Globally optimal univariate spline approximations Article Swipe
Robert Mohr
,
Maximilian Coblenz
,
Peter Kirst
·
YOU?
·
· 2023
· Open Access
·
· DOI: https://doi.org/10.1007/s10589-023-00462-7
YOU?
·
· 2023
· Open Access
·
· DOI: https://doi.org/10.1007/s10589-023-00462-7
We revisit the problem of computing optimal spline approximations for univariate least-squares splines from a combinatorial optimization perspective. In contrast to most approaches from the literature we aim at globally optimal coefficients as well as a globally optimal placement of a fixed number of knots for a discrete variant of this problem. To achieve this, two different possibilities are developed. The first approach that we present is the formulation of the problem as a mixed-integer quadratically constrained problem, which can be solved using commercial optimization solvers. The second method that we propose is a branch-and-bound algorithm tailored specifically to the combinatorial formulation. We compare our algorithmic approaches empirically on both, real and synthetic curve fitting data sets from the literature. The numerical experiments show that our approach to tackle the least-squares spline approximation problem with free knots is able to compute solutions to problems of realistic sizes within reasonable computing times.
Related Topics
Concepts
Mathematics
Univariate
Mathematical optimization
Spline (mechanical)
Optimization problem
Integer (computer science)
Quadratic growth
Least-squares function approximation
Applied mathematics
Algorithm
Computer science
Multivariate statistics
Statistics
Structural engineering
Engineering
Programming language
Estimator
Metadata
- Type
- article
- Language
- en
- Landing Page
- https://doi.org/10.1007/s10589-023-00462-7
- https://link.springer.com/content/pdf/10.1007/s10589-023-00462-7.pdf
- OA Status
- hybrid
- Cited By
- 1
- References
- 28
- Related Works
- 10
- OpenAlex ID
- https://openalex.org/W4322624663
All OpenAlex metadata
Raw OpenAlex JSON
- OpenAlex ID
-
https://openalex.org/W4322624663Canonical identifier for this work in OpenAlex
- DOI
-
https://doi.org/10.1007/s10589-023-00462-7Digital Object Identifier
- Title
-
Globally optimal univariate spline approximationsWork title
- Type
-
articleOpenAlex work type
- Language
-
enPrimary language
- Publication year
-
2023Year of publication
- Publication date
-
2023-02-28Full publication date if available
- Authors
-
Robert Mohr, Maximilian Coblenz, Peter KirstList of authors in order
- Landing page
-
https://doi.org/10.1007/s10589-023-00462-7Publisher landing page
- PDF URL
-
https://link.springer.com/content/pdf/10.1007/s10589-023-00462-7.pdfDirect link to full text PDF
- Open access
-
YesWhether a free full text is available
- OA status
-
hybridOpen access status per OpenAlex
- OA URL
-
https://link.springer.com/content/pdf/10.1007/s10589-023-00462-7.pdfDirect OA link when available
- Concepts
-
Mathematics, Univariate, Mathematical optimization, Spline (mechanical), Optimization problem, Integer (computer science), Quadratic growth, Least-squares function approximation, Applied mathematics, Algorithm, Computer science, Multivariate statistics, Statistics, Structural engineering, Engineering, Programming language, EstimatorTop concepts (fields/topics) attached by OpenAlex
- Cited by
-
1Total citation count in OpenAlex
- Citations by year (recent)
-
2025: 1Per-year citation counts (last 5 years)
- References (count)
-
28Number of works referenced by this work
- Related works (count)
-
10Other works algorithmically related by OpenAlex
Full payload
| id | https://openalex.org/W4322624663 |
|---|---|
| doi | https://doi.org/10.1007/s10589-023-00462-7 |
| ids.doi | https://doi.org/10.1007/s10589-023-00462-7 |
| ids.openalex | https://openalex.org/W4322624663 |
| fwci | 0.33600649 |
| type | article |
| title | Globally optimal univariate spline approximations |
| biblio.issue | 2 |
| biblio.volume | 85 |
| biblio.last_page | 439 |
| biblio.first_page | 409 |
| topics[0].id | https://openalex.org/T11245 |
