Convergent Outer Approximation Algorithms for Solving Unary Programs

Interesting cutting plane approaches for solving certain difficult multiextremal global optimization problems can fail to converge. Examples include the concavity cut method for concave minimization and Ramana's recent outer approximation method for unary programs which are linear programming p...

Ausführliche Beschreibung

Gespeichert in:
Bibliographische Detailangaben
Veröffentlicht in:Journal of global optimization Jg. 13; H. 2; S. 123 - 149
Hauptverfasser: Horst, Reiner, Raber, Ulrich
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Dordrecht Springer Nature B.V 01.09.1998
Schlagworte:
Algorithms
Approximation
Linear programming
Optimization
ISSN:0925-5001, 1573-2916
Online-Zugang:Volltext
Tags: Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
Abstract Interesting cutting plane approaches for solving certain difficult multiextremal global optimization problems can fail to converge. Examples include the concavity cut method for concave minimization and Ramana's recent outer approximation method for unary programs which are linear programming problems with an additional constraint requiring that an affine mapping becomes unary. For the latter problem class, new convergent outer approximation algorithms are proposed which are based on sufficiently deep l∞-norm or quadratic cuts. Implementable versions construct optimal simplicial inner approximations of Euclidean balls and of intersections of Euclidean balls with halfspaces, which are of general interest in computational convexity. Computational behavior of the algorithms depends crucially on the matrices involved in the unary condition. Potential applications to the global minimization of indefinite quadratic functions subject to indefinite quadratic constraints are shown to be practical only for very small problem sizes.
AbstractList Interesting cutting plane approaches for solving certain difficult multiextremal global optimization problems can fail to converge. Examples include the concavity cut method for concave minimization and Ramana's recent outer approximation method for unary programs which are linear programming problems with an additional constraint requiring that an affine mapping becomes unary. For the latter problem class, new convergent outer approximation algorithms are proposed which are based on sufficiently deep l∞-norm or quadratic cuts. Implementable versions construct optimal simplicial inner approximations of Euclidean balls and of intersections of Euclidean balls with halfspaces, which are of general interest in computational convexity. Computational behavior of the algorithms depends crucially on the matrices involved in the unary condition. Potential applications to the global minimization of indefinite quadratic functions subject to indefinite quadratic constraints are shown to be practical only for very small problem sizes.
Author Horst, Reiner
Raber, Ulrich
Author_xml – sequence: 1
  givenname: Reiner
  surname: Horst
  fullname: Horst, Reiner
– sequence: 2
  givenname: Ulrich
  surname: Raber
  fullname: Raber, Ulrich
BookMark eNp1kD1PwzAQhi1UJNrCzBqxh57tOLHZovIpVSoSdI4cxw6pUrvYbkX_PYGygMR0y_vc3ftM0Mg6qxG6xHCNgdBZeYMBOOEZkJxwcYLGmBU0JQLnIzQGQVjKAPAZmoSwBgDBGRmj27mze-1bbWOy3EXtk3K79e6j28jYOZuUfet8F982ITHOJy-u33e2TVZW-kPy7F3r5Saco1Mj-6AvfuYUre7vXueP6WL58DQvF6kieR5TYpgWmstaU9NkXNVUZoUuwOC6biDHeUYVB6yaosGSKqxp3bBMNMzgzKhG0im6Ou4dPnzf6RCrtdt5O5yssMjyYujLhxA7hpR3IXhtKtXF7zLRy66vMFRfvqqy-uVr4GZ_uK0fLPjDv8QnNLVu5Q
CitedBy_id crossref_primary_10_1023_A_1020383700229
crossref_primary_10_1016_j_amc_2006_08_028
crossref_primary_10_1016_j_mcm_2008_04_004
crossref_primary_10_1023_A_1008377529330
Cites_doi 10.1007/978-3-662-03199-5
10.1007/BF01099462
10.1137/1038003
10.2140/pjm.1969.31.795
10.1007/BF02574058
10.1017/CBO9780511810817
10.1007/BF01100205
ContentType Journal Article
Copyright Kluwer Academic Publishers 1998
Copyright_xml – notice: Kluwer Academic Publishers 1998
DBID AAYXX
CITATION
3V.
