On the complexity of testing attainment of the optimal value in nonlinear optimization
We prove that unless P = NP , there exists no polynomial time (or even pseudo-polynomial time) algorithm that can test whether the optimal value of a nonlinear optimization problem where the objective and constraints are given by low-degree polynomials is attained. If the degrees of these polynomial...
Gespeichert in:
| Veröffentlicht in: | Mathematical programming Jg. 184; H. 1-2; S. 221 - 241 |
|---|---|
| Hauptverfasser: | , |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.11.2020
Springer Nature B.V |
| Schlagworte: | |
| ISSN: | 0025-5610, 1436-4646 |
| Online-Zugang: | Volltext |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Abstract | We prove that unless
P
=
NP
, there exists no polynomial time (or even pseudo-polynomial time) algorithm that can test whether the optimal value of a nonlinear optimization problem where the objective and constraints are given by low-degree polynomials is attained. If the degrees of these polynomials are fixed, our results along with previously-known “Frank–Wolfe type” theorems imply that exactly one of two cases can occur: either the optimal value is attained on every instance, or it is strongly NP-hard to distinguish attainment from non-attainment. We also show that testing for some well-known sufficient conditions for attainment of the optimal value, such as coercivity of the objective function and closedness and boundedness of the feasible set, is strongly NP-hard. As a byproduct, our proofs imply that testing the Archimedean property of a quadratic module is strongly NP-hard, a property that is of independent interest to the convergence of the Lasserre hierarchy. Finally, we give semidefinite programming (SDP)-based sufficient conditions for attainment of the optimal value, in particular a new characterization of coercive polynomials that lends itself to an SDP hierarchy. |
|---|---|
| AbstractList | We prove that unless P=NP, there exists no polynomial time (or even pseudo-polynomial time) algorithm that can test whether the optimal value of a nonlinear optimization problem where the objective and constraints are given by low-degree polynomials is attained. If the degrees of these polynomials are fixed, our results along with previously-known “Frank–Wolfe type” theorems imply that exactly one of two cases can occur: either the optimal value is attained on every instance, or it is strongly NP-hard to distinguish attainment from non-attainment. We also show that testing for some well-known sufficient conditions for attainment of the optimal value, such as coercivity of the objective function and closedness and boundedness of the feasible set, is strongly NP-hard. As a byproduct, our proofs imply that testing the Archimedean property of a quadratic module is strongly NP-hard, a property that is of independent interest to the convergence of the Lasserre hierarchy. Finally, we give semidefinite programming (SDP)-based sufficient conditions for attainment of the optimal value, in particular a new characterization of coercive polynomials that lends itself to an SDP hierarchy. We prove that unless P = NP , there exists no polynomial time (or even pseudo-polynomial time) algorithm that can test whether the optimal value of a nonlinear optimization problem where the objective and constraints are given by low-degree polynomials is attained. If the degrees of these polynomials are fixed, our results along with previously-known “Frank–Wolfe type” theorems imply that exactly one of two cases can occur: either the optimal value is attained on every instance, or it is strongly NP-hard to distinguish attainment from non-attainment. We also show that testing for some well-known sufficient conditions for attainment of the optimal value, such as coercivity of the objective function and closedness and boundedness of the feasible set, is strongly NP-hard. As a byproduct, our proofs imply that testing the Archimedean property of a quadratic module is strongly NP-hard, a property that is of independent interest to the convergence of the Lasserre hierarchy. Finally, we give semidefinite programming (SDP)-based sufficient conditions for attainment of the optimal value, in particular a new characterization of coercive polynomials that lends itself to an SDP hierarchy. |
| Author | Ahmadi, Amir Ali Zhang, Jeffrey |
| Author_xml | – sequence: 1 givenname: Amir Ali orcidid: 0000-0003-1590-7281 surname: Ahmadi fullname: Ahmadi, Amir Ali email: a_a_a@princeton.edu organization: Department of Operations Research and Financial Engineering, Princeton University – sequence: 2 givenname: Jeffrey surname: Zhang fullname: Zhang, Jeffrey organization: Department of Operations Research and Financial Engineering, Princeton University |
| BookMark | eNp9kEtLAzEUhYNUsK3-AVcB16O5ecxjKcUXFNyo25CZydSUaTImqVh_vWlHEFx0ERJuzrn33G-GJtZZjdAlkGsgpLgJQIAUGYEqHQ6QwQmaAmd5xnOeT9CUECoykQM5Q7MQ1oQQYGU5RW_PFsd3jRu3GXr9ZeIOuw5HHaKxK6xiVMZutI2HatK5IZqN6vGn6rcaG4tTkN5Yrfz4Zb5VNM6eo9NO9UFf_N5z9Hp_97J4zJbPD0-L22XWMF7ErBXAGBcla7qWc6hZkQtNG6p00ba1AlLXXOuyhVqJqm5LVYqO15RUtCPpRdkcXY19B-8-tim1XLutt2mkpFwAT3tWcFRFBRWC5ownFR1VjXcheN3Jwadd_U4CkXvKcqQsE2V5oCz3rct_psbEA4LolemPW9loDWmOXWn_l-qI6wd1NJMp |
| CitedBy_id | crossref_primary_10_1007_s10107_022_01869_6 crossref_primary_10_1007_s10957_022_02068_1 crossref_primary_10_1007_s10107_020_01532_y crossref_primary_10_1007_s10107_022_01878_5 crossref_primary_10_1007_s10898_024_01450_9 crossref_primary_10_1007_s10957_025_02637_0 crossref_primary_10_1137_21M1434945 |
| Cites_doi | 10.1137/S1052623400366802 10.1007/978-0-387-09686-5_7 10.1137/1.9781611972290 10.1007/978-3-7091-9459-1_3 10.4153/CMB-2003-054-7 10.1007/s10107-011-0499-2 10.1137/130931308 10.1137/140980624 10.1002/nav.3800030109 10.2307/1969640 10.1016/j.jsc.2010.06.009 10.1007/s10107-017-1144-5 10.1007/s10107-005-0672-6 10.1023/A:1014813701864 10.1007/s10107-003-0387-5 10.1023/A:1008203903341 10.1007/978-3-662-04648-7 10.1007/BF01362149 10.1145/800133.804350 10.1007/s10107-006-0003-6 10.1145/1993886.1993910 10.1512/iumj.1993.42.42045 10.1007/BF02807438 10.1007/s10957-014-0545-3 10.1007/s10107-013-0680-x 10.1070/RM1961v016n01ABEH004100 10.1023/A:1008652705980 |
| ContentType | Journal Article |
| Copyright | Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2019 Mathematical Programming is a copyright of Springer, (2019). All Rights Reserved. Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2019. |
| Copyright_xml | – notice: Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2019 – notice: Mathematical Programming is a copyright of Springer, (2019). All Rights Reserved. – notice: Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2019. |
| DBID | AAYXX CITATION 3V. 7SC 7WY 7WZ 7XB 87Z 88I 8AL 8AO 8FD 8FE 8FG 8FK 8FL ABJCF ABUWG AFKRA ARAPS AZQEC BENPR BEZIV BGLVJ CCPQU DWQXO FRNLG F~G GNUQQ HCIFZ JQ2 K60 K6~ K7- L.- L6V L7M L~C L~D M0C M0N M2P M7S P5Z P62 PHGZM PHGZT PKEHL PQBIZ PQBZA PQEST PQGLB PQQKQ PQUKI PRINS PTHSS Q9U |
