On the stationarity for nonlinear optimization problems with polyhedral constraints
For polyhedral constrained optimization problems and a feasible point x , it is shown that the projection of the negative gradient on the tangent cone, denoted ∇ Ω f ( x ) , has an orthogonal decomposition of the form β ( x ) + φ ( x ) . At a stationary point, ∇ Ω f ( x ) = 0 so ‖ ∇ Ω f ( x ) ‖ refl...
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| Veröffentlicht in: | Mathematical programming 2024-05, Vol.205 (1-2), p.107-134 |
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| creator | di Serafino, Daniela Hager, William W. Toraldo, Gerardo Viola, Marco |
| description | For polyhedral constrained optimization problems and a feasible point
x
, it is shown that the projection of the negative gradient on the tangent cone, denoted
∇
Ω
f
(
x
)
, has an orthogonal decomposition of the form
β
(
x
)
+
φ
(
x
)
. At a stationary point,
∇
Ω
f
(
x
)
=
0
so
‖
∇
Ω
f
(
x
)
‖
reflects the distance to a stationary point. Away from a stationary point,
‖
β
(
x
)
‖
and
‖
φ
(
x
)
‖
measure different aspects of optimality since
β
(
x
)
only vanishes when the KKT multipliers at
x
have the correct sign, while
φ
(
x
)
only vanishes when
x
is a stationary point in the active manifold. As an application of the theory, an active set algorithm is developed for convex quadratic programs which adapts the flow of the algorithm based on a comparison between
‖
β
(
x
)
‖
and
‖
φ
(
x
)
‖
. |
| doi_str_mv | 10.1007/s10107-023-01979-9 |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_3034029534</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>3034029534</sourcerecordid><originalsourceid>FETCH-LOGICAL-c363t-5519903be0cf00b280b43b7ddc0013a3644534598e54a31814185a1c406a3c1e3</originalsourceid><addsrcrecordid>eNp9kE1LAzEQhoMoWKt_wFPAc3Rmk-xujlL8gkIP6jlkt1mbsk1qkiL117vtCt48zWHe553hIeQa4RYBqruEgFAxKDgDVJVi6oRMUPCSiVKUp2QCUEgmS4RzcpHSGgCQ1_WEvC48zStLUzbZBW-iy3vahUh98L3z1kQattlt3PdxT7cxNL3dJPrl8opuQ79f2WU0PW2DTzka53O6JGed6ZO9-p1T8v748DZ7ZvPF08vsfs5aXvLMpESlgDcW2g6gKWpoBG-q5bI9PGd4KYTkQqraSmE41iiwlgZbAaXhLVo-JTdj7_DU586mrNdhF_1wUnPgAgo18EOqGFNtDClF2-ltdBsT9xpBH-TpUZ4e5OmjPK0GiI9QGsL-w8a_6n-oH0ZIcio</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>3034029534</pqid></control><display><type>article</type><title>On the stationarity for nonlinear optimization problems with polyhedral constraints</title><source>Springer Nature - Complete Springer Journals</source><creator>di Serafino, Daniela ; Hager, William W. ; Toraldo, Gerardo ; Viola, Marco</creator><creatorcontrib>di Serafino, Daniela ; Hager, William W. ; Toraldo, Gerardo ; Viola, Marco</creatorcontrib><description>For polyhedral constrained optimization problems and a feasible point
x
, it is shown that the projection of the negative gradient on the tangent cone, denoted
∇
Ω
f
(
x
)
, has an orthogonal decomposition of the form
β
(
x
)
+
φ
(
x
)
. At a stationary point,
∇
Ω
f
(
x
)
=
0
so
‖
∇
Ω
f
(
x
)
‖
reflects the distance to a stationary point. Away from a stationary point,
‖
β
(
x
)
‖
and
‖
φ
(
x
)
‖
measure different aspects of optimality since
β
(
x
)
only vanishes when the KKT multipliers at
x
have the correct sign, while
φ
(
x
)
only vanishes when
x
is a stationary point in the active manifold. As an application of the theory, an active set algorithm is developed for convex quadratic programs which adapts the flow of the algorithm based on a comparison between
‖
β
(
x
)
‖
and
‖
φ
(
x
)
‖