| topics[0].field.id | https://openalex.org/fields/22 |
| topics[0].field.display_name | Engineering |
| topics[0].score | 0.9984999895095825 |
| topics[0].domain.id | https://openalex.org/domains/3 |
| topics[0].domain.display_name | Physical Sciences |
| topics[0].subfield.id | https://openalex.org/subfields/2206 |
| topics[0].subfield.display_name | Computational Mechanics |
| topics[0].display_name | Advanced Numerical Analysis Techniques |
| is_xpac | False |
| apc_list.value | 2390 |
| apc_list.currency | EUR |
| apc_list.value_usd | 2990 |
| apc_paid.value | 2390 |
| apc_paid.currency | EUR |
| apc_paid.value_usd | 2990 |
| concepts[0].id | https://openalex.org/C33923547 |
| concepts[0].level | 0 |
| concepts[0].score | 0.7807403802871704 |
| concepts[0].wikidata | https://www.wikidata.org/wiki/Q395 |
| concepts[0].display_name | Mathematics |
| concepts[1].id | https://openalex.org/C199163554 |
| concepts[1].level | 3 |
| concepts[1].score | 0.7096355557441711 |
| concepts[1].wikidata | https://www.wikidata.org/wiki/Q1681619 |
| concepts[1].display_name | Univariate |
| concepts[2].id | https://openalex.org/C126255220 |
| concepts[2].level | 1 |
| concepts[2].score | 0.6220459342002869 |
| concepts[2].wikidata | https://www.wikidata.org/wiki/Q141495 |
| concepts[2].display_name | Mathematical optimization |
| concepts[3].id | https://openalex.org/C10390562 |
| concepts[3].level | 2 |
| concepts[3].score | 0.6051914691925049 |
| concepts[3].wikidata | https://www.wikidata.org/wiki/Q581809 |
| concepts[3].display_name | Spline (mechanical) |
| concepts[4].id | https://openalex.org/C137836250 |
| concepts[4].level | 2 |
| concepts[4].score | 0.4480462968349457 |
| concepts[4].wikidata | https://www.wikidata.org/wiki/Q984063 |
| concepts[4].display_name | Optimization problem |
| concepts[5].id | https://openalex.org/C97137487 |
| concepts[5].level | 2 |
| concepts[5].score | 0.4273446798324585 |
| concepts[5].wikidata | https://www.wikidata.org/wiki/Q729138 |
| concepts[5].display_name | Integer (computer science) |
| concepts[6].id | https://openalex.org/C195956108 |
| concepts[6].level | 2 |
| concepts[6].score | 0.4184727072715759 |
| concepts[6].wikidata | https://www.wikidata.org/wiki/Q7268362 |
| concepts[6].display_name | Quadratic growth |
| concepts[7].id | https://openalex.org/C9936470 |
| concepts[7].level | 3 |
| concepts[7].score | 0.4126480221748352 |
| concepts[7].wikidata | https://www.wikidata.org/wiki/Q6510405 |
| concepts[7].display_name | Least-squares function approximation |
| concepts[8].id | https://openalex.org/C28826006 |
| concepts[8].level | 1 |
| concepts[8].score | 0.3937041461467743 |
| concepts[8].wikidata | https://www.wikidata.org/wiki/Q33521 |
| concepts[8].display_name | Applied mathematics |
| concepts[9].id | https://openalex.org/C11413529 |
| concepts[9].level | 1 |
| concepts[9].score | 0.29449766874313354 |
| concepts[9].wikidata | https://www.wikidata.org/wiki/Q8366 |
| concepts[9].display_name | Algorithm |
| concepts[10].id | https://openalex.org/C41008148 |
| concepts[10].level | 0 |
| concepts[10].score | 0.17979800701141357 |
| concepts[10].wikidata | https://www.wikidata.org/wiki/Q21198 |
| concepts[10].display_name | Computer science |
| concepts[11].id | https://openalex.org/C161584116 |
| concepts[11].level | 2 |
| concepts[11].score | 0.09169483184814453 |
| concepts[11].wikidata | https://www.wikidata.org/wiki/Q1952580 |
| concepts[11].display_name | Multivariate statistics |
| concepts[12].id | https://openalex.org/C105795698 |
| concepts[12].level | 1 |
| concepts[12].score | 0.0 |
| concepts[12].wikidata | https://www.wikidata.org/wiki/Q12483 |
| concepts[12].display_name | Statistics |
| concepts[13].id | https://openalex.org/C66938386 |
| concepts[13].level | 1 |
| concepts[13].score | 0.0 |
| concepts[13].wikidata | https://www.wikidata.org/wiki/Q633538 |
| concepts[13].display_name | Structural engineering |
| concepts[14].id | https://openalex.org/C127413603 |
| concepts[14].level | 0 |