| DOI | 10.1007/s10107-019-01411-1 |
| 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) Materials Science & Engineering Collection ProQuest Central (Alumni) ProQuest Central UK/Ireland Advanced Technologies & Computer Science Collection ProQuest Central Essentials ProQuest Central Database Suite (ProQuest) Business Premium Collection ProQuest Technology Collection ProQuest One Community College ProQuest Central Business Premium Collection (Alumni) ABI/INFORM Global (Corporate) ProQuest Central Student SciTech Collection (ProQuest) 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 Computing Database Science Database (ProQuest) Engineering Database Advanced Technologies & Aerospace Database ProQuest Advanced Technologies & Aerospace Collection ProQuest Central Premium ProQuest One Academic (New) ProQuest One Academic Middle East (New) ProQuest One Business 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) 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 ProQuest Pharma Collection ProQuest Central ABI/INFORM Professional Advanced ProQuest Engineering Collection ProQuest Central Korea 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 | Computer and Information Systems Abstracts 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 Mathematics |
| EISSN | 1436-4646 |
| EndPage | 241 |
| ExternalDocumentID | 10_1007_s10107_019_01411_1 |
| GrantInformation_xml | – fundername: Defense Advanced Research Projects Agency grantid: Young Faculty Award funderid: http://dx.doi.org/10.13039/100000185 – fundername: Google grantid: Google Faculty Award funderid: http://dx.doi.org/10.13039/100006785 – fundername: Alfred P. Sloan Foundation grantid: Sloan Fellowship funderid: http://dx.doi.org/10.13039/100000879 – fundername: Air Force Office of Scientific Research grantid: Young Investigator Prize funderid: http://dx.doi.org/10.13039/100000181 – fundername: Directorate for Engineering grantid: CAREER Award funderid: http://dx.doi.org/10.13039/100000084 |
| GroupedDBID | --K --Z -52 -5D -5G -BR -EM -Y2 -~C -~X .4S .86 .DC .VR 06D 0R~ 0VY 199 1B1 1N0 1OL 1SB 203 28- 29M 2J2 2JN 2JY 2KG 2KM 2LR 2P1 2VQ 2~H 30V 3V. 4.4 406 408 409 40D 40E 5GY 5QI 5VS 67Z 6NX 6TJ 78A 7WY 88I 8AO 8FE 8FG 8FL 8TC 8UJ 8VB 95- 95. 95~ 96X AAAVM AABHQ AACDK AAHNG AAIAL AAJBT AAJKR AANZL AARHV AARTL AASML AATNV AATVU AAUYE AAWCG AAYIU AAYQN AAYTO AAYZH ABAKF ABBBX ABBXA ABDBF ABDZT ABECU ABFTV ABHLI ABHQN ABJCF ABJNI ABJOX ABKCH ABKTR ABMNI ABMQK ABNWP ABQBU ABQSL ABSXP ABTEG ABTHY ABTKH ABTMW ABULA ABUWG ABWNU ABXPI ACAOD ACBXY ACDTI ACGFS ACGOD ACHSB ACHXU ACIWK ACKNC ACMDZ ACMLO ACNCT ACOKC ACOMO ACPIV ACUHS ACZOJ ADHHG ADHIR ADIMF ADINQ ADKNI ADKPE ADRFC ADTPH ADURQ ADYFF ADZKW AEBTG AEFIE AEFQL AEGAL AEGNC AEJHL AEJRE AEKMD AEMOZ AEMSY AENEX AEOHA AEPYU AESKC AETLH AEVLU AEXYK AFBBN AFEXP AFFNX AFGCZ AFKRA AFLOW AFQWF AFWTZ AFZKB AGAYW AGDGC AGGDS AGJBK AGMZJ AGQEE AGQMX AGRTI AGWIL AGWZB AGYKE AHAVH AHBYD AHKAY AHQJS AHSBF AHYZX AIAKS AIGIU AIIXL AILAN AITGF AJBLW AJRNO AJZVZ AKVCP ALMA_UNASSIGNED_HOLDINGS ALWAN AMKLP AMXSW AMYLF AMYQR AOCGG ARAPS ARCSS ARMRJ ASPBG AVWKF AXYYD AYJHY AZFZN AZQEC B-. B0M BA0 BAPOH BBWZM BDATZ BENPR BEZIV BGLVJ BGNMA BPHCQ BSONS CAG CCPQU COF CS3 CSCUP DDRTE DL5 DNIVK DPUIP DU5 DWQXO EAD EAP EBA EBLON EBR EBS EBU ECS EDO EIOEI EJD EMI EMK EPL ESBYG EST ESX FEDTE FERAY FFXSO FIGPU FINBP FNLPD FRNLG FRRFC FSGXE FWDCC GGCAI GGRSB GJIRD GNUQQ GNWQR GQ6 GQ7 GQ8 GROUPED_ABI_INFORM_COMPLETE GXS H13 HCIFZ HF~ HG5 HG6 HMJXF