.</description><identifier>ISSN: 0025-5610</identifier><identifier>EISSN: 1436-4646</identifier><identifier>DOI: 10.1007/s10107-023-01979-9</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer Berlin Heidelberg</publisher><subject>Algorithms ; Calculus of Variations and Optimal Control; Optimization ; Combinatorics ; Constraints ; Full Length Paper ; Mathematical and Computational Physics ; Mathematical Methods in Physics ; Mathematics ; Mathematics and Statistics ; Mathematics of Computing ; Numerical Analysis ; Optimization ; Quadratic programming ; Theoretical</subject><ispartof>Mathematical programming, 2024-05, Vol.205 (1-2), p.107-134</ispartof><rights>The Author(s) 2023</rights><rights>The Author(s) 2023. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c363t-5519903be0cf00b280b43b7ddc0013a3644534598e54a31814185a1c406a3c1e3</citedby><cites>FETCH-LOGICAL-c363t-5519903be0cf00b280b43b7ddc0013a3644534598e54a31814185a1c406a3c1e3</cites><orcidid>0000-0002-2140-8094</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s10107-023-01979-9$$EPDF$$P50$$Gspringer$$Hfree_for_read</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s10107-023-01979-9$$EHTML$$P50$$Gspringer$$Hfree_for_read</linktohtml><link.rule.ids>314,776,780,27901,27902,41464,42533,51294</link.rule.ids></links><search><creatorcontrib>di Serafino, Daniela</creatorcontrib><creatorcontrib>Hager, William W.</creatorcontrib><creatorcontrib>Toraldo, Gerardo</creatorcontrib><creatorcontrib>Viola, Marco</creatorcontrib><title>On the stationarity for nonlinear optimization problems with polyhedral constraints</title><title>Mathematical programming</title><addtitle>Math. Program</addtitle><description>For polyhedral constrained optimization problems and a feasible point
x
, it is shown that the projection of the negative gradient on the tangent cone, denoted
∇
Ω
f
(
x
)
, has an orthogonal decomposition of the form
β
(
x
)
+
φ
(
x
)
. At a stationary point,
∇
Ω
f
(
x
)
=
0
so
‖
∇
Ω
f
(
x
)
‖
reflects the distance to a stationary point. Away from a stationary point,
‖
β
(
x
)
‖
and
‖
φ
(
x
)
‖
measure different aspects of optimality since
β
(
x
)
only vanishes when the KKT multipliers at
x
have the correct sign, while
φ
(
x
)
only vanishes when
x
is a stationary point in the active manifold. As an application of the theory, an active set algorithm is developed for convex quadratic programs which adapts the flow of the algorithm based on a comparison between
‖
β
(
x
)
‖
and
‖
φ
(
x
)
‖
.</description><subject>Algorithms</subject><subject>Calculus of Variations and Optimal Control; Optimization</subject><subject>Combinatorics</subject><subject>Constraints</subject><subject>Full Length Paper</subject><subject>Mathematical and Computational Physics</subject><subject>Mathematical Methods in Physics</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Mathematics of Computing</subject><subject>Numerical Analysis</subject><subject>Optimization</subject><subject>Quadratic programming</subject><subject>Theoretical</subject><issn>0025-5610</issn><issn>1436-4646</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><sourceid>C6C</sourceid><recordid>eNp9kE1LAzEQhoMoWKt_wFPAc3Rmk-xujlL8gkIP6jlkt1mbsk1qkiL117vtCt48zWHe553hIeQa4RYBqruEgFAxKDgDVJVi6oRMUPCSiVKUp2QCUEgmS4RzcpHSGgCQ1_WEvC48zStLUzbZBW-iy3vahUh98L3z1kQattlt3PdxT7cxNL3dJPrl8opuQ79f2WU0PW2DTzka53O6JGed6ZO9-p1T8v748DZ7ZvPF08vsfs5aXvLMpESlgDcW2g6gKWpoBG-q5bI9PGd4KYTkQqraSmE41iiwlgZbAaXhLVo-JTdj7_DU586mrNdhF_1wUnPgAgo18EOqGFNtDClF2-ltdBsT9xpBH-TpUZ4e5OmjPK0GiI9QGsL-w8a_6n-oH0ZIcio</recordid><startdate>20240501</startdate><enddate>20240501</enddate><creator>di Serafino, Daniela</creator><creator>Hager, William W.