7SC
7WY
7WZ
7XB
87Z
88I
8AL
8AO
8FD
8FE
8FG
8FK
8FL
8G5
ABJCF
ABUWG
AFKRA
ARAPS
AZQEC
BENPR
BEZIV
BGLVJ
CCPQU
DWQXO
FRNLG
F~G
GNUQQ
GUQSH
HCIFZ
JQ2
K60
K6~
K7-
L.-
L6V
L7M
L~C
L~D
M0C
M0N
M2O
M2P
M7S
MBDVC
P5Z
P62
PHGZM
PHGZT
PKEHL
PQBIZ
PQBZA
PQEST
PQGLB
PQQKQ
PQUKI
PRINS
PTHSS
Q9U
DOI 10.1023/A:1008284026289
DatabaseName CrossRef
ProQuest Central (Corporate)
Computer and Information Systems Abstracts
ABI/INFORM Collection
ABI/INFORM Global (PDF only)
ProQuest Central (purchase pre-March 2016)
ABI/INFORM Collection
Science Database (Alumni Edition)
Computing Database (Alumni Edition)
ProQuest Pharma Collection
Technology Research Database
ProQuest SciTech Collection
ProQuest Technology Collection
ProQuest Central (Alumni) (purchase pre-March 2016)
ABI/INFORM Collection (Alumni Edition)
Research Library (Alumni Edition)
Materials Science & Engineering Collection
ProQuest Central (Alumni)
ProQuest Central UK/Ireland
Advanced Technologies & Computer Science Collection
ProQuest Central Essentials
ProQuest Central
Business Premium Collection
Technology collection
ProQuest One
ProQuest Central
Business Premium Collection (Alumni)
ABI/INFORM Global (Corporate)
ProQuest Central Student
Research Library Prep
SciTech Premium Collection
ProQuest Computer Science Collection
ProQuest Business Collection (Alumni Edition)
ProQuest Business Collection
Computer Science Database
ABI/INFORM Professional Advanced
ProQuest Engineering Collection
Advanced Technologies Database with Aerospace
Computer and Information Systems Abstracts – Academic
Computer and Information Systems Abstracts Professional
ABI/INFORM Global (OCUL)
Computing Database
Research Library
Science Database
Engineering Database
Research Library (Corporate)
Advanced Technologies & Aerospace Database
ProQuest Advanced Technologies & Aerospace Collection
ProQuest Central Premium
ProQuest One Academic (New)
ProQuest One Academic Middle East (New)
One Business (ProQuest)
ProQuest One Business (Alumni)
ProQuest One Academic Eastern Edition (DO NOT USE)
ProQuest One Applied & Life Sciences
ProQuest One Academic (retired)
ProQuest One Academic UKI Edition
ProQuest Central China
Engineering Collection
ProQuest Central Basic
DatabaseTitle CrossRef
ProQuest Business Collection (Alumni Edition)
Research Library Prep
Computer Science Database
ProQuest Central Student
ProQuest Advanced Technologies & Aerospace Collection
ProQuest Central Essentials
ProQuest Computer Science Collection
Computer and Information Systems Abstracts
SciTech Premium Collection
ProQuest Central China
ABI/INFORM Complete
ProQuest One Applied & Life Sciences
ProQuest Central (New)
Engineering Collection
Advanced Technologies & Aerospace Collection
Business Premium Collection