| concepts[14].score | 0.0 |
| concepts[14].wikidata | https://www.wikidata.org/wiki/Q11023 |
| concepts[14].display_name | Engineering |
| concepts[15].id | https://openalex.org/C199360897 |
| concepts[15].level | 1 |
| concepts[15].score | 0.0 |
| concepts[15].wikidata | https://www.wikidata.org/wiki/Q9143 |
| concepts[15].display_name | Programming language |
| concepts[16].id | https://openalex.org/C185429906 |
| concepts[16].level | 2 |
| concepts[16].score | 0.0 |
| concepts[16].wikidata | https://www.wikidata.org/wiki/Q1130160 |
| concepts[16].display_name | Estimator |
| keywords[0].id | https://openalex.org/keywords/mathematics |
| keywords[0].score | 0.7807403802871704 |
| keywords[0].display_name | Mathematics |
| keywords[1].id | https://openalex.org/keywords/univariate |
| keywords[1].score | 0.7096355557441711 |
| keywords[1].display_name | Univariate |
| keywords[2].id | https://openalex.org/keywords/mathematical-optimization |
| keywords[2].score | 0.6220459342002869 |
| keywords[2].display_name | Mathematical optimization |
| keywords[3].id | https://openalex.org/keywords/spline |
| keywords[3].score | 0.6051914691925049 |
| keywords[3].display_name | Spline (mechanical) |
| keywords[4].id | https://openalex.org/keywords/optimization-problem |
| keywords[4].score | 0.4480462968349457 |
| keywords[4].display_name | Optimization problem |
| keywords[5].id | https://openalex.org/keywords/integer |
| keywords[5].score | 0.4273446798324585 |
| keywords[5].display_name | Integer (computer science) |
| keywords[6].id | https://openalex.org/keywords/quadratic-growth |
| keywords[6].score | 0.4184727072715759 |
| keywords[6].display_name | Quadratic growth |
| keywords[7].id | https://openalex.org/keywords/least-squares-function-approximation |
| keywords[7].score | 0.4126480221748352 |
| keywords[7].display_name | Least-squares function approximation |
| keywords[8].id | https://openalex.org/keywords/applied-mathematics |
| keywords[8].score | 0.3937041461467743 |
| keywords[8].display_name | Applied mathematics |
| keywords[9].id | https://openalex.org/keywords/algorithm |
| keywords[9].score | 0.29449766874313354 |
| keywords[9].display_name | Algorithm |
| keywords[10].id | https://openalex.org/keywords/computer-science |
| keywords[10].score | 0.17979800701141357 |
| keywords[10].display_name | Computer science |
| keywords[11].id | https://openalex.org/keywords/multivariate-statistics |
| keywords[11].score | 0.09169483184814453 |
| keywords[11].display_name | Multivariate statistics |
| language | en |
| locations[0].id | doi:10.1007/s10589-023-00462-7 |
| locations[0].is_oa | True |
| locations[0].source.id | https://openalex.org/S39243548 |
| locations[0].source.issn | 0926-6003, 1573-2894 |
| locations[0].source.type | journal |
| locations[0].source.is_oa | False |
| locations[0].source.issn_l | 0926-6003 |
| locations[0].source.is_core | True |
| locations[0].source.is_in_doaj | False |
| locations[0].source.display_name | Computational Optimization and Applications |
| locations[0].source.host_organization | https://openalex.org/P4310319900 |
| locations[0].source.host_organization_name | Springer Science+Business Media |
| locations[0].source.host_organization_lineage | https://openalex.org/P4310319900, https://openalex.org/P4310319965 |
| locations[0].source.host_organization_lineage_names | Springer Science+Business Media, Springer Nature |
| locations[0].license | cc-by |
| locations[0].pdf_url | https://link.springer.com/content/pdf/10.1007/s10589-023-00462-7.pdf |
| locations[0].version | publishedVersion |
| locations[0].raw_type | journal-article |
| locations[0].license_id | https://openalex.org/licenses/cc-by |
| locations[0].is_accepted | True |
| locations[0].is_published | True |
| locations[0].raw_source_name | Computational Optimization and Applications |
| locations[0].landing_page_url | https://doi.org/10.1007/s10589-023-00462-7 |
| locations[1].id | pmh:oai:econstor.eu:10419/307569 |
| locations[1].is_oa | True |
