Non-convex nested Benders decomposition

We propose a new decomposition method to solve multistage non-convex mixed-integer (stochastic) nonlinear programming problems (MINLPs). We call this algorithm non-convex nested Benders decomposition (NC-NBD). NC-NBD is based on solving dynamically improved mixed-integer linear outer approximations...

Full description

Saved in:

Bibliographic Details
Published in:	Mathematical programming Vol. 196; no. 1-2; pp. 987 - 1024
Main Authors:	Füllner, Christian, Rebennack, Steffen
Format:	Journal Article
Language:	English
Published:	Berlin/Heidelberg Springer Berlin Heidelberg 01.11.2022 Springer Springer Nature B.V
Subjects:	Algorithms Approximation Benders decomposition Calculus of Variations and Optimal Control; Optimization Combinatorics Full Length Paper Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Mathematics of Computing Mixed integer Nonlinear programming Numerical Analysis Optimization Regularization Theoretical Unit commitment Mixed-integer nonlinear programming (MINLP) Global optimization Non-convexities 90C11 Nested Benders decomposition 90C26 49M27 Non-convex value functions
ISSN:	0025-5610, 1436-4646
Online Access:	Get full text
Tags:	Add Tag No Tags, Be the first to tag this record!

Abstract	We propose a new decomposition method to solve multistage non-convex mixed-integer (stochastic) nonlinear programming problems (MINLPs). We call this algorithm non-convex nested Benders decomposition (NC-NBD). NC-NBD is based on solving dynamically improved mixed-integer linear outer approximations of the MINLP, obtained by piecewise linear relaxations of nonlinear functions. Those MILPs are solved to global optimality using an enhancement of nested Benders decomposition, in which regularization, dynamically refined binary approximations of the state variables and Lagrangian cut techniques are combined to generate Lipschitz continuous non-convex approximations of the value functions. Those approximations are then used to decide whether the approximating MILP has to be dynamically refined and in order to compute feasible solutions for the original MINLP. We prove that NC-NBD converges to an ε -optimal solution in a finite number of steps. We provide promising computational results for some unit commitment problems of moderate size.
AbstractList	We propose a new decomposition method to solve multistage non-convex mixed-integer (stochastic) nonlinear programming problems (MINLPs). We call this algorithm non-convex nested Benders decomposition (NC-NBD). NC-NBD is based on solving dynamically improved mixed-integer linear outer approximations of the MINLP, obtained by piecewise linear relaxations of nonlinear functions. Those MILPs are solved to global optimality using an enhancement of nested Benders decomposition, in which regularization, dynamically refined binary approximations of the state variables and Lagrangian cut techniques are combined to generate Lipschitz continuous non-convex approximations of the value functions. Those approximations are then used to decide whether the approximating MILP has to be dynamically refined and in order to compute feasible solutions for the original MINLP. We prove that NC-NBD converges to an $$\varepsilon $$ ε -optimal solution in a finite number of steps. We provide promising computational results for some unit commitment problems of moderate size. We propose a new decomposition method to solve multistage non-convex mixed-integer (stochastic) nonlinear programming problems (MINLPs). We call this algorithm non-convex nested Benders decomposition (NC-NBD). NC-NBD is based on solving dynamically improved mixed-integer linear outer approximations of the MINLP, obtained by piecewise linear relaxations of nonlinear functions. Those MILPs are solved to global optimality using an enhancement of nested Benders decomposition, in which regularization, dynamically refined binary approximations of the state variables and Lagrangian cut techniques are combined to generate Lipschitz continuous non-convex approximations of the value functions. Those approximations are then used to decide whether the approximating MILP has to be dynamically refined and in order to compute feasible solutions for the original MINLP. We prove that NC-NBD converges to an ε-optimal solution in a finite number of steps. We provide promising computational results for some unit commitment problems of moderate size. We propose a new decomposition method to solve multistage non-convex mixed-integer (stochastic) nonlinear programming problems (MINLPs). We call this algorithm non-convex nested Benders decomposition (NC-NBD). NC-NBD is based on solving dynamically improved mixed-integer linear outer approximations of the MINLP, obtained by piecewise linear relaxations of nonlinear functions. Those MILPs are solved to global optimality using an enhancement of nested Benders decomposition, in which regularization, dynamically refined binary approximations of the state variables and Lagrangian cut techniques are combined to generate Lipschitz continuous non-convex approximations of the value functions. Those approximations are then used to decide whether the approximating MILP has to be dynamically refined and in order to compute feasible solutions for the original MINLP. We prove that NC-NBD converges to an [Formula omitted]-optimal solution in a finite number of steps. We provide promising computational results for some unit commitment problems of moderate size. We propose a new decomposition method to solve multistage non-convex mixed-integer (stochastic) nonlinear programming problems (MINLPs). We call this algorithm non-convex nested Benders decomposition (NC-NBD). NC-NBD is based on solving dynamically improved mixed-integer linear outer approximations of the MINLP, obtained by piecewise linear relaxations of nonlinear functions. Those MILPs are solved to global optimality using an enhancement of nested Benders decomposition, in which regularization, dynamically refined binary approximations of the state variables and Lagrangian cut techniques are combined to generate Lipschitz continuous non-convex approximations of the value functions. Those approximations are then used to decide whether the approximating MILP has to be dynamically refined and in order to compute feasible solutions for the original MINLP. We prove that NC-NBD converges to an ε -optimal solution in a finite number of steps. We provide promising computational results for some unit commitment problems of moderate size.