ABI/INFORM Global
Engineering Database
ProQuest Science Journals (Alumni Edition)
ProQuest One Academic Eastern Edition
ProQuest Technology Collection
ProQuest Business Collection
ProQuest One Academic UKI Edition
ProQuest One Academic
ProQuest One Academic (New)
ABI/INFORM Global (Corporate)
ProQuest One Business
Technology Collection
Technology Research Database
Computer and Information Systems Abstracts – Academic
ProQuest One Academic Middle East (New)
ProQuest Central (Alumni Edition)
ProQuest One Community College
Research Library (Alumni Edition)
ProQuest Pharma Collection
ProQuest Central
ABI/INFORM Professional Advanced
ProQuest Engineering Collection
ProQuest Central Korea
ProQuest Research Library
Advanced Technologies Database with Aerospace
ABI/INFORM Complete (Alumni Edition)
ProQuest Computing
ABI/INFORM Global (Alumni Edition)
ProQuest Central Basic
ProQuest Science Journals
ProQuest Computing (Alumni Edition)
ProQuest SciTech Collection
Computer and Information Systems Abstracts Professional
Advanced Technologies & Aerospace Database
Materials Science & Engineering Collection
ProQuest One Business (Alumni)
ProQuest Central (Alumni)
Business Premium Collection (Alumni)
DatabaseTitleList ProQuest Business Collection (Alumni Edition)
Database_xml – sequence: 1
  dbid: BENPR
  name: ProQuest Central
  url: https://www.proquest.com/central
  sourceTypes: Aggregation Database
DeliveryMethod fulltext_linktorsrc
Discipline Engineering
Sciences (General)
EISSN 1573-2916
EndPage 149
ExternalDocumentID 945079931
10_1023_A_1008284026289
GroupedDBID -Y2
-~C
.4S
.86
.DC
.VR
06D
0R~
0VY
199
1N0
1SB
2.D
203
28-
29K
2J2
2JN
2JY
2KG
2LR
2P1
2VQ
2~H
30V
4.4
406
408
409
40D
40E
5GY
5QI
5VS
67Z
6NX
78A
7WY
88I
8AO
8FE
8FG
8FL
8G5
8TC
8UJ
95-
95.
95~
96X
AABHQ
AACDK
AAHNG
AAIAL
AAJBT
AAJKR
AANZL
AAPKM
AARHV
AARTL
AASML
AATNV
AATVU
AAUYE
AAWCG
AAYIU
AAYQN
AAYTO
AAYXX
ABAKF
ABBBX
ABBRH
ABBXA
ABDBE
ABDZT
ABECU
ABFSG
ABFTV
ABHLI
ABHQN
ABJCF
ABJNI
ABJOX
ABKCH
ABKTR
ABMNI
ABMQK
ABNWP
ABQBU
ABQSL
ABRTQ
ABSXP
ABTEG
ABTHY
ABTKH
ABTMW
ABULA
ABUWG
ABWNU
ABXPI
ACAOD
ACBXY
ACDTI
ACGFS
ACGOD
ACHSB
ACHXU
ACIWK
ACKNC
ACMDZ
ACMLO
ACOKC
ACOMO
ACPIV
ACSTC
ACZOJ
ADHHG
ADHIR
ADHKG
ADIMF
ADKNI
ADKPE
ADRFC
ADTPH
ADURQ
ADYFF
ADZKW
AEBTG
AEFIE
AEFQL
AEGAL
AEGNC
AEJHL
AEJRE
AEKMD
AEMSY
AENEX
AEOHA
AEPYU
AETLH
AEVLU
AEXYK
AEZWR
AFBBN
AFDZB
AFEXP
AFFHD
AFGCZ
AFHIU
AFKRA
AFLOW
AFOHR
AFQWF
AFWTZ
AFZKB
AGAYW
AGDGC
AGGDS
AGJBK
AGMZJ
AGQEE
AGQMX
AGQPQ
AGRTI
AGWIL
AGWZB
AGYKE
AHAVH
AHBYD
AHKAY
AHPBZ
AHSBF
AHWEU
AHYZX
AI.
AIAKS
AIGIU
AIIXL
AILAN
AITGF
AIXLP
AJBLW
AJRNO
AJZVZ
ALMA_UNASSIGNED_HOLDINGS
ALWAN
AMKLP
AMVHM
AMXSW
AMYLF
AMYQR
AOCGG
ARAPS
ARCSS
ARMRJ
ASPBG
ATHPR
AVWKF
AXYYD
AYFIA
AYQZM
AZFZN
AZQEC
B-.