| locations[1].source.id | https://openalex.org/S4306401696 |
| locations[1].source.issn | |
| locations[1].source.type | repository |
| locations[1].source.is_oa | False |
| locations[1].source.issn_l | |
| locations[1].source.is_core | False |
| locations[1].source.is_in_doaj | False |
| locations[1].source.display_name | Econstor (Econstor) |
| locations[1].source.host_organization | |
| locations[1].source.host_organization_name | |
| locations[1].license | cc-by |
| locations[1].pdf_url | |
| locations[1].version | submittedVersion |
| locations[1].raw_type | doc-type:article |
| locations[1].license_id | https://openalex.org/licenses/cc-by |
| locations[1].is_accepted | False |
| locations[1].is_published | False |
| locations[1].raw_source_name | |
| locations[1].landing_page_url | https://hdl.handle.net/10419/307569 |
| locations[2].id | pmh:oai:library.wur.nl:wurpubs/615309 |
| locations[2].is_oa | True |
| locations[2].source.id | https://openalex.org/S4306400096 |
| locations[2].source.issn | |
| locations[2].source.type | repository |
| locations[2].source.is_oa | False |
| locations[2].source.issn_l | |
| locations[2].source.is_core | False |
| locations[2].source.is_in_doaj | False |
| locations[2].source.display_name | Wageningen University and Researchcenter Publications (Wageningen University & Research) |
| locations[2].source.host_organization | https://openalex.org/I913481162 |
| locations[2].source.host_organization_name | Wageningen University & Research |
| locations[2].source.host_organization_lineage | https://openalex.org/I913481162 |
| locations[2].license | cc-by |
| locations[2].pdf_url | |
| locations[2].version | submittedVersion |
| locations[2].raw_type | Article/Letter to editor |
| locations[2].license_id | https://openalex.org/licenses/cc-by |
| locations[2].is_accepted | False |
| locations[2].is_published | False |
| locations[2].raw_source_name | Computational Optimization and Applications 85 (2023) 2 |
| locations[2].landing_page_url | https://research.wur.nl/en/publications/globally-optimal-univariate-spline-approximations |
| indexed_in | crossref |
| authorships[0].author.id | https://openalex.org/A5037025465 |
| authorships[0].author.orcid | https://orcid.org/0009-0005-2536-5019 |
| authorships[0].author.display_name | Robert Mohr |
| authorships[0].countries | DE |
| authorships[0].affiliations[0].institution_ids | https://openalex.org/I102335020 |
| authorships[0].affiliations[0].raw_affiliation_string | Institute for Operations Research (IOR), Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany |
| authorships[0].institutions[0].id | https://openalex.org/I102335020 |
| authorships[0].institutions[0].ror | https://ror.org/04t3en479 |
| authorships[0].institutions[0].type | education |
| authorships[0].institutions[0].lineage | https://openalex.org/I102335020, https://openalex.org/I1305996414 |
| authorships[0].institutions[0].country_code | DE |
| authorships[0].institutions[0].display_name | Karlsruhe Institute of Technology |
| authorships[0].author_position | first |
| authorships[0].raw_author_name | Robert Mohr |
| authorships[0].is_corresponding | False |
| authorships[0].raw_affiliation_strings | Institute for Operations Research (IOR), Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany |
| authorships[1].author.id | https://openalex.org/A5031525875 |
| authorships[1].author.orcid | https://orcid.org/0000-0001-9464-3856 |
| authorships[1].author.display_name | Maximilian Coblenz |
| authorships[1].countries | DE |
| authorships[1].affiliations[0].institution_ids | https://openalex.org/I230073098 |
| authorships[1].affiliations[0].raw_affiliation_string | Department of Services and Consulting, Ludwigshafen University of Business and Society, Ludwigshafen, Germany |
| authorships[1].institutions[0].id | https://openalex.org/I230073098 |
| authorships[1].institutions[0].ror | https://ror.org/042xxdh20 |
| authorships[1].institutions[0].type | education |
| authorships[1].institutions[0].lineage | https://openalex.org/I230073098 |
