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Evaluating approximations of the semidefinite cone with trace normalized distance
We evaluate the dual cone of the set of diagonally dominant matrices (resp., scaled diagonally dominant matrices), namely DD n ∗ (resp., SDD n ∗ ), as an approximation of the semidefinite cone. We prove that the norm normalized distance, proposed by Blekherman et al. [ 5 ], between a set S and the s...
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| Published in: | Optimization letters 2023-05, Vol.17 (4), p.917-934 |
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| Main Authors: | , |
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| Language: | English |
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| container_end_page | 934 |
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| container_title | Optimization letters |
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| creator | Wang, Yuzhu Yoshise, Akiko |
| description | We evaluate the dual cone of the set of diagonally dominant matrices (resp., scaled diagonally dominant matrices), namely
DD
n
∗
(resp.,
SDD
n
∗
), as an approximation of the semidefinite cone. We prove that the norm normalized distance, proposed by Blekherman et al. [
5
], between a set
S
and the semidefinite cone has the same value whenever
SDD
n
∗
⊆
S
⊆
DD
n
∗
. This implies that the norm normalized distance is not a sufficient measure to evaluate these approximations. As a new measure to compensate for the weakness of that distance, we propose a new distance, called the trace normalized distance. We prove that the trace normalized distance between
DD
n
∗
and
S
+
n
has a different value from the one between
SDD
n
∗
and
S
+
n
and give the exact values of these distances. |
| doi_str_mv | 10.1007/s11590-022-01908-3 |
| format | article |
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DD
n
∗
(resp.,
SDD
n
∗
), as an approximation of the semidefinite cone. We prove that the norm normalized distance, proposed by Blekherman et al. [
5
], between a set
S
and the semidefinite cone has the same value whenever
SDD
n
∗
⊆
S
⊆
DD
n
∗
. This implies that the norm normalized distance is not a sufficient measure to evaluate these approximations. As a new measure to compensate for the weakness of that distance, we propose a new distance, called the trace normalized distance. We prove that the trace normalized distance between
DD
n
∗
and
S
+
n
has a different value from the one between
SDD
n
∗
and
S
+
n
and give the exact values of these distances.</description><identifier>ISSN: 1862-4472</identifier><identifier>EISSN: 1862-4480</identifier><identifier>DOI: 10.1007/s11590-022-01908-3</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer Berlin Heidelberg</publisher><subject>Computational Intelligence ; Mathematics ; Mathematics and Statistics ; Numerical and Computational Physics ; Operations Research/Decision Theory ; Optimization ; Original Paper ; Simulation</subject><ispartof>Optimization letters, 2023-05, Vol.17 (4), p.917-934</ispartof><rights>The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2022</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c242t-62ec2c425011a1bc4f400f7b8555fd6086c1c42a779fa6f30e963d98bb35b90f3</cites><orcidid>0000-0002-1073-9138</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,776,780,27901,27902</link.rule.ids></links><search><creatorcontrib>Wang, Yuzhu</creatorcontrib><creatorcontrib>Yoshise, Akiko</creatorcontrib><title>Evaluating approximations of the semidefinite cone with trace normalized distance</title><title>Optimization letters</title><addtitle>Optim Lett</addtitle><description>We evaluate the dual cone of the set of diagonally dominant matrices (resp., scaled diagonally dominant matrices), namely
DD
n
∗
(resp.,
SDD
n
∗
), as an approximation of the semidefinite cone. We prove that the norm normalized distance, proposed by Blekherman et al. [
5
], between a set
S
and the semidefinite cone has the same value whenever
SDD
n
∗
⊆
S
⊆
DD
n
∗
. This implies that the norm normalized distance is not a sufficient measure to evaluate these approximations. As a new measure to compensate for the weakness of that distance, we propose a new distance, called the trace normalized distance. We prove that the trace normalized distance between
