026, Vol. 74 Issue 3, p1626-1647. 22p. – Name: Subject Label: Subject Terms Group: Su Data: *<searchLink fieldCode="DE" term="%22Mathematical+optimization%22">Mathematical optimization</searchLink><br /><searchLink fieldCode="DE" term="%22Optimization+algorithms%22">Optimization algorithms</searchLink><br /><searchLink fieldCode="DE" term="%22Matrix+decomposition%22">Matrix decomposition</searchLink><br /><searchLink fieldCode="DE" term="%22Semidefinite+programming%22">Semidefinite programming</searchLink><br /><searchLink fieldCode="DE" term="%22Convex+programming%22">Convex programming</searchLink><br /><searchLink fieldCode="DE" term="%22Quadratic+programming%22">Quadratic programming</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: Low-Rank Approaches to Large-Scale DNN Optimization Doubly nonnegative (DNN) relaxations are a powerful tool for approximating large-scale mixed-binary quadratic programs, but their size—often involving millions of constraints—makes them difficult to solve. We propose RiNNAL, a Riemannian augmented Lagrangian method that exploits the low-rank structure often present in optimal solutions. By applying a low-rank decomposition, RiNNAL reformulates most quadratic constraints into simpler affine ones, reducing problem complexity and mitigating issues caused by violations of Slater's condition. A key innovation is showing that the required metric projection onto a certain algebraic variety, although nonconvex in form, can be solved as a convex optimization problem under mild regularity conditions, enforced through constraint reformulation. RiNNAL is versatile, handling not only DNN relaxations but also general semidefinite programs with polyhedral constraints. Extensive experiments on challenging benchmarks confirm its efficiency and robustness, making RiNNAL a promising solver for large-scale optimization problems. Doubly nonnegative (DNN) programming problems are challenging to solve because of their huge number of Ω(n2) constraints and Ω(n2) variables. In this work, we introduce RiNNAL, a method for solving DNN relaxations of large-scale mixed-binary quadratic programs by leveraging their solutions' possible low-rank property. RiNNAL is a globally convergent Riemannian augmented Lagrangian method (ALM) that penalizes the nonnegativity and complementarity constraints while preserving all other constraints as an algebraic variety. After applying the low-rank decomposition to the ALM subproblem, the resulting feasible region becomes an algebraic variety with favorable geometric properties. Our low-rank decomposition model improves upon the standard Burer-Monteiro (BM) model by equivalently reformulating most quadratic constraints after the BM decomposition into fewer, more manageable affine constraints, which also helps to alleviate Slater's condition violations in the primal DNN problem. Moreover, we make the crucial step to show that the metric projection onto the algebraic variety, although nonconvex, can be transformed into a solvable convex optimization problem under certain regularity conditions, which can be ensured by a constraint reformulation strategy. RiNNAL is able to handle general semidefinite programming (SDP) with additional polyhedral cone constraints, thus serving as a prototype algorithm for solving general DNN problems. Numerous numerical experiments are conducted to validate the efficiency of the proposed RiNNAL method. Funding: This research was supported by the Ministry of Education, Singapore, under its Academic Research Fund Tier 3 grant call [Grant MOE-2019-T3-1-010]. Supplemental Material: All supplemental materials, including the code, data, and files required to reproduce the results, are available at https://doi.org/10.1287/opre.2024.1137. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Operations Research is the property of INFORMS: Institute for Operations Research & the Management Sciences 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.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1287/opre.2024.1137 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 22 StartPage: 1626 Subjects: – SubjectFull: Mathematical optimization Type: general – SubjectFull: Optimization algorithms Type: general – SubjectFull: Matrix decomposition Type: general – SubjectFull: Semidefinite programming Type: general – SubjectFull: Convex programming Type: general – SubjectFull: Quadratic programming Type: general Titles: – TitleFull: A Low-Rank Augmented Lagrangian Method for Doubly Nonnegative Relaxations of Mixed-Binary Quadratic Programs. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Hou, Di – PersonEntity: Name: NameFull: Tang, Tianyun – PersonEntity: Name: NameFull: Toh, Kim-Chuan IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 05 Text: May/Jun2026 Type: published Y: 2026 Identifiers: – Type: issn-print Value: 0030364X Numbering: – Type: volume Value: 74 – Type: issue Value: 3 Titles: – TitleFull: Operations Research Type: main |
| ResultId | 1 |