</creator><creator>Toraldo, Gerardo</creator><creator>Viola, Marco</creator><general>Springer Berlin Heidelberg</general><general>Springer Nature B.V</general><scope>C6C</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>8FD</scope><scope>JQ2</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope><orcidid>https://orcid.org/0000-0002-2140-8094</orcidid></search><sort><creationdate>20240501</creationdate><title>On the stationarity for nonlinear optimization problems with polyhedral constraints</title><author>di Serafino, Daniela ; Hager, William W. ; Toraldo, Gerardo ; Viola, Marco</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c363t-5519903be0cf00b280b43b7ddc0013a3644534598e54a31814185a1c406a3c1e3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Algorithms</topic><topic>Calculus of Variations and Optimal Control; Optimization</topic><topic>Combinatorics</topic><topic>Constraints</topic><topic>Full Length Paper</topic><topic>Mathematical and Computational Physics</topic><topic>Mathematical Methods in Physics</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Mathematics of Computing</topic><topic>Numerical Analysis</topic><topic>Optimization</topic><topic>Quadratic programming</topic><topic>Theoretical</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>di Serafino, Daniela</creatorcontrib><creatorcontrib>Hager, William W.</creatorcontrib><creatorcontrib>Toraldo, Gerardo</creatorcontrib><creatorcontrib>Viola, Marco</creatorcontrib><collection>Springer Nature OA Free Journals</collection><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Technology Research Database</collection><collection>ProQuest Computer Science 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><jtitle>Mathematical programming</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>di Serafino, Daniela</au><au>Hager, William W.</au><au>Toraldo, Gerardo</au><au>Viola, Marco</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On the stationarity for nonlinear optimization problems with polyhedral constraints</atitle><jtitle>Mathematical programming</jtitle><stitle>Math. Program</stitle><date>2024-05-01</date><risdate>2024</risdate><volume>205</volume><issue>1-2</issue><spage>107</spage><epage>134</epage><pages>107-134</pages><issn>0025-5610</issn><eissn>1436-4646</eissn><abstract>For polyhedral constrained optimization problems and a feasible point
x
, it is shown that the projection of the negative gradient on the tangent cone, denoted
∇
Ω
f
(
x
)
, has an orthogonal decomposition of the form
β
(
x
)
+
φ
(
x
)
. At a stationary point,
∇
Ω
f
(
x
)
=
0
so
‖
∇
Ω
f
(
x
)
‖
reflects the distance to a stationary point. Away from a stationary point,
‖
β
(
x
)
‖
and
‖
φ
(
x
)
‖
measure different aspects of optimality since
β
(
x
)
only vanishes when the KKT multipliers at
x
have the correct sign, while
φ
(
x
)
only vanishes when
x
is a stationary point in the active manifold. As an application of the theory, an active set algorithm is developed for convex quadratic programs which adapts the flow of the algorithm based on a comparison between
‖
β
(
x
)
‖
and
‖
φ
(
x
)
‖
.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/s10107-023-01979-9</doi><tpages>28</tpages><orcidid>https://orcid.org/0000-0002-2140-8094</orcidid><oa>free_for_read</oa></addata></record> |
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| issn | 0025-5610 1436-4646 |
| language | eng |
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| source | Springer Nature - Complete Springer Journals |
| subjects | Algorithms Calculus of Variations and Optimal Control Optimization Combinatorics Constraints Full Length Paper Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Optimization Quadratic programming Theoretical |
| title | On the stationarity for nonlinear optimization problems with polyhedral constraints |
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