Audience	Academic
Author	Rebennack, Steffen Füllner, Christian
Author_xml	– sequence: 1 givenname: Christian orcidid: 0000-0003-2485-6199 surname: Füllner fullname: Füllner, Christian email: christian.fuellner@kit.edu organization: Institute for Operations Research (IOR), Stochastic Optimization (SOP), Karlsruhe Institute of Technology (KIT) – sequence: 2 givenname: Steffen surname: Rebennack fullname: Rebennack, Steffen organization: Institute for Operations Research (IOR), Stochastic Optimization (SOP), Karlsruhe Institute of Technology (KIT)
BookMark	eNp9kE9LAzEQxYNUsK1-AU8FD55SJ8lms3usxX9Q9KLnsLuZlC1tUpOt6Lc36wqChzKHMOH9Zua9CRk575CQSwZzBqBuIgMGigJnFJjKgMIJGbNM5DTLs3xExgBcUpkzOCOTGDcAwERRjMn1s3e08e4DP2cOY4dmdovOYIgzg43f7X1su9a7c3Jqq23Ei993St7u716Xj3T18vC0XKxokwHvaGlrK2UpKlWCMlj0PdSQVwUXtUitqUsFvMYSbc3zQkpmrGoaNExkloOYkqth7j7490M6SG_8Ibi0UnMlhCgUFyqp5oNqXW1Rt876LlRNKoO7NrlB26b_heIyV5DLIgHFADTBxxjQ6qbtqt5YAtutZqD7HPWQo0456p8cdX8R_4fuQ7urwtdxSAxQTGK3xvBn4wj1DQyUhaA
CitedBy_id	crossref_primary_10_1007_s10898_025_01480_x crossref_primary_10_1016_j_ejor_2024_05_013 crossref_primary_10_1515_revce_2024_0064 crossref_primary_10_1016_j_apenergy_2025_126052 crossref_primary_10_1007_s10107_022_01876_7 crossref_primary_10_1287_ijoc_2024_0755 crossref_primary_10_1007_s12667_023_00645_5 crossref_primary_10_3390_a15040103 crossref_primary_10_1109_TSTE_2022_3211865 crossref_primary_10_1049_rpg2_12859 crossref_primary_10_1016_j_ejor_2024_05_011 crossref_primary_10_1016_j_ejor_2025_05_022
Cites_doi	10.1287/ijoc.2020.0987 10.1007/BF02592154 10.1007/s10107-019-01368-1 10.1007/s10107-005-0581-8 10.1021/ie201262f 10.1287/opre.33.5.989 10.1007/s00186-016-0546-0 10.1007/BF01582895 10.1080/10556788.2017.1350178 10.1080/10556780903087124 10.1109/TSTE.2018.2805164 10.1007/BF01386316 10.1287/ijoc.2019.0890 10.1007/s10107-022-01875-8 10.1007/s10898-014-0166-2 10.1016/j.ejor.2011.11.040 10.1007/BF00934810 10.1007/BF01262932 10.1109/TPWRS.2018.2880996 10.1007/s10957-014-0687-3 10.1007/s10107-015-0884-3 10.1016/j.ejor.2016.08.006 10.1137/15M1020575 10.1137/0117061 10.1007/s10898-017-0559-0 10.1007/s10479-012-1102-9 10.1007/s10107-009-0295-4 10.1007/978-1-4939-0808-0_14 10.1287/mnsc.22.4.455 10.1007/s10107-016-1012-8 10.1057/jors.1990.166 10.1007/s10107-018-1249-5 10.1007/s10107-020-01569-z 10.1007/s10957-011-9888-1 10.1137/141000671 10.1007/s10898-019-00786-x 10.1007/s10898-019-00816-8 10.1007/s10107-002-0308-z 10.1017/S0962492913000032 10.1287/opre.1090.0721 10.1080/10556788.2018.1556661
ContentType	Journal Article
Copyright	The Author(s) 2022 COPYRIGHT 2022 Springer The Author(s) 2022. 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.