BA0
BAPOH
BBWZM
BDATZ
BENPR
BEZIV
BGLVJ
BGNMA
BPHCQ
BSONS
CAG
CCPQU
CITATION
COF
CS3
CSCUP
D-I
DDRTE
DL5
DNIVK
DPUIP
DU5
DWQXO
EBLON
EBS
EDO
EIOEI
EJD
ESBYG
FEDTE
FERAY
FFXSO
FIGPU
FINBP
FNLPD
FRNLG
FRRFC
FSGXE
FWDCC
GGCAI
GGRSB
GJIRD
GNUQQ
GNWQR
GQ7
GQ8
GUQSH
GXS
H13
HCIFZ
HF~
HG5
HG6
HMJXF
HQYDN
HRMNR
HVGLF
HZ~
I09
IAO
IHE
IJ-
IKXTQ
ITC
ITM
IWAJR
IXC
IZIGR
IZQ
I~X
I~Z
J-C
J0Z
JBSCW
JCJTX
JZLTJ
K60
K6V
K6~
K7-
KDC
KOV
KOW
L6V
LAK
LLZTM
M0C
M2O
M2P
M4Y
M7S
MA-
N2Q
N9A
NB0
NDZJH
NPVJJ
NQJWS
NU0
O9-
O93
O9G
O9I
O9J
OAM
OVD
P19
P2P
P62
P9M
PF0
PHGZM
PHGZT
PQBIZ
PQBZA
PQGLB
PQQKQ
PROAC
PT4
PT5
PTHSS
Q2X
QOK
QOS
R4E
R89
R9I
RHV
RNI
RNS
ROL
RPX
RSV
RZC
RZE
RZK
S16
S1Z
S26
S27
S28
S3B
SAP
SBE
SCLPG
SDD
SDH
SDM
SHX
SISQX
SJYHP
SNE
SNPRN
SNX
SOHCF
SOJ
SPISZ
SRMVM
SSLCW
STPWE
SZN
T13
T16
TEORI
TN5
TSG
TSK
TSV
TUC
TUS
U2A
UG4
UOJIU
UTJUX
UZXMN
VC2
VFIZW
VH1
W23
W48
WK8
YLTOR
Z45
ZMTXR
ZWQNP
ZY4
~EX
3V.
7SC
7XB
8AL
8FD
8FK
AESKC
JQ2
L.-
L7M
L~C
L~D
M0N
MBDVC
PKEHL
PQEST
PQUKI
PRINS
Q9U
ID FETCH-LOGICAL-c266t-2f5e9e8abe3fd48cb3a47e70f1bbd061643c801cd7d1a3c1e3bd549d5f14fcda3
IEDL.DBID M2P
ISICitedReferencesCount 4
ISICitedReferencesURI http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000076350500002&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D
ISSN 0925-5001
IngestDate Tue Nov 04 16:55:58 EST 2025
Sat Nov 29 07:50:52 EST 2025
Tue Nov 18 20:44:46 EST 2025
IsPeerReviewed true
IsScholarly true
Issue 2
Language English
License https://www.springernature.com/gp/researchers/text-and-data-mining
LinkModel DirectLink
MergedId FETCHMERGED-LOGICAL-c266t-2f5e9e8abe3fd48cb3a47e70f1bbd061643c801cd7d1a3c1e3bd549d5f14fcda3
Notes ObjectType-Article-1
SourceType-Scholarly Journals-1
content type line 14
PQID 194675738
PQPubID 29930
PageCount 27
ParticipantIDs proquest_journals_194675738
crossref_citationtrail_10_1023_A_1008284026289
crossref_primary_10_1023_A_1008284026289
PublicationCentury 1900
PublicationDate 1998-09-01
PublicationDateYYYYMMDD 1998-09-01
PublicationDate_xml – month: 09
  year: 1998
  text: 1998-09-01
  day: 01
PublicationDecade 1990
PublicationPlace Dordrecht
PublicationPlace_xml – name: Dordrecht
PublicationTitle Journal of global optimization
PublicationYear 1998
Publisher Springer Nature B.V
Publisher_xml – name: Springer Nature B.V
References S. Poljak (152824_CR6) 1995; 7
R. Horn (152824_CR4) 1985
D.M.Y. Sommerville (152824_CR11) 1929
P. Gritzmann (152824_CR3) 1995; 13
M. Ramana (152824_CR8) 1993
R. Horst (152824_CR5) 1996
152824_CR7
F. Al-Khayyal (152824_CR1) 1945; 6
D. Slepan (152824_CR10) 1969; 31
N.Z. Shor (152824_CR9) 1987; 25