| authorships[1].institutions[0].country_code | DE |
| authorships[1].institutions[0].display_name | Ludwigshafen University of Business and Society |
| authorships[1].author_position | middle |
| authorships[1].raw_author_name | Maximilian Coblenz |
| authorships[1].is_corresponding | True |
| authorships[1].raw_affiliation_strings | Department of Services and Consulting, Ludwigshafen University of Business and Society, Ludwigshafen, Germany |
| authorships[2].author.id | https://openalex.org/A5064366789 |
| authorships[2].author.orcid | https://orcid.org/0000-0002-8472-3569 |
| authorships[2].author.display_name | Peter Kirst |
| authorships[2].countries | NL |
| authorships[2].affiliations[0].institution_ids | https://openalex.org/I913481162 |
| authorships[2].affiliations[0].raw_affiliation_string | Operations Research and Logistics (ORL), Wageningen University and Research (WUR), Wageningen, The Netherlands |
| authorships[2].institutions[0].id | https://openalex.org/I913481162 |
| authorships[2].institutions[0].ror | https://ror.org/04qw24q55 |
| authorships[2].institutions[0].type | education |
| authorships[2].institutions[0].lineage | https://openalex.org/I913481162 |
| authorships[2].institutions[0].country_code | NL |
| authorships[2].institutions[0].display_name | Wageningen University & Research |
| authorships[2].author_position | last |
| authorships[2].raw_author_name | Peter Kirst |
| authorships[2].is_corresponding | False |
| authorships[2].raw_affiliation_strings | Operations Research and Logistics (ORL), Wageningen University and Research (WUR), Wageningen, The Netherlands |
| has_content.pdf | True |
| has_content.grobid_xml | True |
| is_paratext | False |
| open_access.is_oa | True |
| open_access.oa_url | https://link.springer.com/content/pdf/10.1007/s10589-023-00462-7.pdf |
| open_access.oa_status | hybrid |
| open_access.any_repository_has_fulltext | False |
| created_date | 2025-10-10T00:00:00 |
| display_name | Globally optimal univariate spline approximations |
| has_fulltext | True |
| is_retracted | False |
| updated_date | 2025-11-06T03:46:38.306776 |
| primary_topic.id | https://openalex.org/T11245 |
| primary_topic.field.id | https://openalex.org/fields/22 |
| primary_topic.field.display_name | Engineering |
| primary_topic.score | 0.9984999895095825 |
| primary_topic.domain.id | https://openalex.org/domains/3 |
| primary_topic.domain.display_name | Physical Sciences |
| primary_topic.subfield.id | https://openalex.org/subfields/2206 |
| primary_topic.subfield.display_name | Computational Mechanics |
| primary_topic.display_name | Advanced Numerical Analysis Techniques |
| related_works | https://openalex.org/W2084086280, https://openalex.org/W2027198738, https://openalex.org/W2397767885, https://openalex.org/W1828158523, https://openalex.org/W2047547195, https://openalex.org/W237421766, https://openalex.org/W204175656, https://openalex.org/W1993992974, https://openalex.org/W2478648414, https://openalex.org/W1633071293 |
| cited_by_count | 1 |
| counts_by_year[0].year | 2025 |
| counts_by_year[0].cited_by_count | 1 |
| locations_count | 3 |
| best_oa_location.id | doi:10.1007/s10589-023-00462-7 |
| best_oa_location.is_oa | True |
| best_oa_location.source.id | https://openalex.org/S39243548 |
| best_oa_location.source.issn | 0926-6003, 1573-2894 |
| best_oa_location.source.type | journal |
| best_oa_location.source.is_oa | False |
| best_oa_location.source.issn_l | 0926-6003 |
| best_oa_location.source.is_core | True |
| best_oa_location.source.is_in_doaj | False |
| best_oa_location.source.display_name | Computational Optimization and Applications |
| best_oa_location.source.host_organization | https://openalex.org/P4310319900 |
| best_oa_location.source.host_organization_name | Springer Science+Business Media |
| best_oa_location.source.host_organization_lineage | https://openalex.org/P4310319900, https://openalex.org/P4310319965 |
| best_oa_location.source.host_organization_lineage_names | Springer Science+Business Media, Springer Nature |
