Mathematical programs with vanishing constraints: critical point theory

We study mathematical programs with vanishing constraints (MPVCs) from a topological point of view. We introduce the new concept of a T-stationary point for MPVC. Under the Linear Independence Constraint Qualification we derive an equivariant Morse Lemma at nondegenerate T-stationary points. Then, t...

Ausführliche Beschreibung

Gespeichert in:
Bibliographische Detailangaben
Veröffentlicht in:Journal of global optimization 2012-03, Vol.52 (3), p.591-605
Hauptverfasser: Dorsch, Dominik, Shikhman, Vladimir, Stein, Oliver
Format: Artikel
Sprache:eng
Schlagworte:
Online-Zugang:Volltext
Tags: Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
container_end_page 605
container_issue 3
container_start_page 591
container_title Journal of global optimization
container_volume 52
creator Dorsch, Dominik
Shikhman, Vladimir
Stein, Oliver
description We study mathematical programs with vanishing constraints (MPVCs) from a topological point of view. We introduce the new concept of a T-stationary point for MPVC. Under the Linear Independence Constraint Qualification we derive an equivariant Morse Lemma at nondegenerate T-stationary points. Then, two basic theorems from Morse Theory (deformation theorem and cell-attachment theorem) are proved. Outside the T-stationary point set, continuous deformation of lower level sets can be performed. As a consequence, the topological data (such as the number of connected components) then remain invariant. However, when passing a T-stationary level, the topology of the lower level set changes via the attachment of a q -dimensional cell. The dimension q equals the stationary T-index of the (nondegenerate) T-stationary point. The stationary T-index depends on both the restricted Hessian of the Lagrangian and the number of bi-active vanishing constraints. Further, we prove that all T-stationary points are generically nondegenerate. The latter property is shown to be stable under C 2 -perturbations of the defining functions. Finally, some relations with other stationarity concepts, such as strong, weak, and M-stationarity, are discussed.
doi_str_mv 10.1007/s10898-011-9805-z
format Article
fullrecord <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_926561696</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2603939811</sourcerecordid><originalsourceid>FETCH-LOGICAL-c315t-889e0cd91b92c912333199dc36ecbf66a2a7b228e38267372a7b2d06c4c3adf43</originalsourceid><addsrcrecordid>eNp1kMFKAzEQhoMoWKsP4G3xHp1JutnEmxStQsWLnkOazbYp7W5NUqV9elO34MnTMMP3zwwfIdcItwhQ3UUEqSQFRKoklHR_QgZYVpwyheKUDECxkpYAeE4uYlwCgJIlG5DJq0kLtzbJW7MqNqGbB7OOxbdPi-LLtD4ufDsvbNfGFIxvU7wvbPBHusuDIse7sLskZ41ZRXd1rEPy8fT4Pn6m07fJy_hhSi3HMlEplQNbK5wpZhUyzjkqVVsunJ01Qhhmqhlj0nHJRMWr37YGYUeWm7oZ8SG56ffmVz-3Lia97LahzSe1YqIUKJTIEPaQDV2MwTV6E_zahJ1G0Addutelsy590KX3OcP6TMxsO3fhb_H_oR87jG5_</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>926561696</pqid></control><display><type>article</type><title>Mathematical programs with vanishing constraints: critical point theory</title><source>SpringerLink Journals - AutoHoldings</source><creator>Dorsch, Dominik ; Shikhman, Vladimir ; Stein, Oliver</creator><creatorcontrib>Dorsch, Dominik ; Shikhman, Vladimir ; Stein, Oliver</creatorcontrib><description>We study mathematical programs with vanishing constraints (MPVCs) from a topological point of view. We introduce the new concept of a T-stationary point for MPVC. Under the Linear Independence Constraint Qualification we derive an equivariant Morse Lemma at nondegenerate T-stationary points. Then, two basic theorems from Morse Theory (deformation theorem and cell-attachment theorem) are proved. Outside the T-stationary point set, continuous deformation of lower level sets can be performed. As a consequence, the topological data (such as the number of connected components) then remain invariant. However, when passing a T-stationary level, the topology of the lower level set changes via the attachment of a q -dimensional cell. The dimension q equals the stationary T-index of the (nondegenerate) T-stationary point. The stationary T-index depends on both the restricted Hessian of the Lagrangian and the number of bi-active vanishing constraints. Further, we prove that all T-stationary points are generically nondegenerate. The latter property is shown to be stable under C 2 -perturbations of the defining functions. Finally, some relations with other stationarity concepts, such as strong, weak, and M-stationarity, are discussed.