HQYDN HRMNR HVGLF HZ~ H~9 I-F I09 IAO IHE IJ- IKXTQ ITM IWAJR IXC IZIGR IZQ I~X I~Z J-C J0Z JBSCW JCJTX JZLTJ K1G K60 K6V K6~ K7- KDC KOV KOW L6V LAS LLZTM M0C M0N M2P M4Y M7S MA- N2Q N9A NB0 NDZJH NPVJJ NQ- NQJWS NU0 O9- O93 O9G O9I O9J OAM P19 P2P P62 P9R PF0 PQBIZ PQBZA PQQKQ PROAC PT4 PT5 PTHSS Q2X QOK QOS QWB R4E R89 R9I RHV RIG RNI RNS ROL RPX RPZ RSV RZK S16 S1Z S26 S27 S28 S3B SAP SCLPG SDD SDH SDM SHX SISQX SJYHP SMT SNE SNPRN SNX SOHCF SOJ SPISZ SRMVM SSLCW STPWE SZN T13 T16 TH9 TN5 TSG TSK TSV TUC TUS U2A UG4 UOJIU UTJUX UZXMN VC2 VFIZW W23 W48 WH7 WK8 XPP YLTOR Z45 Z5O Z7R Z7S Z7X Z7Y Z7Z Z81 Z83 Z86 Z88 Z8M Z8N Z8R Z8T Z8W Z92 ZL0 ZMTXR ZWQNP ~02 ~8M ~EX AAPKM AAYXX ABBRH ABDBE ABFSG ABRTQ ACSTC ADHKG ADXHL AEZWR AFDZB AFFHD AFHIU AFOHR AGQPQ AHPBZ AHWEU AIXLP AMVHM ATHPR AYFIA CITATION PHGZM PHGZT PQGLB 7SC 7XB 8AL 8FD 8FK JQ2 L.- L7M L~C L~D PKEHL PQEST PQUKI PRINS Q9U |
| ID | FETCH-LOGICAL-c347t-d51334583cfd441b3765e2c2ae7ddba10bb4ee8d1ba59bd8a85f4b2092f05f423 |
| IEDL.DBID | RSV |
| ISICitedReferencesCount | 7 |
| ISICitedReferencesURI | http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000578114700007&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| ISSN | 0025-5610 |
| IngestDate | Thu Sep 25 00:40:27 EDT 2025 Thu Sep 25 00:48:06 EDT 2025 Sat Nov 29 03:34:01 EST 2025 Tue Nov 18 21:43:46 EST 2025 Fri Feb 21 02:32:35 EST 2025 |
| IsPeerReviewed | true |
| IsScholarly | true |
| Issue | 1-2 |
| Keywords | Semidefinite programming 90C30 Frank–Wolfe type theorems Archimedean quadratic modules Existence of solutions in mathematical programs 90C60 Computational complexity Coercive polynomials |
| Language | English |
| LinkModel | DirectLink |
| MergedId | FETCHMERGED-LOGICAL-c347t-d51334583cfd441b3765e2c2ae7ddba10bb4ee8d1ba59bd8a85f4b2092f05f423 |
| Notes | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ORCID | 0000-0003-1590-7281 |
| PQID | 2252552634 |
| PQPubID | 25307 |
| PageCount | 21 |
| ParticipantIDs | proquest_journals_2451400091 proquest_journals_2252552634 crossref_primary_10_1007_s10107_019_01411_1 crossref_citationtrail_10_1007_s10107_019_01411_1 springer_journals_10_1007_s10107_019_01411_1 |
| PublicationCentury | 2000 |
| PublicationDate | 2020-11-01 |
| PublicationDateYYYYMMDD | 2020-11-01 |
| PublicationDate_xml | – month: 11 year: 2020 text: 2020-11-01 day: 01 |
| PublicationDecade | 2020 |
| PublicationPlace | Berlin/Heidelberg |
| PublicationPlace_xml | – name: Berlin/Heidelberg – name: Heidelberg |
| PublicationSubtitle | A Publication of the Mathematical Optimization Society |
| PublicationTitle | Mathematical programming |
| PublicationTitleAbbrev | Math. Program |
| PublicationYear | 2020 |
| Publisher | Springer Berlin Heidelberg Springer Nature B.V |
| Publisher_xml | – name: Springer Berlin Heidelberg – name: Springer Nature B.V |
| References | Jeyakumar, Lasserre, Li (CR16) 2014; 163 Prestel, Delzell (CR27) 2001 Gorin (CR13) 1961; 16 CR19 CR14 Nie (CR23) 2014; 146 Belousov, Klatte (CR7) 2002; 22 CR33 CR32 Lasserre (CR18) 2001; 11 Putinar (CR28) 1993; 42 Garey, Johnson (CR12) 2002 Bertsekas, Tseng (CR9) 2007; 110 Krivine (CR17) 1964; 12 Bertsekas (CR8) 1999 Luo, Zhang (CR21) 1999; 13 Parrilo (CR25) 2003; 96 CR4 Ahmadi, Hall (CR1) 2015; 169 Stengle (CR31) 1974; 207 Basu, Roy (CR6) 2010; 45 Porkolab, Khachiyan (CR26) 1997; 10 CR29 Ahmadi, Olshevsky, Parrilo, Tsitsiklis (CR2) 2013; 137 Nie, Demmel, Sturmfels (CR24) 2006; 106 CR20 Blekherman, Parrilo, Thomas (CR10) 2012 Bajbar, Stein (CR5) 2015; 