| best_oa_location.license | cc-by |
| best_oa_location.pdf_url | https://link.springer.com/content/pdf/10.1007/s10589-023-00462-7.pdf |
| best_oa_location.version | publishedVersion |
| best_oa_location.raw_type | journal-article |
| best_oa_location.license_id | https://openalex.org/licenses/cc-by |
| best_oa_location.is_accepted | True |
| best_oa_location.is_published | True |
| best_oa_location.raw_source_name | Computational Optimization and Applications |
| best_oa_location.landing_page_url | https://doi.org/10.1007/s10589-023-00462-7 |
| primary_location.id | doi:10.1007/s10589-023-00462-7 |
| primary_location.is_oa | True |
| primary_location.source.id | https://openalex.org/S39243548 |
| primary_location.source.issn | 0926-6003, 1573-2894 |
| primary_location.source.type | journal |
| primary_location.source.is_oa | False |
| primary_location.source.issn_l | 0926-6003 |
| primary_location.source.is_core | True |
| primary_location.source.is_in_doaj | False |
| primary_location.source.display_name | Computational Optimization and Applications |
| primary_location.source.host_organization | https://openalex.org/P4310319900 |
| primary_location.source.host_organization_name | Springer Science+Business Media |
| primary_location.source.host_organization_lineage | https://openalex.org/P4310319900, https://openalex.org/P4310319965 |
| primary_location.source.host_organization_lineage_names | Springer Science+Business Media, Springer Nature |
| primary_location.license | cc-by |
| primary_location.pdf_url | https://link.springer.com/content/pdf/10.1007/s10589-023-00462-7.pdf |
| primary_location.version | publishedVersion |
| primary_location.raw_type | journal-article |
| primary_location.license_id | https://openalex.org/licenses/cc-by |
| primary_location.is_accepted | True |
| primary_location.is_published | True |
| primary_location.raw_source_name | Computational Optimization and Applications |
| primary_location.landing_page_url | https://doi.org/10.1007/s10589-023-00462-7 |
| publication_date | 2023-02-28 |
| publication_year | 2023 |
| referenced_works | https://openalex.org/W2024941149, https://openalex.org/W2052260341, https://openalex.org/W2128860595, https://openalex.org/W2068120653, https://openalex.org/W2027293138, https://openalex.org/W2036050714, https://openalex.org/W2018089423, https://openalex.org/W1981937516, https://openalex.org/W2067596735, https://openalex.org/W1973409314, https://openalex.org/W2100352587, https://openalex.org/W1986096508, https://openalex.org/W2072081687, https://openalex.org/W1979449478, https://openalex.org/W1964080509, https://openalex.org/W2505968561, https://openalex.org/W2029190273, https://openalex.org/W2151380144, https://openalex.org/W2027106381, https://openalex.org/W2024690006, https://openalex.org/W2167897348, https://openalex.org/W2770291243, https://openalex.org/W2002869784, https://openalex.org/W3125630184, https://openalex.org/W1976601779, https://openalex.org/W1762430620, https://openalex.org/W2162870748, https://openalex.org/W2069323107 |
| referenced_works_count | 28 |
| abstract_inverted_index.a | 15, 36, 41, 47, 74, 94 |
| abstract_inverted_index.In | 19 |
| abstract_inverted_index.To | 53 |
| abstract_inverted_index.We | 1, 103 |
| abstract_inverted_index.as | 33, 35, 73 |
| abstract_inverted_index.at | 29 |
| abstract_inverted_index.be | 81 |
| abstract_inverted_index.is | 67, 93, 138 |
| abstract_inverted_index.of | 5, 40, 44, 50, 70, 145 |
| abstract_inverted_index.on | 109 |
| abstract_inverted_index.to | 21, 99, 128, 140, 143 |
| abstract_inverted_index.we | 27, 65, 91 |
| abstract_inverted_index.The | 61, 87, 121 |
| abstract_inverted_index.aim | 28 |
| abstract_inverted_index.and | 112 |
| abstract_inverted_index.are | 59 |
| abstract_inverted_index.can | 80 |
| abstract_inverted_index.for | 10, 46 |
| abstract_inverted_index.our | 105, 126 |
| abstract_inverted_index.the | 3, 25, 68, 71, 100, 119, 130 |
| abstract_inverted_index.two | 56 |