152824_CR2
L. Vandenberghe (152824_CR12) 1996; 38
References_xml – volume-title: Global Optimization – Deterministic Methods
  year: 1996
  ident: 152824_CR5
  doi: 10.1007/978-3-662-03199-5
– volume: 6
  start-page: 215
  year: 1945
  ident: 152824_CR1
  publication-title: J. Global Optimization
  doi: 10.1007/BF01099462
– ident: 152824_CR2
– volume-title: An Introduction to the Geometry of N Dimensions
  year: 1929
  ident: 152824_CR11
– volume-title: An Algorithmic Analysis of Multiquadratic and Semidefinite Programming Problems
  year: 1993
  ident: 152824_CR8
– volume: 25
  start-page: 1
  year: 1987
  ident: 152824_CR9
  publication-title: Soviet J. Computer and Systems Sciences
– volume: 38
  start-page: 49
  year: 1996
  ident: 152824_CR12
  publication-title: SIAM Review
  doi: 10.1137/1038003
– ident: 152824_CR7
– volume: 31
  start-page: 795
  year: 1969
  ident: 152824_CR10
  publication-title: Pacific J. Math.
  doi: 10.2140/pjm.1969.31.795
– volume: 13
  start-page: 477
  year: 1995
  ident: 152824_CR3
  publication-title: Discrete Comput. Geom.
  doi: 10.1007/BF02574058
– volume-title: Matrix Analysis
  year: 1985
  ident: 152824_CR4
  doi: 10.1017/CBO9780511810817
– volume: 7
  start-page: 51
  year: 1995
  ident: 152824_CR6
  publication-title: J. Global Optimization
  doi: 10.1007/BF01100205
SSID ssj0009852
Score 1.5195649
Snippet Interesting cutting plane approaches for solving certain difficult multiextremal global optimization problems can fail to converge. Examples include the...
SourceID proquest
crossref
SourceType Aggregation Database
Enrichment Source
Index Database
StartPage 123
SubjectTerms Algorithms
Approximation
Linear programming
Optimization
Title Convergent Outer Approximation Algorithms for Solving Unary Programs
URI https://www.proquest.com/docview/194675738
Volume 13
WOSCitedRecordID wos000076350500002&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D
hasFullText 1
inHoldings 1
isFullTextHit
isPrint
journalDatabaseRights – providerCode: PRVPQU
  databaseName: ABI/INFORM Collection
  customDbUrl:
  eissn: 1573-2916
  dateEnd: 20171231
  omitProxy: false
  ssIdentifier: ssj0009852
  issn: 0925-5001
  databaseCode: 7WY
  dateStart: 19970101
  isFulltext: true
  titleUrlDefault: https://www.proquest.com/abicomplete
  providerName: ProQuest
– providerCode: PRVPQU
  databaseName: ABI/INFORM Global (OCUL)
  customDbUrl:
  eissn: 1573-2916
  dateEnd: 20171231
  omitProxy: false
  ssIdentifier: ssj0009852
  issn: 0925-5001
  databaseCode: M0C
  dateStart: 19970101
  isFulltext: true
  titleUrlDefault: https://search.proquest.com/abiglobal
