Loading…
How to project onto the intersection of a closed affine subspace and a hyperplane
Affine subspaces are translates of linear subspaces, and hyperplanes are well-known instances of affine subspaces. In basic linear algebra, one encounters the explicit formula for projecting onto a hyperplane. An interesting—and relevant for applications—question is whether or not there is a formula...
Saved in:
Published in: | Mathematical methods of operations research (Heidelberg, Germany) Germany), 2024-10, Vol.100 (2), p.411-435 |
---|---|
Main Authors: | , , |
Format: | Article |
Language: | English |
Subjects: | |
Citations: | Items that this one cites |
Online Access: | Get full text |
Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
cited_by | |
---|---|
cites | cdi_FETCH-LOGICAL-c258t-bfcd497787e33a5845cb77be7e60029265f1e1506328b18f09a8c6c86ab283ed3 |
container_end_page | 435 |
container_issue | 2 |
container_start_page | 411 |
container_title | Mathematical methods of operations research (Heidelberg, Germany) |
container_volume | 100 |
creator | Bauschke, Heinz H. Mao, Dayou Moursi, Walaa M. |
description | Affine subspaces are translates of linear subspaces, and hyperplanes are well-known instances of affine subspaces. In basic linear algebra, one encounters the explicit formula for projecting onto a hyperplane. An interesting—and relevant for applications—question is whether or not there is a formula for projecting onto the intersection of two hyperplanes. The answer turns out to be yes, as demonstrated recently by Behling, Bello-Cruz, and Santos, by López, by Needell and Ward, and by Ouyang. Most of these authors also provided formulas for projecting onto the intersection of an affine subspace and a hyperplane. In this note, we present an alternative approach which has the advantage of being more explicit and more elementary. Our results also provide useful information in the case when the two sets don’t intersect. Luckily, the material is fully accessible to readers with a basic background in linear algebra and analysis. Finally, we demonstrate the computational efficiency of our formula when applied to an image reconstruction problem arising in Computed Tomography, and we also present a new formula for the projection onto the set of generalized bistochastic matrices with a moment constraint. |
doi_str_mv | 10.1007/s00186-024-00866-z |
format | article |
fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_3116751789</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>3116751789</sourcerecordid><originalsourceid>FETCH-LOGICAL-c258t-bfcd497787e33a5845cb77be7e60029265f1e1506328b18f09a8c6c86ab283ed3</originalsourceid><addsrcrecordid>eNp9kE9LAzEQxYMoWKtfwFPAczR_dpPsUYpaoSCCnkM2ndgtNVmTLdJ-elO34M3TzDx-b2Z4CF0zessoVXeZUqYlobwilGopyf4ETVglOKk5U6fHnjdNdY4ucl7TwlcVn6DXefzGQ8R9imtwA46hDMMKcBcGSLlIXQw4emyx28QMS2y97wLgvG1zbx1gG4qGV7seUr-xAS7RmbebDFfHOkXvjw9vszlZvDw9z-4XxPFaD6T1blk1SmkFQthaV7VrlWpBgaSUN1zWngGrqRRct0x72ljtpNPStlwLWIopuhn3lte_tpAHs47bFMpJIxiTqmZKN4XiI-VSzDmBN33qPm3aGUbNITozRmdKdOY3OrMvJjGacoHDB6S_1f-4fgBdpnFu</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>3116751789</pqid></control><display><type>article</type><title>How to project onto the intersection of a closed affine subspace and a hyperplane</title><source>Springer Link</source><creator>Bauschke, Heinz H. ; Mao, Dayou ; Moursi, Walaa M.</creator><creatorcontrib>Bauschke, Heinz H. ; Mao, Dayou ; Moursi, Walaa M.</creatorcontrib><description>Affine subspaces are translates of linear subspaces, and hyperplanes are well-known instances of affine subspaces. In basic linear algebra, one encounters the explicit formula for projecting onto a hyperplane. An interesting—and relevant for applications—question is whether or not there is a formula for projecting onto the intersection of two hyperplanes. The answer turns out to be yes, as demonstrated recently by Behling, Bello-Cruz, and Santos, by López, by Needell and Ward, and by Ouyang. Most of these authors also provided formulas for projecting onto the intersection of an affine subspace and a hyperplane. In this note, we present an alternative approach which has the advantage of being more explicit and more elementary. Our results also provide useful information in the case when the two sets don’t intersect. Luckily, the material is fully accessible to readers with a basic background in linear algebra and analysis. Finally, we demonstrate the computational efficiency of our formula when applied to an image reconstruction problem arising in Computed Tomography, and we also present a new formula for the projection onto the set of generalized bistochastic matrices with a moment constraint.