| abstract_inverted_index.able | 139 |
| abstract_inverted_index.data | 116 |
| abstract_inverted_index.free | 136 |
| abstract_inverted_index.from | 14, 24, 118 |
| abstract_inverted_index.most | 22 |
| abstract_inverted_index.real | 111 |
| abstract_inverted_index.sets | 117 |
| abstract_inverted_index.show | 124 |
| abstract_inverted_index.that | 64, 90, 125 |
| abstract_inverted_index.this | 51 |
| abstract_inverted_index.well | 34 |
| abstract_inverted_index.with | 135 |
| abstract_inverted_index.both, | 110 |
| abstract_inverted_index.curve | 114 |
| abstract_inverted_index.first | 62 |
| abstract_inverted_index.fixed | 42 |
| abstract_inverted_index.knots | 45, 137 |
| abstract_inverted_index.sizes | 147 |
| abstract_inverted_index.this, | 55 |
| abstract_inverted_index.using | 83 |
| abstract_inverted_index.which | 79 |
| abstract_inverted_index.method | 89 |
| abstract_inverted_index.number | 43 |
| abstract_inverted_index.second | 88 |
| abstract_inverted_index.solved | 82 |
| abstract_inverted_index.spline | 8, 132 |
| abstract_inverted_index.tackle | 129 |
| abstract_inverted_index.times. | 151 |
| abstract_inverted_index.within | 148 |
| abstract_inverted_index.achieve | 54 |
| abstract_inverted_index.compare | 104 |
| abstract_inverted_index.compute | 141 |
| abstract_inverted_index.fitting | 115 |
| abstract_inverted_index.optimal | 7, 31, 38 |
| abstract_inverted_index.present | 66 |
| abstract_inverted_index.problem | 4, 72, 134 |
| abstract_inverted_index.propose | 92 |
| abstract_inverted_index.revisit | 2 |
| abstract_inverted_index.splines | 13 |
| abstract_inverted_index.variant | 49 |
| abstract_inverted_index.Abstract | 0 |
| abstract_inverted_index.approach | 63, 127 |
| abstract_inverted_index.contrast | 20 |
| abstract_inverted_index.discrete | 48 |
| abstract_inverted_index.globally | 30, 37 |
| abstract_inverted_index.problem, | 78 |
| abstract_inverted_index.problem. | 52 |
| abstract_inverted_index.problems | 144 |
| abstract_inverted_index.solvers. | 86 |
| abstract_inverted_index.tailored | 97 |
| abstract_inverted_index.algorithm | 96 |
| abstract_inverted_index.computing | 6, 150 |
| abstract_inverted_index.different | 57 |
| abstract_inverted_index.numerical | 122 |
| abstract_inverted_index.placement | 39 |
| abstract_inverted_index.realistic | 146 |
| abstract_inverted_index.solutions | 142 |
| abstract_inverted_index.synthetic | 113 |
| abstract_inverted_index.approaches | 23, 107 |
| abstract_inverted_index.commercial | 84 |
| abstract_inverted_index.developed. | 60 |
| abstract_inverted_index.literature | 26 |
| abstract_inverted_index.reasonable | 149 |
| abstract_inverted_index.univariate | 11 |
| abstract_inverted_index.algorithmic | 106 |
| abstract_inverted_index.constrained | 77 |
| abstract_inverted_index.empirically | 108 |
| abstract_inverted_index.experiments | 123 |
| abstract_inverted_index.formulation | 69 |
| abstract_inverted_index.literature. | 120 |
| abstract_inverted_index.coefficients | 32 |
| abstract_inverted_index.formulation. | 102 |
| abstract_inverted_index.optimization | 17, 85 |
| abstract_inverted_index.perspective. | 18 |
| abstract_inverted_index.specifically | 98 |
| abstract_inverted_index.approximation | 133 |
| abstract_inverted_index.combinatorial | 16, 101 |
| abstract_inverted_index.least-squares | 12, 131 |
| abstract_inverted_index.mixed-integer | 75 |
| abstract_inverted_index.possibilities | 58 |
| abstract_inverted_index.quadratically | 76 |
| abstract_inverted_index.approximations | 9 |
| abstract_inverted_index.branch-and-bound | 95 |
| cited_by_percentile_year.max | 95 |
| cited_by_percentile_year.min | 91 |
| corresponding_author_ids | https://openalex.org/A5031525875 |
| countries_distinct_count | 2 |
| institutions_distinct_count | 3 |
| corresponding_institution_ids | https://openalex.org/I230073098 |
| citation_normalized_percentile.value | 0.45824244 |
| citation_normalized_percentile.is_in_top_1_percent | False |
| citation_normalized_percentile.is_in_top_10_percent | False |