Smoothed analysis of condition numbers and complexity implications for linear programming
We perform a smoothed analysis of Renegar’s condition number for linear programming by analyzing the distribution of the distance to ill-posedness of a linear program subject to a slight Gaussian perturbation. In particular, we show that for every n -by- d matrix Ā, n -vector , and d -vector satisfy...
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| Veröffentlicht in: | Mathematical programming 2011-02, Vol.126 (2), p.315-350 |
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| creator | Dunagan, John Spielman, Daniel A. Teng, Shang-Hua |
| description | We perform a smoothed analysis of Renegar’s condition number for linear programming by analyzing the distribution of the distance to ill-posedness of a linear program subject to a slight Gaussian perturbation. In particular, we show that for every
n
-by-
d
matrix Ā,
n
-vector
, and
d
-vector
satisfying
and every
σ
≤ 1,
where
A
,
b
and
c
are Gaussian perturbations of Ā,
and
of variance
σ
2
and C (
A
,
b
,
c
) is the condition number of the linear program defined by (
A
,
b
,
c
). From this bound, we obtain a smoothed analysis of interior point algorithms. By combining this with the smoothed analysis of finite termination of Spielman and Teng (Math. Prog. Ser. B, 2003), we show that the smoothed complexity of interior point algorithms for linear programming is
O
(
n
3
log(
nd
/
σ
)). |
| doi_str_mv | 10.1007/s10107-009-0278-5 |
| format | Article |
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n
-by-
d
matrix Ā,
n
-vector
, and
d
-vector
satisfying
and every
σ
≤ 1,
where
A
,
b
and
c
are Gaussian perturbations of Ā,
and
of variance
σ
2
and C (
A
,
b
,
c
) is the condition number of the linear program defined by (
A
,
b
,
c
). From this bound, we obtain a smoothed analysis of interior point algorithms. By combining this with the smoothed analysis of finite termination of Spielman and Teng (Math. Prog. Ser. B, 2003), we show that the smoothed complexity of interior point algorithms for linear programming is
O
(
n
3
log(
nd
/
σ
)).</description><identifier>ISSN: 0025-5610</identifier><identifier>EISSN: 1436-4646</identifier><identifier>DOI: 10.1007/s10107-009-0278-5</identifier><identifier>CODEN: MHPGA4</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer-Verlag</publisher><subject>Algorithms ; Applied sciences ; C (programming language) ; Calculus of Variations and Optimal Control; Optimization ; Combinatorics ; Complexity ; Computer science ; Data smoothing ; Exact sciences and technology ; Full Length Paper ; Gaussian ; Linear programming ; Mathematical analysis ; Mathematical and Computational Physics ; Mathematical Methods in Physics ; Mathematical programming ; Mathematics ; Mathematics and Statistics ; Mathematics of Computing ; Numerical Analysis ; Operational research and scientific management ; Operational research. Management science ; Perturbation methods ; Random variables ; Simplex method ; Studies ; Theoretical</subject><ispartof>Mathematical programming, 2011-02, Vol.126 (2), p.315-350</ispartof><rights>Springer and Mathematical Programming Society 2009</rights><rights>2015 INIST-CNRS</rights><rights>Springer and Mathematical Optimization Society 2011</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c377t-83e72df369d546d0f9c3300dfddeca787fcb5924f8070648965ba4c2084e11223</citedby><cites>FETCH-LOGICAL-c377t-83e72df369d546d0f9c3300dfddeca787fcb5924f8070648965ba4c2084e11223</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/s10107-009-0278-5$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s10107-009-0278-5$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27924,27925,41488,42557,51319</link.rule.ids><backlink>$$Uhttp://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=23790124$$DView record in Pascal Francis$$Hfree_for_read</backlink></links><search><creatorcontrib>Dunagan, John</creatorcontrib><creatorcontrib>Spielman, Daniel A.</creatorcontrib><creatorcontrib>Teng, Shang-Hua</creatorcontrib><title>Smoothed analysis of condition numbers and complexity implications for linear programming</title><title>Mathematical programming</title><addtitle>Math. Program</addtitle><description>We perform a smoothed analysis of Renegar’s condition number for linear programming by analyzing the distribution of the distance to ill-posedness of a linear program subject to a slight Gaussian perturbation. In particular, we show that for every