25 Andronov, Belousov, Shironin (CR3) 1982; 4 Greuet, Safey El Din (CR15) 2014; 24 Marshall (CR22) 2003; 46 Seidenberg (CR30) 1954; 60 Frank, Wolfe (CR11) 1956; 3 A Seidenberg (1411_CR30) 1954; 60 M Marshall (1411_CR22) 2003; 46 G Blekherman (1411_CR10) 2012 1411_CR19 EG Belousov (1411_CR7) 2002; 22 1411_CR14 Z-Q Luo (1411_CR21) 1999; 13 EA Gorin (1411_CR13) 1961; 16 1411_CR33 V Andronov (1411_CR3) 1982; 4 J Nie (1411_CR24) 2006; 106 1411_CR32 AA Ahmadi (1411_CR1) 2015; 169 PA Parrilo (1411_CR25) 2003; 96 G Stengle (1411_CR31) 1974; 207 T Bajbar (1411_CR5) 2015; 25 MR Garey (1411_CR12) 2002 A Prestel (1411_CR27) 2001 M Putinar (1411_CR28) 1993; 42 L Porkolab (1411_CR26) 1997; 10 S Basu (1411_CR6) 2010; 45 AA Ahmadi (1411_CR2) 2013; 137 1411_CR29 V Jeyakumar (1411_CR16) 2014; 163 1411_CR4 DP Bertsekas (1411_CR9) 2007; 110 J-L Krivine (1411_CR17) 1964; 12 DP Bertsekas (1411_CR8) 1999 M Frank (1411_CR11) 1956; 3 1411_CR20 A Greuet (1411_CR15) 2014; 24 J Nie (1411_CR23) 2014; 146 JB Lasserre (1411_CR18) 2001; 11 |
| References_xml | – volume: 24 start-page: 1313 issue: 3 year: 2014 end-page: 1343 ident: CR15 article-title: Probabilistic algorithm for polynomial optimization over a real algebraic set publication-title: SIAM J. Optim. – volume: 4 start-page: 194 year: 1982 end-page: 197 ident: CR3 article-title: On solvability of the problem of polynomial programming publication-title: Izvestija Akadem. Nauk SSSR Tekhnicheskaja Kibernetika – year: 2012 ident: CR10 publication-title: Semidefinite Optimization and Convex Algebraic Geometry – volume: 169 start-page: 69 year: 2015 end-page: 94 ident: CR1 article-title: DC decomposition of nonconvex polynomials with algebraic techniques publication-title: Math. Program. – volume: 13 start-page: 87 issue: 1–3 year: 1999 end-page: 110 ident: CR21 article-title: On extensions of the Frank–Wolfe theorems publication-title: Comput. Optim. Appl. – volume: 96 start-page: 293 issue: 2 year: 2003 end-page: 320 ident: CR25 article-title: Semidefinite programming relaxations for semialgebraic problems publication-title: Math. Program. – ident: CR4 – ident: CR14 – volume: 11 start-page: 796 issue: 3 year: 2001 end-page: 817 ident: CR18 article-title: Global optimization with polynomials and the problem of moments publication-title: SIAM J. Optim. – volume: 163 start-page: 707 issue: 3 year: 2014 end-page: 718 ident: CR16 article-title: On polynomial optimization over non-compact semi-algebraic sets publication-title: J. Optim. Theory Appl. – volume: 60 start-page: 365 year: 1954 end-page: 374 ident: CR30 article-title: A new decision method for elementary algebra publication-title: Ann. Math. – volume: 46 start-page: 575 issue: 4 year: 2003 end-page: 587 ident: CR22 article-title: Optimization of polynomial functions publication-title: Can. Math. Bull. – ident: CR33 – volume: 25 start-page: 1542 issue: 3 year: 2015 end-page: 1570 ident: CR5 article-title: Coercive polynomials and their Newton polytopes publication-title: SIAM J. Optim. – volume: 3 start-page: 95 issue: 1–2 year: 1956 end-page: 110 ident: CR11 article-title: An algorithm for quadratic programming publication-title: Naval Res. Logist. (NRL) – volume: 106 start-page: 587 issue: 3 year: 2006 end-page: 606 ident: CR24 article-title: Minimizing polynomials via sum of squares over the gradient ideal publication-title: Math. Program. – ident: CR29 – volume: 207 start-page: 87 issue: 2 year: 1974 end-page: 97 ident: CR31 article-title: A Nullstellensatz and a