DD
n
∗
and
S
+
n
has a different value from the one between
SDD
n
∗
and
S
+
n
and give the exact values of these distances.</description><subject>Computational Intelligence</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Numerical and Computational Physics</subject><subject>Operations Research/Decision Theory</subject><subject>Optimization</subject><subject>Original Paper</subject><subject>Simulation</subject><issn>1862-4472</issn><issn>1862-4480</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2023</creationdate><recordtype>article</recordtype><recordid>eNp9kNtOAyEQhonRxFp9Aa94AXSAXXa5NE09JE2MiV4TloWWpoUGqKenF63x0quZycw3-fMhdEnhigJ015nSVgIBxghQCT3hR2hCe8FI0_Rw_Nd37BSd5bwGEJRKOUFP81e92eviwxLr3S7Fd7-tUwwZR4fLyuJst360zgdfLDYxWPzmywqXpI3FIaat3vhPO-LR56KDsefoxOlNthe_dYpebufPs3uyeLx7mN0siGENK0Qwa5hpWAuUajqYxjUArhv6tm3dKKAXhta17jrptHAcrBR8lP0w8HaQ4PgUscNfk2LOyTq1SzV7-lAU1LcUdZCiqhT1I0XxCvEDlOtxWNqk1nGfQs35H_UFD7lmLQ</recordid><startdate>20230501</startdate><enddate>20230501</enddate><creator>Wang, Yuzhu</creator><creator>Yoshise, Akiko</creator><general>Springer Berlin Heidelberg</general><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0002-1073-9138</orcidid></search><sort><creationdate>20230501</creationdate><title>Evaluating approximations of the semidefinite cone with trace normalized distance</title><author>Wang, Yuzhu ; Yoshise, Akiko</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c242t-62ec2c425011a1bc4f400f7b8555fd6086c1c42a779fa6f30e963d98bb35b90f3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2023</creationdate><topic>Computational Intelligence</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Numerical and Computational Physics</topic><topic>Operations Research/Decision Theory</topic><topic>Optimization</topic><topic>Original Paper</topic><topic>Simulation</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Wang, Yuzhu</creatorcontrib><creatorcontrib>Yoshise, Akiko</creatorcontrib><collection>CrossRef</collection><jtitle>Optimization letters</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Wang, Yuzhu</au><au>Yoshise, Akiko</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Evaluating approximations of the semidefinite cone with trace normalized distance</atitle><jtitle>Optimization letters</jtitle><stitle>Optim Lett</stitle><date>2023-05-01</date><risdate>2023</risdate><volume>17</volume><issue>4</issue><spage>917</spage><epage>934</epage><pages>917-934</pages><issn>1862-4472</issn><eissn>1862-4480</eissn><abstract>We evaluate the dual cone of the set of diagonally dominant matrices (resp., scaled diagonally dominant matrices), namely
DD
n
∗
(resp.,
SDD
n
∗
), as an approximation of the semidefinite cone. We prove that the norm normalized distance, proposed by Blekherman et al. [
5
], between a set
S
and the semidefinite cone has the same value whenever
SDD
n
∗
⊆
S
⊆
DD
n
∗
. This implies that the norm normalized distance is not a sufficient measure to evaluate these approximations. As a new measure to compensate for the weakness of that distance, we propose a new distance, called the trace normalized distance. We prove that the trace normalized distance between
DD
n
∗
and
S
+
n
has a different value from the one between
SDD
n
∗
and
S
+
n
and give the exact values of these distances.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/s11590-022-01908-3</doi><tpages>18</tpages><orcidid>https://orcid.org/0000-0002-1073-9138</orcidid></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 1862-4472 |
| ispartof | Optimization letters, 2023-05, Vol.17 (4), p.917-934 |
| issn | 1862-4472 1862-4480 |
| language | eng |
| recordid | cdi_crossref_primary_10_1007_s11590_022_01908_3 |
| source | Springer Nature |
| subjects | Computational Intelligence Mathematics Mathematics and Statistics Numerical and Computational Physics Operations Research/Decision Theory Optimization Original Paper Simulation |
| title | Evaluating approximations of the semidefinite cone with trace normalized distance |
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