Copyright_xml	– notice: The Author(s) 2022 – notice: COPYRIGHT 2022 Springer – notice: The Author(s) 2022. 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.
DBID	C6C AAYXX CITATION 7SC 8FD JQ2 L7M L~C L~D
DOI	10.1007/s10107-021-01740-0
DatabaseName	Springer Nature OA Free Journals CrossRef Computer and Information Systems Abstracts Technology Research Database ProQuest Computer Science Collection Advanced Technologies Database with Aerospace Computer and Information Systems Abstracts Academic Computer and Information Systems Abstracts Professional
DatabaseTitle	CrossRef Computer and Information Systems Abstracts Technology Research Database Computer and Information Systems Abstracts – Academic Advanced Technologies Database with Aerospace ProQuest Computer Science Collection Computer and Information Systems Abstracts Professional
DatabaseTitleList	CrossRef Computer and Information Systems Abstracts
DeliveryMethod	fulltext_linktorsrc
Discipline	Engineering Mathematics
EISSN	1436-4646
EndPage	1024
ExternalDocumentID	A725670658 10_1007_s10107_021_01740_0
GrantInformation_xml	– fundername: Deutsche Forschungsgemeinschaft grantid: 445857709 funderid: http://dx.doi.org/10.13039/501100001659
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 C6C 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 8FD JQ2 L7M L~C L~D
ID	FETCH-LOGICAL-c402t-9fbf5593a7907de89fbf0b06a823b389fdb9702be9efb268551df7cced134f203
IEDL.DBID	RSV
ISICitedReferencesCount	16
ISICitedReferencesURI	http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000740163700002&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 00:40:45 EDT 2025 Sat Nov 29 11:02:17 EST 2025 Tue Nov 18 22:03:02 EST 2025 Sat Nov 29 03:34:03 EST 2025 Fri Feb 21 02:44:42 EST 2025
IsDoiOpenAccess	true
IsOpenAccess	true
IsPeerReviewed	true
IsScholarly	true
Issue	1-2
Keywords	Mixed-integer nonlinear programming (MINLP) Global optimization Non-convexities 90C11 Nested Benders decomposition 90C26 49M27 Non-convex value functions
Language	English
LinkModel	DirectLink
MergedId	FETCHMERGED-LOGICAL-c402t-9fbf5593a7907de89fbf0b06a823b389fdb9702be9efb268551df7cced134f203
Notes	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14
ORCID	0000-0003-2485-6199
OpenAccessLink	https://link.springer.com/10.1007/s10107-021-01740-0
PQID	2733387237
PQPubID	25307
PageCount	38
ParticipantIDs	proquest_journals_2733387237 gale_infotracacademiconefile_A725670658 crossref_citationtrail_10_1007_s10107_021_01740_0 crossref_primary_10_1007_s10107_021_01740_0 springer_journals_10_1007_s10107_021_01740_0
PublicationCentury	2000
PublicationDate	2022-11-01
PublicationDateYYYYMMDD	2022-11-01
PublicationDate_xml	– month: 11 year: 2022 text: 2022-11-01 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	2022
Publisher	Springer Berlin Heidelberg Springer Springer Nature B.V
Publisher_xml	– name: Springer Berlin Heidelberg – name: Springer – name: Springer Nature B.V
References	PereiraMVFPintoLMVGMulti-stage stochastic optimization applied to energy planningMath. Program.1991521–3359375112617610.1007/BF015828950749.90057 Philpott, A.B., Wahid, F., Bonnans, F.: MIDAS: a mixed integer dynamic approximation scheme. Math. Program. (2019) CerisolaSLatorreJMRamosAStochastic dual dynamic programming applied to nonconvex hydrothermal modelsEur. J. Oper. Res.20122183687697288174110.1016/j.ejor.2011.11.0401244.90173 ZhouKKilinçMRChenXSahinidisNVAn efficient strategy for the activation of mip relaxations in a multicore global minlp solverJ. Global Optim.2018703497516376461610.1007/s10898-017-0559-01393.90076 BeasleyJEOR-library: distributing test problems by electronic mailJ. Oper. Res. Soc.19904111106910.1057/jors.1990.166 GAMS Software GmbH. GAMS.jl. Code released on GitHub