</description><identifier>ISSN: 1432-2994</identifier><identifier>EISSN: 1432-5217</identifier><identifier>DOI: 10.1007/s00186-024-00866-z</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer Berlin Heidelberg</publisher><subject>Business and Management ; Calculus of Variations and Optimal Control; Optimization ; Computed tomography ; Hyperplanes ; Image reconstruction ; Intersections ; Linear algebra ; Mathematics ; Mathematics and Statistics ; Operations Research/Decision Theory ; OR for the classroom ; Subspaces</subject><ispartof>Mathematical methods of operations research (Heidelberg, Germany), 2024-10, Vol.100 (2), p.411-435</ispartof><rights>The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2024. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c258t-bfcd497787e33a5845cb77be7e60029265f1e1506328b18f09a8c6c86ab283ed3</cites><orcidid>0000-0002-0113-9309</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,780,784,27924,27925</link.rule.ids></links><search><creatorcontrib>Bauschke, Heinz H.</creatorcontrib><creatorcontrib>Mao, Dayou</creatorcontrib><creatorcontrib>Moursi, Walaa M.</creatorcontrib><title>How to project onto the intersection of a closed affine subspace and a hyperplane</title><title>Mathematical methods of operations research (Heidelberg, Germany)</title><addtitle>Math Meth Oper Res</addtitle><description>Affine subspaces are translates of linear subspaces, and hyperplanes are well-known instances of affine subspaces. In basic linear algebra, one encounters the explicit formula for projecting onto a hyperplane. An interesting—and relevant for applications—question is whether or not there is a formula for projecting onto the intersection of two hyperplanes. The answer turns out to be yes, as demonstrated recently by Behling, Bello-Cruz, and Santos, by López, by Needell and Ward, and by Ouyang. Most of these authors also provided formulas for projecting onto the intersection of an affine subspace and a hyperplane. In this note, we present an alternative approach which has the advantage of being more explicit and more elementary. Our results also provide useful information in the case when the two sets don’t intersect. Luckily, the material is fully accessible to readers with a basic background in linear algebra and analysis. Finally, we demonstrate the computational efficiency of our formula when applied to an image reconstruction problem arising in Computed Tomography, and we also present a new formula for the projection onto the set of generalized bistochastic matrices with a moment constraint.</description><subject>Business and Management</subject><subject>Calculus of Variations and Optimal Control; Optimization</subject><subject>Computed tomography</subject><subject>Hyperplanes</subject><subject>Image reconstruction</subject><subject>Intersections</subject><subject>Linear algebra</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Operations Research/Decision Theory</subject><subject>OR for the classroom</subject><subject>Subspaces</subject><issn>1432-2994</issn><issn>1432-5217</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><recordid>eNp9kE9LAzEQxYMoWKtfwFPAczR_dpPsUYpaoSCCnkM2ndgtNVmTLdJ-elO34M3TzDx-b2Z4CF0zessoVXeZUqYlobwilGopyf4ETVglOKk5U6fHnjdNdY4ucl7TwlcVn6DXefzGQ8R9imtwA46hDMMKcBcGSLlIXQw4emyx28QMS2y97wLgvG1zbx1gG4qGV7seUr-xAS7RmbebDFfHOkXvjw9vszlZvDw9z-4XxPFaD6T1blk1SmkFQthaV7VrlWpBgaSUN1zWngGrqRRct0x72ljtpNPStlwLWIopuhn3lte_tpAHs47bFMpJIxiTqmZKN4XiI-VSzDmBN33qPm3aGUbNITozRmdKdOY3OrMvJjGacoHDB6S_1f-4fgBdpnFu</recordid><startdate>20241001</startdate><enddate>20241001</enddate><creator>Bauschke, Heinz H.</creator><creator>Mao, Dayou</creator><creator>Moursi, Walaa M.</creator><general>Springer Berlin Heidelberg</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>7TB</scope><scope>8FD</scope><scope>FR3</scope><scope>JQ2</scope><scope>KR7</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope><orcidid>https://orcid.org/0000-0002-0113-9309</orcidid></search><sort><creationdate>20241001</creationdate><title>How to project onto the intersection of a closed affine subspace and a hyperplane</title><author>Bauschke, Heinz H. ; Mao, Dayou ; Moursi, Walaa M.