  providerName: ProQuest
– providerCode: PRVPQU
  databaseName: Advanced Technologies & Aerospace Database
  customDbUrl:
  eissn: 1573-2916
  dateEnd: 20171231
  omitProxy: false
  ssIdentifier: ssj0009852
  issn: 0925-5001
  databaseCode: P5Z
  dateStart: 19970101
  isFulltext: true
  titleUrlDefault: https://search.proquest.com/hightechjournals
  providerName: ProQuest
– providerCode: PRVPQU
  databaseName: Computer Science Database
  customDbUrl:
  eissn: 1573-2916
  dateEnd: 20171231
  omitProxy: false
  ssIdentifier: ssj0009852
  issn: 0925-5001
  databaseCode: K7-
  dateStart: 19970101
  isFulltext: true
  titleUrlDefault: http://search.proquest.com/compscijour
  providerName: ProQuest
– providerCode: PRVPQU
  databaseName: Engineering Database
  customDbUrl:
  eissn: 1573-2916
  dateEnd: 20171231
  omitProxy: false
  ssIdentifier: ssj0009852
  issn: 0925-5001
  databaseCode: M7S
  dateStart: 19970101
  isFulltext: true
  titleUrlDefault: http://search.proquest.com
  providerName: ProQuest
– providerCode: PRVPQU
  databaseName: ProQuest Central
  customDbUrl:
  eissn: 1573-2916
  dateEnd: 20171231
  omitProxy: false
  ssIdentifier: ssj0009852
  issn: 0925-5001
  databaseCode: BENPR
  dateStart: 19970101
  isFulltext: true
  titleUrlDefault: https://www.proquest.com/central
  providerName: ProQuest
– providerCode: PRVPQU
  databaseName: Research Library
  customDbUrl:
  eissn: 1573-2916
  dateEnd: 20171231
  omitProxy: false
  ssIdentifier: ssj0009852
  issn: 0925-5001
  databaseCode: M2O
  dateStart: 19970101
  isFulltext: true
  titleUrlDefault: https://search.proquest.com/pqrl
  providerName: ProQuest
– providerCode: PRVPQU
  databaseName: Science Database
  customDbUrl:
  eissn: 1573-2916
  dateEnd: 20171231
  omitProxy: false
  ssIdentifier: ssj0009852
  issn: 0925-5001
  databaseCode: M2P
  dateStart: 19970101
  isFulltext: true
  titleUrlDefault: https://search.proquest.com/sciencejournals
  providerName: ProQuest
– providerCode: PRVAVX
  databaseName: SpringerLINK Contemporary 1997-Present
  customDbUrl:
  eissn: 1573-2916
  dateEnd: 99991231
  omitProxy: false
  ssIdentifier: ssj0009852
  issn: 0925-5001
  databaseCode: RSV
  dateStart: 19970101
  isFulltext: true
  titleUrlDefault: https://link.springer.com/search?facet-content-type=%22Journal%22
  providerName: Springer Nature