n
-by-
d
matrix Ā,
n
-vector
, and
d
-vector
satisfying
and every
σ
≤ 1,
where
A
,
b
and
c
are Gaussian perturbations of Ā,
and
of variance
σ
2
and C (
A
,
b
,
c
) is the condition number of the linear program defined by (
A
,
b
,
c
). From this bound, we obtain a smoothed analysis of interior point algorithms. By combining this with the smoothed analysis of finite termination of Spielman and Teng (Math. Prog. Ser. B, 2003), we show that the smoothed complexity of interior point algorithms for linear programming is
O
(
n
3
log(
nd
/
σ
)).</description><subject>Algorithms</subject><subject>Applied sciences</subject><subject>C (programming language)</subject><subject>Calculus of Variations and Optimal Control; Optimization</subject><subject>Combinatorics</subject><subject>Complexity</subject><subject>Computer science</subject><subject>Data smoothing</subject><subject>Exact sciences and technology</subject><subject>Full Length Paper</subject><subject>Gaussian</subject><subject>Linear programming</subject><subject>Mathematical analysis</subject><subject>Mathematical and Computational Physics</subject><subject>Mathematical Methods in Physics</subject><subject>Mathematical programming</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Mathematics of Computing</subject><subject>Numerical Analysis</subject><subject>Operational research and scientific management</subject><subject>Operational research. Management science</subject><subject>Perturbation methods</subject><subject>Random variables</subject><subject>Simplex method</subject><subject>Studies</subject><subject>Theoretical</subject><issn>0025-5610</issn><issn>1436-4646</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2011</creationdate><recordtype>article</recordtype><sourceid>ABUWG</sourceid><sourceid>AFKRA</sourceid><sourceid>AZQEC</sourceid><sourceid>BENPR</sourceid><sourceid>CCPQU</sourceid><sourceid>DWQXO</sourceid><sourceid>GNUQQ</sourceid><recordid>eNp1kEtrFEEQxxsx4Jr4AbwNgniaWP2Yfhwl-AgEPJgcPDW9_Vg7zHSvXbPgfnt72aAgeKqi6lf_qvoT8prCNQVQ75ECBTUCmBGY0uP0jGyo4HIUUsjnZAPApnGSFF6Ql4iPAEC51hvy_dtS6_ojhsEVNx8x41DT4GsJec21DOWwbGPD3g29uuzn-CuvxyH3LHt3QnBItQ1zLtG1Yd_qrrllyWV3RS6SmzG-eoqX5OHTx_ubL-Pd18-3Nx_uRs-VWkfNo2IhcWnCJGSAZDznACGFEL1TWiW_nQwTSYMCKbSR09YJz0CLSClj_JK8O-v23T8PEVe7ZPRxnl2J9YDWgDKcS2k6-eYf8rEeWn8breZUUsk0dIieId8qYovJ7lteXDtaCvZktT1bbbvV9mS1nfrM2ydhh97NqbniM_4ZZFwZoEx0jp057K2yi-3vAf8X_w2QI45-</recordid><startdate>20110201</startdate><enddate>20110201</enddate><creator>Dunagan, John</creator><creator>Spielman, Daniel A.</creator><creator>Teng, Shang-Hua</creator><general>Springer-Verlag</general><general>Springer</general><general>Springer Nature B.V</general><scope>IQODW</scope><scope>AAYXX</scope><scope>CITATION</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>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>HCIFZ</scope><scope>JQ2</scope><scope>K60</scope><scope>K6~</scope><scope>K7-</scope><scope>L.-</scope><scope>L6V</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope><scope>M0C</scope><scope>M0N</scope><scope>M2P</scope><scope>M7S</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>20110201</creationdate><title>Smoothed analysis of condition numbers and complexity implications for linear programming</title><author>Dunagan, John ; Spielman, Daniel A. ; Teng, Shang-Hua</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c377t-83e72df369d546d0f9c3300dfddeca787fcb5924f8070648965ba4c2084e11223</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2011</creationdate><topic>Algorithms</topic><topic>Applied sciences</topic><topic>C (programming language)</topic><topic>Calculus of Variations and Optimal Control; Optimization</topic><topic>Combinatorics</topic><topic>Complexity</topic><topic>Computer science</topic><topic>Data smoothing</topic><topic>Exact sciences and technology</topic><topic>Full Length Paper</topic><topic>Gaussian</topic><topic>Linear programming</topic><topic>Mathematical