</description><identifier>ISSN: 0925-5001</identifier><identifier>EISSN: 1573-2916</identifier><identifier>DOI: 10.1007/s10898-011-9805-z</identifier><language>eng</language><publisher>Boston: Springer US</publisher><subject>Computer Science ; Equality ; Mathematical programming ; Mathematics ; Mathematics and Statistics ; Operations Research/Decision Theory ; Optimization ; Real Functions ; Regularization methods ; Studies</subject><ispartof>Journal of global optimization, 2012-03, Vol.52 (3), p.591-605</ispartof><rights>Springer Science+Business Media, LLC. 2011</rights><rights>Springer Science+Business Media, LLC. 2012</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c315t-889e0cd91b92c912333199dc36ecbf66a2a7b228e38267372a7b2d06c4c3adf43</citedby><cites>FETCH-LOGICAL-c315t-889e0cd91b92c912333199dc36ecbf66a2a7b228e38267372a7b2d06c4c3adf43</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s10898-011-9805-z$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s10898-011-9805-z$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27924,27925,41488,42557,51319</link.rule.ids></links><search><creatorcontrib>Dorsch, Dominik</creatorcontrib><creatorcontrib>Shikhman, Vladimir</creatorcontrib><creatorcontrib>Stein, Oliver</creatorcontrib><title>Mathematical programs with vanishing constraints: critical point theory</title><title>Journal of global optimization</title><addtitle>J Glob Optim</addtitle><description>We study mathematical programs with vanishing constraints (MPVCs) from a topological point of view. We introduce the new concept of a T-stationary point for MPVC. Under the Linear Independence Constraint Qualification we derive an equivariant Morse Lemma at nondegenerate T-stationary points. Then, two basic theorems from Morse Theory (deformation theorem and cell-attachment theorem) are proved. Outside the T-stationary point set, continuous deformation of lower level sets can be performed. As a consequence, the topological data (such as the number of connected components) then remain invariant. However, when passing a T-stationary level, the topology of the lower level set changes via the attachment of a q -dimensional cell. The dimension q equals the stationary T-index of the (nondegenerate) T-stationary point. The stationary T-index depends on both the restricted Hessian of the Lagrangian and the number of bi-active vanishing constraints. Further, we prove that all T-stationary points are generically nondegenerate. The latter property is shown to be stable under C 2 -perturbations of the defining functions. Finally, some relations with other stationarity concepts, such as strong, weak, and M-stationarity, are discussed.</description><subject>Computer Science</subject><subject>Equality</subject><subject>Mathematical programming</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Operations Research/Decision Theory</subject><subject>Optimization</subject><subject>Real Functions</subject><subject>Regularization methods</subject><subject>Studies</subject><issn>0925-5001</issn><issn>1573-2916</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2012</creationdate><recordtype>article</recordtype><sourceid>8G5</sourceid><sourceid>ABUWG</sourceid><sourceid>AFKRA</sourceid><sourceid>AZQEC</sourceid><sourceid>BENPR</sourceid><sourceid>CCPQU</sourceid><sourceid>DWQXO</sourceid><sourceid>GNUQQ</sourceid><sourceid>GUQSH</sourceid><sourceid>M2O</sourceid><recordid>eNp1kMFKAzEQhoMoWKsP4G3xHp1JutnEmxStQsWLnkOazbYp7W5NUqV9elO34MnTMMP3zwwfIdcItwhQ3UUEqSQFRKoklHR_QgZYVpwyheKUDECxkpYAeE4uYlwCgJIlG5DJq0kLtzbJW7MqNqGbB7OOxbdPi-LLtD4ufDsvbNfGFIxvU7wvbPBHusuDIse7sLskZ41ZRXd1rEPy8fT4Pn6m07fJy_hhSi3HMlEplQNbK5wpZhUyzjkqVVsunJ01Qhhmqhlj0nHJRMWr37YGYUeWm7oZ8SG56ffmVz-3Lia97LahzSe1YqIUKJTIEPaQDV2MwTV6E_zahJ1G0Addutelsy590KX3OcP6TMxsO3fhb_H_oR87jG5_</recordid><startdate>20120301</startdate><enddate>20120301</enddate><creator>Dorsch, Dominik</creator><creator>Shikhman, Vladimir</creator><creator>Stein, Oliver</creator><general>Springer US</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><scope>0U~</scope><scope>1-H</scope><scope>3V.