Academic Journal
McCormick envelopes in mixed-integer PDE-constrained optimization.
| Τίτλος: | McCormick envelopes in mixed-integer PDE-constrained optimization. |
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| Συγγραφείς: | Leyffer, Sven, Manns, Paul |
| Πηγή: | Mathematical Programming; Jul2026, Vol. 218 Issue 1/2, p245-290, 46p |
| Θεματικοί όροι: | Partial differential equations, Nonlinear programming, Mixed integer linear programming, Mathematical optimization, Optimization algorithms, Branch & bound algorithms |
| Περίληψη: | McCormick envelopes are a standard tool for deriving convex relaxations of optimization problems that involve polynomial terms. Such McCormick relaxations provide lower bounds, for example, in branch-and-bound procedures for mixed-integer nonlinear programs but have not gained much attention in PDE-constrained optimization so far. This lack of attention may be due to the distributed nature of such problems, which on the one hand leads to infinitely many linear constraints (generally state constraints that may be difficult to handle) in addition to the state equation for a pointwise formulation of the McCormick envelopes and renders bound-tightening procedures that successively improve the resulting convex relaxations computationally intractable. We analyze McCormick envelopes for a model problem class that is governed by a semilinear PDE involving a bilinearity and integrality constraints. We approximate the nonlinearity and in turn the McCormick envelopes by averaging the involved terms over the cells of a partition of the computational domain on which the PDE is defined. This yields convex relaxations that underestimate the original problem up to an a priori error estimate that depends on the mesh size of the discretization. These approximate McCormick relaxations can be improved by means of an optimization-based bound-tightening procedure. We show that their minimizers converge to minimizers to a limit problem with a pointwise formulation of the McCormick envelopes when driving the mesh size to zero. We provide a computational example, for which we certify all of our imposed assumptions. The results point to both the potential of the methodology and the gaps in the research that need to be closed. Our methodology provides a framework first for obtaining pointwise underestimators for nonconvexities and second for approximating them with finitely many linear inequalities in an infinite-dimensional setting. [ABSTRACT FROM AUTHOR] |
| Copyright of Mathematical Programming 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.) | |
| Βάση Δεδομένων: | Complementary Index |
| FullText | Links: – Type: other Text: Availability: 0 CustomLinks: – Url: https://dx.doi.org/doi:10.1007/s10107-024-02181-1 Name: EDS - Springer Nature Journals (s7799221) Category: fullText Text: View record at Springer |
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| Items | – Name: Title Label: Title Group: Ti Data: McCormick envelopes in mixed-integer PDE-constrained optimization. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Leyffer%2C+Sven%22">Leyffer, Sven</searchLink><br /><searchLink fieldCode="AR" term="%22Manns%2C+Paul%22">Manns, Paul</searchLink> – Name: TitleSource Label: Source Group: Src Data: Mathematical Programming; Jul2026, Vol. 218 Issue 1/2, p245-290, 46p – Name: Subject Label: Subject Terms Group: Su Data: <searchLink fieldCode="DE" term="%22Partial+differential+equations%22">Partial differential equations</searchLink><br /><searchLink fieldCode="DE" term="%22Nonlinear+programming%22">Nonlinear programming</searchLink><br /><searchLink fieldCode="DE" term="%22Mixed+integer+linear+programming%22">Mixed integer linear programming</searchLink><br /><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="%22Branch+%26+bound+algorithms%22">Branch & bound algorithms</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: McCormick envelopes are a standard tool for deriving convex relaxations of optimization problems that involve polynomial terms. Such McCormick relaxations provide lower bounds, for example, in branch-and-bound procedures for mixed-integer nonlinear programs but have not gained much attention in PDE-constrained optimization so far. This lack of attention may be due to the distributed nature of such problems, which on the one hand leads to infinitely many linear constraints (generally state constraints that may be difficult to handle) in addition to the state equation for a pointwise formulation of the McCormick envelopes and renders bound-tightening procedures that successively improve the resulting convex relaxations computationally intractable. We analyze McCormick envelopes for a model problem class that is governed by a semilinear PDE involving a bilinearity and integrality constraints. We approximate the nonlinearity and in turn the McCormick envelopes by averaging the involved terms over the cells of a partition of the computational domain on which the PDE is defined. This yields convex relaxations that underestimate the original problem up to an a priori error estimate that depends on the mesh size of the discretization. These approximate McCormick relaxations can be improved by means of an optimization-based bound-tightening procedure. We show that their minimizers converge to minimizers to a limit problem with a pointwise formulation of the McCormick envelopes when driving the mesh size to zero. We provide a computational example, for which we certify all of our imposed assumptions. The results point to both the potential of the methodology and the gaps in the research that need to be closed. Our methodology provides a framework first for obtaining pointwise underestimators for nonconvexities and second for approximating them with finitely many linear inequalities in an infinite-dimensional setting. [ABSTRACT FROM AUTHOR] – Name: Abstract Label: Group: Ab Data: <i>Copyright of Mathematical Programming 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.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1007/s10107-024-02181-1 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 46 StartPage: 245 Subjects: – SubjectFull: Partial differential equations Type: general – SubjectFull: Nonlinear programming Type: general – SubjectFull: Mixed integer linear programming Type: general – SubjectFull: Mathematical optimization Type: general – SubjectFull: Optimization algorithms Type: general – SubjectFull: Branch & bound algorithms Type: general Titles: – TitleFull: McCormick envelopes in mixed-integer PDE-constrained optimization. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Leyffer, Sven – PersonEntity: Name: NameFull: Manns, Paul IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 07 Text: Jul2026 Type: published Y: 2026 Identifiers: – Type: issn-print Value: 00255610 Numbering: – Type: volume Value: 218 – Type: issue Value: 1/2 Titles: – TitleFull: Mathematical Programming Type: main |
| ResultId | 1 |