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c258t-bfcd497787e33a5845cb77be7e60029265f1e1506328b18f09a8c6c86ab283ed3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Business and Management</topic><topic>Calculus of Variations and Optimal Control; Optimization</topic><topic>Computed tomography</topic><topic>Hyperplanes</topic><topic>Image reconstruction</topic><topic>Intersections</topic><topic>Linear algebra</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Operations Research/Decision Theory</topic><topic>OR for the classroom</topic><topic>Subspaces</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Bauschke, Heinz H.</creatorcontrib><creatorcontrib>Mao, Dayou</creatorcontrib><creatorcontrib>Moursi, Walaa M.</creatorcontrib><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Mechanical & Transportation Engineering Abstracts</collection><collection>Technology Research Database</collection><collection>Engineering Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Civil Engineering Abstracts</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><jtitle>Mathematical methods of operations research (Heidelberg, Germany)</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Bauschke, Heinz H.</au><au>Mao, Dayou</au><au>Moursi, Walaa M.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>How to project onto the intersection of a closed affine subspace and a hyperplane</atitle><jtitle>Mathematical methods of operations research (Heidelberg, Germany)</jtitle><stitle>Math Meth Oper Res</stitle><date>2024-10-01</date><risdate>2024</risdate><volume>100</volume><issue>2</issue><spage>411</spage><epage>435</epage><pages>411-435</pages><issn>1432-2994</issn><eissn>1432-5217</eissn><abstract>Affine subspaces are translates of linear subspaces, and hyperplanes are well-known instances of affine subspaces. In basic linear algebra, one encounters the explicit formula for projecting onto a hyperplane. An interesting—and relevant for applications—question is whether or not there is a formula for projecting onto the intersection of two hyperplanes. The answer turns out to be yes, as demonstrated recently by Behling, Bello-Cruz, and Santos, by López, by Needell and Ward, and by Ouyang. Most of these authors also provided formulas for projecting onto the intersection of an affine subspace and a hyperplane. In this note, we present an alternative approach which has the advantage of being more explicit and more elementary. Our results also provide useful information in the case when the two sets don’t intersect. Luckily, the material is fully accessible to readers with a basic background in linear algebra and analysis. Finally, we demonstrate the computational efficiency of our formula when applied to an image reconstruction problem arising in Computed Tomography, and we also present a new formula for the projection onto the set of generalized bistochastic matrices with a moment constraint.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/s00186-024-00866-z</doi><tpages>25</tpages><orcidid>https://orcid.org/0000-0002-0113-9309</orcidid></addata></record> |
fulltext | fulltext |
identifier | ISSN: 1432-2994 |
ispartof | Mathematical methods of operations research (Heidelberg, Germany), 2024-10, Vol.100 (2), p.411-435 |
issn | 1432-2994 1432-5217 |
language | eng |
recordid | cdi_proquest_journals_3116751789 |
source | Springer Link |
subjects | Business and Management Calculus of Variations and Optimal Control Optimization Computed tomography Hyperplanes Image reconstruction Intersections Linear algebra Mathematics Mathematics and Statistics Operations Research/Decision Theory OR for the classroom Subspaces |
title | How to project onto the intersection of a closed affine subspace and a hyperplane |
url | http://sfxeu10.hosted.exlibrisgroup.com/loughborough?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2024-12-27T20%3A39%3A38IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-proquest_cross&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=How%20to%20project%20onto%20the%20intersection%20of%20a%20closed%20affine%20subspace%20and%20a%20hyperplane&rft.jtitle=Mathematical%20methods%20of%20operations%20research%20(Heidelberg,%20Germany)&rft.au=Bauschke,%20Heinz%20H.&rft.date=2024-10-01&rft.volume=100&rft.issue=2&rft.spage=411&rft.epage=435&rft.pages=411-435&rft.issn=1432-2994&rft.eissn=1432-5217&rft_id=info:doi/10.1007/s00186-024-00866-z&rft_dat=%3Cproquest_cross%3E3116751789%3C/proquest_cross%3E%3Cgrp_id%3Ecdi_FETCH-LOGICAL-c258t-bfcd497787e33a5845cb77be7e60029265f1e1506328b18f09a8c6c86ab283ed3%3C/grp_id%3E%3Coa%3E%3C/oa%3E%3Curl%3E%3C/url%3E&rft_id=info:oai/&rft_pqid=3116751789&rft_id=info:pmid/&rfr_iscdi=true |