</scope><scope>7SC</scope><scope>7WY</scope><scope>7WZ</scope><scope>7XB</scope><scope>87Z</scope><scope>88I</scope><scope>8AL</scope><scope>8AO</scope><scope>8FD</scope><scope>8FE</scope><scope>8FG</scope><scope>8FK</scope><scope>8FL</scope><scope>8G5</scope><scope>ABJCF</scope><scope>ABUWG</scope><scope>AFKRA</scope><scope>ARAPS</scope><scope>AZQEC</scope><scope>BENPR</scope><scope>BEZIV</scope><scope>BGLVJ</scope><scope>CCPQU</scope><scope>DWQXO</scope><scope>FRNLG</scope><scope>F~G</scope><scope>GNUQQ</scope><scope>GUQSH</scope><scope>HCIFZ</scope><scope>JQ2</scope><scope>K60</scope><scope>K6~</scope><scope>K7-</scope><scope>L.-</scope><scope>L.0</scope><scope>L6V</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope><scope>M0C</scope><scope>M0N</scope><scope>M2O</scope><scope>M2P</scope><scope>M7S</scope><scope>MBDVC</scope><scope>P5Z</scope><scope>P62</scope><scope>PQBIZ</scope><scope>PQBZA</scope><scope>PQEST</scope><scope>PQQKQ</scope><scope>PQUKI</scope><scope>PTHSS</scope><scope>Q9U</scope></search><sort><creationdate>20120301</creationdate><title>Mathematical programs with vanishing constraints: critical point theory</title><author>Dorsch, Dominik ; Shikhman, Vladimir ; Stein, Oliver</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c315t-889e0cd91b92c912333199dc36ecbf66a2a7b228e38267372a7b2d06c4c3adf43</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2012</creationdate><topic>Computer Science</topic><topic>Equality</topic><topic>Mathematical programming</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Operations Research/Decision Theory</topic><topic>Optimization</topic><topic>Real Functions</topic><topic>Regularization methods</topic><topic>Studies</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Dorsch, Dominik</creatorcontrib><creatorcontrib>Shikhman, Vladimir</creatorcontrib><creatorcontrib>Stein, Oliver</creatorcontrib><collection>CrossRef</collection><collection>Global News &amp; ABI/Inform Professional</collection><collection>Trade PRO</collection><collection>ProQuest Central (Corporate)</collection><collection>Computer and Information Systems Abstracts</collection><collection>Access via ABI/INFORM (ProQuest)</collection><collection>ABI/INFORM Global (PDF only)</collection><collection>ProQuest Central (purchase pre-March 2016)</collection><collection>ABI/INFORM Global (Alumni Edition)</collection><collection>Science Database (Alumni Edition)</collection><collection>Computing Database (Alumni Edition)</collection><collection>ProQuest Pharma Collection</collection><collection>Technology Research Database</collection><collection>ProQuest SciTech Collection</collection><collection>ProQuest Technology Collection</collection><collection>ProQuest Central (Alumni) (purchase pre-March 2016)</collection><collection>ABI/INFORM Collection (Alumni Edition)</collection><collection>Research Library (Alumni Edition)</collection><collection>Materials Science &amp; Engineering Collection</collection><collection>ProQuest Central (Alumni Edition)</collection><collection>ProQuest Central UK/Ireland</collection><collection>Advanced Technologies &amp; Aerospace Collection</collection><collection>ProQuest Central Essentials</collection><collection>ProQuest Central</collection><collection>Business Premium Collection</collection><collection>Technology Collection</collection><collection>ProQuest One Community College</collection><collection>ProQuest Central Korea</collection><collection>Business Premium Collection (Alumni)</collection><collection>ABI/INFORM Global (Corporate)</collection><collection>ProQuest Central Student</collection><collection>Research Library Prep</collection><collection>SciTech Premium Collection</collection><collection>ProQuest Computer Science Collection</collection><collection>ProQuest Business Collection (Alumni Edition)</collection><collection>ProQuest Business Collection</collection><collection>Computer Science Database</collection><collection>ABI/INFORM