analysis</topic><topic>Mathematical and Computational Physics</topic><topic>Mathematical Methods in Physics</topic><topic>Mathematical programming</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Mathematics of Computing</topic><topic>Numerical Analysis</topic><topic>Operational research and scientific management</topic><topic>Operational research. Management science</topic><topic>Perturbation methods</topic><topic>Random variables</topic><topic>Simplex method</topic><topic>Studies</topic><topic>Theoretical</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Dunagan, John</creatorcontrib><creatorcontrib>Spielman, Daniel A.</creatorcontrib><creatorcontrib>Teng, Shang-Hua</creatorcontrib><collection>Pascal-Francis</collection><collection>CrossRef</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 Collection</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>Materials Science & Engineering Collection</collection><collection>ProQuest Central (Alumni)</collection><collection>ProQuest Central</collection><collection>Advanced Technologies & 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</collection><collection>Business Premium Collection (Alumni)</collection><collection>ABI/INFORM Global (Corporate)</collection><collection>ProQuest Central Student</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>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>Science Database</collection><collection>Engineering Database</collection><collection>Advanced Technologies & Aerospace Database</collection><collection>ProQuest Advanced Technologies & 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>Mathematical programming</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Dunagan, John</au><au>Spielman, Daniel A.</au><au>Teng, Shang-Hua</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Smoothed analysis of condition numbers and complexity implications for linear programming</atitle><jtitle>Mathematical programming</jtitle><stitle>Math. Program</stitle><date>2011-02-01</date><risdate>2011</risdate><volume>126</volume><issue>2</issue><spage>315</spage><epage>350</epage><pages>315-350</pages><issn>0025-5610</issn><eissn>1436-4646</eissn><coden>MHPGA4</coden><abstract>We perform a smoothed analysis of Renegar’s condition number for linear programming by analyzing the distribution of the distance to ill-posedness of a linear program subject to a slight Gaussian perturbation. In particular, we show that for every
n
-by-
d
matrix Ā,
n
-vector
, and
d
-vector
satisfying
and every
σ
≤ 1,
where
A
,
b
and
c
are Gaussian perturbations of Ā,
and
of variance
σ
2
and C (
A
,
b
,
c
) is the condition number of the linear program defined by (
A
,
b
,
c
). From this bound, we obtain a smoothed analysis of interior point algorithms. By combining this with the smoothed analysis of finite termination of Spielman and Teng (Math. Prog. Ser. B, 2003), we show that the smoothed complexity of interior point algorithms for linear programming is
O
(
n
3
log(
nd
/
σ
)).</abstract><cop>Berlin/Heidelberg</cop><pub>Springer-Verlag</pub><doi>10.1007/s10107-009-0278-5</doi><tpages>36</tpages></addata></record> |
| fulltext | fulltext |
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| ispartof | Mathematical programming, 2011-02, Vol.126 (2), p.315-350 |
| issn | 0025-5610 1436-4646 |
| language | eng |
| recordid | cdi_proquest_miscellaneous_907933669 |
| source | SpringerNature Journals; EBSCOhost Business Source Complete |
| subjects | Algorithms Applied sciences C (programming language) Calculus of Variations and Optimal Control Optimization Combinatorics Complexity Computer science Data smoothing Exact sciences and technology Full Length Paper Gaussian Linear programming Mathematical analysis Mathematical and Computational Physics Mathematical Methods in Physics Mathematical programming Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Operational research and scientific management Operational research. Management science Perturbation methods Random variables Simplex method Studies Theoretical |
| title | Smoothed analysis of condition numbers and complexity implications for linear programming |
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