Positivstellensatz in semialgebraic geometry publication-title: Math. Ann. – volume: 146 start-page: 97 issue: 1–2 year: 2014 end-page: 121 ident: CR23 article-title: Optimality conditions and finite convergence of the Lasserre hierarchy publication-title: Math. Program. – volume: 16 start-page: 93 issue: 1 year: 1961 end-page: 119 ident: CR13 article-title: Asymptotic properties of polynomials and algebraic functions of several variables publication-title: Russ. Math. Surv. – volume: 45 start-page: 1270 issue: 12 year: 2010 end-page: 1279 ident: CR6 article-title: Bounding the radii of balls meeting every connected component of semi-algebraic sets publication-title: J. Symb. Comput. – ident: CR19 – volume: 10 start-page: 351 issue: 4 year: 1997 end-page: 365 ident: CR26 article-title: On the complexity of semidefinite programs publication-title: J. Glob. Optim. – volume: 12 start-page: 307 issue: 1 year: 1964 end-page: 326 ident: CR17 article-title: Anneaux préordonnés publication-title: Journal d’analyse mathématique – year: 2002 ident: CR12 publication-title: Computers and Intractability – year: 2001 ident: CR27 publication-title: Positive Polynomials: From Hilbert’s 17th Problem to Real Algebra. Springer Monographs in Mathematics – year: 1999 ident: CR8 publication-title: Nonlinear Programming – volume: 22 start-page: 37 issue: 1 year: 2002 end-page: 48 ident: CR7 article-title: A Frank–Wolfe type theorem for convex polynomial programs publication-title: Comput. Optim. Appl. – ident: CR32 – volume: 42 start-page: 969 issue: 3 year: 1993 end-page: 984 ident: CR28 article-title: Positive polynomials on compact semi-algebraic sets publication-title: Indiana Univ. Math. J. – volume: 110 start-page: 287 issue: 2 year: 2007 end-page: 314 ident: CR9 article-title: Set intersection theorems and existence of optimal solutions publication-title: Math. Program. – volume: 137 start-page: 453 issue: 1–2 year: 2013 end-page: 476 ident: CR2 article-title: NP-hardness of deciding convexity of quartic polynomials and related problems publication-title: Math. Program. – ident: CR20 – volume: 11 start-page: 796 issue: 3 year: 2001 ident: 1411_CR18 publication-title: SIAM J. Optim. doi: 10.1137/S1052623400366802 – ident: 1411_CR19 doi: 10.1007/978-0-387-09686-5_7 – volume-title: Semidefinite Optimization and Convex Algebraic Geometry year: 2012 ident: 1411_CR10 doi: 10.1137/1.9781611972290 – ident: 1411_CR32 doi: 10.1007/978-3-7091-9459-1_3 – volume: 46 start-page: 575 issue: 4 year: 2003 ident: 1411_CR22 publication-title: Can. Math. Bull. doi: 10.4153/CMB-2003-054-7 – volume: 137 start-page: 453 issue: 1–2 year: 2013 ident: 1411_CR2 publication-title: Math. Program. doi: 10.1007/s10107-011-0499-2 – ident: 1411_CR33 – volume: 24 start-page: 1313 issue: 3 year: 2014 ident: 1411_CR15 publication-title: SIAM J. Optim. doi: 10.1137/130931308 – volume-title: Nonlinear Programming year: 1999 ident: 1411_CR8 – volume: 25 start-page: 1542 issue: 3 year: 2015 ident: 1411_CR5 publication-title: SIAM J. Optim. doi: 10.1137/140980624 – volume: 3 start-page: 95 issue: 1–2 year: 1956 ident: 1411_CR11 publication-title: Naval Res. Logist. (NRL) doi: 10.1002/nav.3800030109 – volume: 60 start-page: 365 year: 1954 ident: 1411_CR30 publication-title: Ann. Math. doi: 10.2307/1969640 – ident: 1411_CR20 – volume-title: Computers and Intractability year: 2002 ident: 1411_CR12 – volume: 45 start-page: 1270 issue: 12 year: 2010 ident: 1411_CR6 publication-title: J. Symb. Comput. doi: 