Professional Advanced</collection><collection>ABI/INFORM Professional Standard</collection><collection>ProQuest Engineering Collection</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts – Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><collection>ABI/INFORM Global</collection><collection>Computing Database</collection><collection>Research Library</collection><collection>Science Database</collection><collection>Engineering Database</collection><collection>Research Library (Corporate)</collection><collection>Advanced Technologies &amp; Aerospace Database</collection><collection>ProQuest Advanced Technologies &amp; Aerospace Collection</collection><collection>ProQuest One Business</collection><collection>ProQuest One Business (Alumni)</collection><collection>ProQuest One Academic Eastern Edition (DO NOT USE)</collection><collection>ProQuest One Academic</collection><collection>ProQuest One Academic UKI Edition</collection><collection>Engineering Collection</collection><collection>ProQuest Central Basic</collection><jtitle>Journal of global optimization</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Dorsch, Dominik</au><au>Shikhman, Vladimir</au><au>Stein, Oliver</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Mathematical programs with vanishing constraints: critical point theory</atitle><jtitle>Journal of global optimization</jtitle><stitle>J Glob Optim</stitle><date>2012-03-01</date><risdate>2012</risdate><volume>52</volume><issue>3</issue><spage>591</spage><epage>605</epage><pages>591-605</pages><issn>0925-5001</issn><eissn>1573-2916</eissn><abstract>We study mathematical programs with vanishing constraints (MPVCs) from a topological point of view. We introduce the new concept of a T-stationary point for MPVC. Under the Linear Independence Constraint Qualification we derive an equivariant Morse Lemma at nondegenerate T-stationary points. Then, two basic theorems from Morse Theory (deformation theorem and cell-attachment theorem) are proved. Outside the T-stationary point set, continuous deformation of lower level sets can be performed. As a consequence, the topological data (such as the number of connected components) then remain invariant. However, when passing a T-stationary level, the topology of the lower level set changes via the attachment of a q -dimensional cell. The dimension q equals the stationary T-index of the (nondegenerate) T-stationary point. The stationary T-index depends on both the restricted Hessian of the Lagrangian and the number of bi-active vanishing constraints. Further, we prove that all T-stationary points are generically nondegenerate. The latter property is shown to be stable under C 2 -perturbations of the defining functions. Finally, some relations with other stationarity concepts, such as strong, weak, and M-stationarity, are discussed.</abstract><cop>Boston</cop><pub>Springer US</pub><doi>10.1007/s10898-011-9805-z</doi><tpages>15</tpages></addata></record>
fulltext fulltext
identifier ISSN: 0925-5001
ispartof Journal of global optimization, 2012-03, Vol.52 (3), p.591-605
issn 0925-5001
1573-2916
language eng
recordid cdi_proquest_journals_926561696
source SpringerLink Journals - AutoHoldings
subjects Computer Science
Equality
Mathematical programming
Mathematics
Mathematics and Statistics
Operations Research/Decision Theory
Optimization
Real Functions
Regularization methods
Studies
title Mathematical programs with vanishing constraints: critical point theory
url https://sfx.bib-bvb.de/sfx_tum?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2024-12-23T15%3A25%3A22IST&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=Mathematical%20programs%20with%20vanishing%20constraints:%20critical%20point%20theory&rft.jtitle=Journal%20of%20global%20optimization&rft.au=Dorsch,%20Dominik&rft.date=2012-03-01&rft.volume=52&rft.issue=3&rft.spage=591&rft.epage=605&rft.pages=591-605&rft.issn=0925-5001&rft.eissn=1573-2916&rft_id=info:doi/10.1007/s10898-011-9805-z&rft_dat=%3Cproquest_cross%3E2603939811%3C/proquest_cross%3E%3Curl%3E%3C/url%3E&disable_directlink=true&sfx.directlink=off&sfx.report_link=0&rft_id=info:oai/&rft_pqid=926561696&rft_id=info:pmid/&rfr_iscdi=true