link http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV1LT9wwEB4B2wM9lLdKlyIfeqAHt3Fs4-RULVsQUsUSlaJCL5HjR4u03aUkoP58PImXl9ReuIwUJVGiTDwztr_5PoB3UmomKseoSoSjIkyZaV5lkgptMym89T4xrdiEGo2ys7O8iNicOsIqZzGxDdR2anCN_CNDLXipePbp8g9F0SjcXI0KGvPQC4UNQ0TXUVrcc-5mreBOkqeSyhCOnzD7IHlbmD3tpijx_jApPY7JbaI5WHrmKy7Dq1hhkkH3S6zAnJuswssHvIOrsBJHdE12Iu30-zX4PEQAOvZiNuQYhR7IAPnG_150zY1kMP4Zntb8-l2TUOiSk-kY1yLIKfbzkqKDedXrcHqw_214SKPGAjUhNTc09dLlLtOV496KzFRcC-VU4llV2ZDrQ8FiQhIzVlmmuWGOVzZMKa30THhjNd-Ahcl04l4DEaJKrNLCOR-KGrabu1RbrrjWWe4Sm2zCh9lHLk0kIEcdjHHZboSnvByUj7yyCTt3N1x23Bv_vrQ_c0kZB2Fd3vnjzX_P9mGxazRE4NgWLDRX1-4tvDA3zUV9tQ3z6vv5NvT29kfF13D0RdFgj5Ih2vS4tQVadRJsIX_cAtX42lQ
linkProvider ProQuest
linkToHtml http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMw1V3JTuQwEC2xScCBGTaxzDA-gAQHQxLbJDmgUYtFIKBpCZC4BccLtNR0Awkz8FH8I64sbBLcOHBOYsmpp1rsqvcAFoWQPk-NT0OPG8pdyUzjNBKUSx0JbrW1nirEJsJmMzo7i1t98FjPwmBbZe0TC0etewrPyNd81IIXIYv-Xt9QFI3Cy9VaQaNExb55-O8qtmxjb8uZdykIdrZPNndpJSpAlYtFOQ2sMLGJZGqY1TxSKZM8NKFn_TTVLri5CK2c11Y61L5kyjcs1a6G0sL63CotmVu3HwY5Eothp2DQeuH4jQqBHy8OBBXO_b9jEkKyOFetrQcoKf86CL6NAUVg2_nxzX7JTxirMmjSKCE_Dn2mOwGjr3gVJ2C88lgZWa5otVcmYWsTG-xx1jQnRyhkQRrIp37fLoc3SaNz4XaXX15lxCXy5LjXwbMWcorzyqRVtrFlU3D6JXubhoFur2tmgHCeejqU3BjrkjZ_PTaB1CxkUkax8bQ3C6u1URNVEayjzkcnKS76A5Y0kjcomIXl5w-uS26Rj1-dryGQVE4mS57tP_fp0z8wvHtyeJAc7DX352GkHKrEJrlfMJDf3pnfMKT-5e3sdqGAM4HzrwbLEy9BM4k
linkToPdf http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMw1V3LThsxFL2CgFC7gPKomgaoFyDBwmRmbOOZRVVFCVERKEQCJMRm8PgBkUKSMtMCn9a_qz0PXhLsWHQ9D8nj43t8PfeeA7DBmPBpon3MPaoxtSkzjpKQYSpUyKhRxngyN5vgvV54dhb1p-Bv1QvjyiqrmJgHajWW7oy86TsveMZJ2DRlVUS_0_0x-YWdgZT70Vq5aRQIOdD3tzZ7S7_vd-xUbwZBd--k_ROXBgNYWl7KcGCYjnQoEk2MoqFMiKBcc8_4SaIs0Vm2ljaCS8WVL4j0NUmUzacUMz41Ugli3zsNM9ymmK6asM_OH_V-w9zsx4sChpmlgheqQk44zmZuu4Gzl39KiM_5ICe57sJ__Hk-wXy5s0atYikswpQeLcHHJ3qLS7BYRrIUbZVy29vL0Gm7wnvXg5qhI2dwgVpOZ_1uUDR1otbw0o4uu7pOkd3go-Px0J3BoFPXx4z6RXlbugKn7zK2z1AbjUf6CyBKE09xQbU2djPn70Y6EIpwIkQYaU95ddipJjiWpfC68_8YxnkBQEDiVvwMEXXYenhgUmiOvH5ro4JDXAafNH7Awtc3r36DOYuR-HC_d9CAD0WvpaudW4VadvNbr8Gs_JMN0pv1HNkILt4bK_8AOVo8dQ
openUrl ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Convergent+Outer+Approximation+Algorithms+for+Solving+Unary+Programs&rft.jtitle=Journal+of+global+optimization&rft.au=Reiner+Horst&rft.au=Raber%2C+Ulrich&rft.date=1998-09-01&rft.pub=Springer+Nature+B.V&rft.issn=0925-5001&rft.eissn=1573-2916&rft.volume=13&rft.issue=2&rft.spage=123&rft_id=info:doi/10.1023%2FA%3A1008284026289&rft.externalDBID=HAS_PDF_LINK&rft.externalDocID=945079931
thumbnail_l http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0925-5001&client=summon
thumbnail_m http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0925-5001&client=summon
thumbnail_s http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0925-5001&client=summon