10.1016/j.jsc.2010.06.009 – volume: 169 start-page: 69 year: 2015 ident: 1411_CR1 publication-title: Math. Program. doi: 10.1007/s10107-017-1144-5 – ident: 1411_CR4 – volume: 106 start-page: 587 issue: 3 year: 2006 ident: 1411_CR24 publication-title: Math. Program. doi: 10.1007/s10107-005-0672-6 – volume: 22 start-page: 37 issue: 1 year: 2002 ident: 1411_CR7 publication-title: Comput. Optim. Appl. doi: 10.1023/A:1014813701864 – volume: 96 start-page: 293 issue: 2 year: 2003 ident: 1411_CR25 publication-title: Math. Program. doi: 10.1007/s10107-003-0387-5 – volume: 10 start-page: 351 issue: 4 year: 1997 ident: 1411_CR26 publication-title: J. Glob. Optim. doi: 10.1023/A:1008203903341 – volume-title: Positive Polynomials: From Hilbert’s 17th Problem to Real Algebra. Springer Monographs in Mathematics year: 2001 ident: 1411_CR27 doi: 10.1007/978-3-662-04648-7 – volume: 4 start-page: 194 year: 1982 ident: 1411_CR3 publication-title: Izvestija Akadem. Nauk SSSR Tekhnicheskaja Kibernetika – volume: 207 start-page: 87 issue: 2 year: 1974 ident: 1411_CR31 publication-title: Math. Ann. doi: 10.1007/BF01362149 – ident: 1411_CR29 doi: 10.1145/800133.804350 – volume: 110 start-page: 287 issue: 2 year: 2007 ident: 1411_CR9 publication-title: Math. Program. doi: 10.1007/s10107-006-0003-6 – ident: 1411_CR14 doi: 10.1145/1993886.1993910 – volume: 42 start-page: 969 issue: 3 year: 1993 ident: 1411_CR28 publication-title: Indiana Univ. Math. J. doi: 10.1512/iumj.1993.42.42045 – volume: 12 start-page: 307 issue: 1 year: 1964 ident: 1411_CR17 publication-title: Journal d’analyse mathématique doi: 10.1007/BF02807438 – volume: 163 start-page: 707 issue: 3 year: 2014 ident: 1411_CR16 publication-title: J. Optim. Theory Appl. doi: 10.1007/s10957-014-0545-3 – volume: 146 start-page: 97 issue: 1–2 year: 2014 ident: 1411_CR23 publication-title: Math. Program. doi: 10.1007/s10107-013-0680-x – volume: 16 start-page: 93 issue: 1 year: 1961 ident: 1411_CR13 publication-title: Russ. Math. Surv. doi: 10.1070/RM1961v016n01ABEH004100 – volume: 13 start-page: 87 issue: 1–3 year: 1999 ident: 1411_CR21 publication-title: Comput. Optim. Appl. doi: 10.1023/A:1008652705980 |
| SSID | ssj0001388 |
| Score | 2.3464062 |
| Snippet | We prove that unless
P
=
NP
, there exists no polynomial time (or even pseudo-polynomial time) algorithm that can test whether the optimal value of a nonlinear... We prove that unless P=NP, there exists no polynomial time (or even pseudo-polynomial time) algorithm that can test whether the optimal value of a nonlinear... |
| SourceID | proquest crossref springer |
| SourceType | Aggregation Database Enrichment Source Index Database Publisher |
| StartPage | 221 |
| SubjectTerms | Algorithms Calculus of Variations and Optimal Control; Optimization Coercivity Combinatorics Full Length Paper Job shops Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Optimization Polynomials Semidefinite programming Theoretical |
| SummonAdditionalLinks | – databaseName: Advanced Technologies & Aerospace Database dbid: P5Z link: http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV07T8MwELagMMDAG1EoyAMbRNiJ0zgTQoiKhdIBUMUS-RVUCdLSBMTP5-w6LSDowhbHti7KXc5f7LvvEDoOBUuJjHUA2LodMMAAAQdnCbZsVEwMTYzOXbGJpNvl_X7a8xtupQ-rrH2ic9R6qOwe-RnYHaDfsB2x89FrYKtG2dNVX0JjES1ZlgRbuqEXP049MY04r0u2Wpzgk2Z86hx1QZc2WojZsK7vC9MMbf44IHXrTmf9v0-8gdY84sQXExPZRAum2EKrX3gIoXUzJW8tt9HDbYGhiV20ufkAmI6HOa4sHUfxhEVVhxC4uzBuCF7nBSRY3nCDBwUuJvQbYjzp8pmeO-i-c3V3eR348guBilhSBdqWfrHHqirXAJokuKLYhCoUJtFaCkqkZMZwTaWIU6m54HHOZEjSMCdwFUa7qAECzR7CRAqANiqngnDG4iiNiW7ztM0pTQVJVBPR-t1nynOT2xIZz9mMVdnqKwN9ZU5fGW2ik-mc0YSZY-7oVq2kzH-lZTbT0O_dDNCkBaEw-7S2gln338L25ws7QCuh_Wt3GY0t1KjGb-YQLav3alCOj5wJfwJ35_Q- priority: 102 providerName: ProQuest |
| Title | On the complexity of testing attainment of the optimal value in nonlinear optimization |
| URI | https://link.springer.com/article/10.1007/s10107-019-01411-1 https://www.proquest.com/docview/2252552634 https://www.proquest.com/docview/2451400091 |
| Volume | 184 |
| WOSCitedRecordID | wos000578114700007&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: PRVAVX databaseName: SpringerLINK Contemporary 1997-Present customDbUrl: eissn: 1436-4646 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0001388 issn: 0025-5610 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/eLvHCXMwnV3PT8IwFG4UvHjwtxGDpAdvuqTdOtodlUBMDEhAEb0s7doZEh0GpvHP93VsgAZN9PKyru26vHav39L3vofQqStZQJSvHcDWdYcBBnAEGEtYyybyiaHc6DhLNsE7HTEcBt08KGxaeLsXR5KZpV4KdqOZm6T172HWEWsdlWG749aRr9cfzO0v9YQoErVadJCHyqx-xtftaIExvx2LZrtNa_t_77mDtnJ0iS9my2EXrZlkD20ucQ5CqT0nap3uo8FNgqGIM89y8wGQHI9jnFrqjeQJy7RwF8juQrsxWJgXGMFyhBs8SnAyo9qQk1lVHtV5gO5azdvGlZOnWnAij_HU0TbNiz1CjWINAEmB2fGNG7nScK2VpEQpZozQVEk_UFpI4cdMuSRwYwJXrneISjCgOUKYKAkwJoqpJIIx3wt8ousiqAtKA0l4VEG00HgY5TzkNh3Gc7hgULYaDEGDYabBkFbQ2bzP64yF49fW1WIiw_yLnIZgt-Dvya17bHU1A-RoASf0Pi_mdVH982DHf2teRaV08mZO0Eb0no6mkxpa5_cPNVS-bHa6PShdcwdkmzSsdLtW8j7Irv9Yy9b5J8Uh7z0 |
| linkProvider | Springer Nature |
| linkToHtml | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMw1V3PT9RAFH7Z7JqIBxXFuLDqHOQkDTPT6XZ6MAYUwgZYCUHCrcyvmk2gu-xW1H_Kv9E3_bErRrhx8NZ2pnlp55vXbzrvfQ_gLVcioTqyAXLrfiCQAwQSnSVi2ZmIOhY7m5XFJuLhUJ6dJUct-NXkwviwysYnlo7ajo3_R76JuEP2y_uh-DC5CnzVKL-72pTQqGCx735-xyXb7P3gE47vOue7Oycf94K6qkBgQhEXgfUVTfxuockscgGNMyxy3HDlYmu1YlRr4Zy0TKso0VYqGWVCc5rwjOKRFzpAl98RIbqYNnS2d4ZHx3Pfz0IpmyKxnpnUaTp1sh4rwzx9fJLwgWQ3P4ULfvvXlmz5pdt98r-9o6fwuObUZKuaBMvQcvkzePSH0iKeHc7laWfP4fRzTvCUlPH07gcuRMg4I4UXHMm_ElU0QRLlVew3Rr96iRa8Mrojo5zklcCImlZNdS7rCny5l8d8AW006F4CoVoheTMZU1QKEYVJRG1fJn3JWKJobLrAmrFOTa2-7ouAXKQL3WiPjxTxkZb4SFkX3s3vmVTaI3f27jWgSGs_NEsXiPh3s0C-7Gk23r3RoG7RfLux1buNvYGHeyeHB-nBYLi_Bkvc_6Mo8zd70C6m39wreGCui9Fs-rqeQATO7xuPvwEo1FOm |
| linkToPdf | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV3PS8MwFA46RfTgb3E6NQdvWpa06ZoeRR2KOgfq2K0kTSoD7cZWxT_fl7TdpkxBvDVN0pSX9PUL-d73EDp2BQuJ9JUD2LrhMMAADgdnCWtZxz7RNNAqsckmglaLd7theyqK37LdyyPJPKbBqDSlWX2gkvpU4Bu1lEnD9WGGlDWPFpihy5n9-kNn7Iupx3mZtNUghSJsZvYzvv6aJnjz2xGp_fM01_7_zutotUCd-CxfJhtoTqebaGVKixBKd2MB19EW6tynGIrYMs71B0B13E9wZiQ50mcsspJGYO9Cuz54nlcYwWiHa9xLcZpLcIhhXlVEe26jp-bl4_mVU6RgcGKPBZmjTPoXc7QaJwqAkwR35Gs3doUOlJKCEimZ1lxRKfxQKi64nzDpktBNCFy53g6qwIB6F2EiBcCbOKGCcMZ8L_SJavCwwSkNBQniKqKl9aO40Cc3aTJeoomysrFgBBaMrAUjWkUn4z6DXJ3j19a1clKj4ksdReDPYFflNjw2u5oBojRAFHqflnM8qf55sL2_NT9CS-2LZnR73brZR8uu2dTbgMcaqmTDN32AFuP3rDcaHtr1_Qnda_XH |
| 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=On+the+complexity+of+testing+attainment+of+the+optimal+value+in+nonlinear+optimization&rft.jtitle=Mathematical+programming&rft.au=Ahmadi%2C+Amir+Ali&rft.au=Zhang%2C+Jeffrey&rft.date=2020-11-01&rft.issn=0025-5610&rft.eissn=1436-4646&rft.volume=184&rft.issue=1-2&rft.spage=221&rft.epage=241&rft_id=info:doi/10.1007%2Fs10107-019-01411-1&rft.externalDBID=n%2Fa&rft.externalDocID=10_1007_s10107_019_01411_1 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0025-5610&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0025-5610&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0025-5610&client=summon |