https://github.com/GAMS-dev/gams.jl (2020) MisenerRFloudasCAAntigone: algorithms for continuous/integer global optimization of nonlinear equationsJ. Global Optim.2014592–3503526321669010.1007/s10898-014-0166-21301.90063 Dowson, O., Kapelevich, L.: SDDP.jl: a Julia package for stochastic dual dynamic programming. INFORMS J. Comput. (2020) LiXTomasgardABartonPINonconvex generalized Benders decomposition for stochastic separable mixed-integer nonlinear programsJ. Optim. Theory Appl.2011151425454285122410.1007/s10957-011-9888-11245.90079 Geißler, B.: Towards globally optimal solutions for MINLPs by discretization techniques with applications in gas network optimization. PhD thesis, Friedrich-Alexander-Universität Erlangen-Nürnberg (2011) ZouJAhmedSSunXAStochastic dual dynamic integer programmingMath. Program.20191751–2461502394289710.1007/s10107-018-1249-51412.90101 Pedroso, J. P., Kubo, M., Viana, A.: Unit commitment with valve-point loading effect. Technical report, Universidade do Porto (2014) Rebennack, S., Krasko, V.: Piecewise linear function fitting via mixed-integer linear programming. INFORMS J. Comput. (2020) VielmaJPNemhauserGLModeling disjunctive constraints with a logarithmic number of binary variables and constraintsMath. Program.20111281–24972281095210.1007/s10107-009-0295-41218.90137 Meyer, R.R.: Integer and mixed-integer programming models: general properties. J. Optim. Theory Appl. 3(4) (1975) Zou, J., Ahmed, S., Sun, X.: Multistage stochastic unit commitment using stochastic dual dynamic integer programming. IEEE Trans. Power Syst. 34(3) (2019) Schnetter, E.: Delaunay.jl. Code released on GitHub https://github.com/eschnett/Delaunay.jl (2020) RebennackSKallrathJContinuous piecewise linear delta-approximations for univariate functions: computing minimal breakpoint systemsJ. Optim. Theory Appl.20151672617643341245310.1007/s10957-014-0687-31327.90245 Schrage, L.: LindoSystems: LindoAPI (2004) OgbeELiXA joint decomposition method for global optimization of multiscenario nonconvex mixed-integer nonlinear programsJ. Global Optim.201875595629401996110.1007/s10898-019-00786-x1432.90092 TawarmalaniMSahinidisNConvex extensions and envelopes of lower semi-continuous functionsMath. Program.200293247263195265310.1007/s10107-002-0308-z1065.90062 TawarmalaniMSahinidisNVA polyhedral branch-and-cut approach to global optimizationMath. Program.2005103225249214637910.1007/s10107-005-0581-81099.90047 BelottiPLeeJLibertiLMargotFWächterABranching and bounds tightening techniques for non-convex minlpOptim. Methods Softw.200924597634255490210.1080/105567809030871241179.90237 Gamrath, G., Anderson, D., Bestuzheva, K., Chen, W.-K., Eifler, L., Gasse, M., Gemander, P., Gleixner, A., Gottwald, L., Halbig, K., Hendel, G., Hojny, C., Koch, T., Le Bodic, P., Maher, S. J., Matter, F., Miltenberger, M., Mühmer, E., Müller, B., Pfetsch, M. E., Schlösser, F., Serrano, F., Shinano, Y., Tawfik, C., Vigerske, S., Wegscheider, F., Weninger, D., Witzig, J.: The SCIP optimization suite 7.0. Technical report, Optimization Online (2020) Sahinidis, N.V.: BARON 17.8.9: global optimization of mixed-integer nonlinear programs, User’s Manual (2017) Kannan, R.: Algorithms, analysis and software for the global optimization of two-stage stochastic programs. PhD thesis, Massachusetts Institute of Technology (2018) Füllner, C.: NCNBD.jl. Code released on GitHub https://github.com/ChrisFuelOR/NCNBD.jl (2021) LiXChenYBartonPINonconvex generalized Benders decomposition with piecewise convex relaxations for global optimization of integrated process design and operation problemsInd. Eng. Chem. Res.201251217287729910.1021/ie201262f RebennackSCombining sampling-based and scenario-based nested Benders decomposition methods: application to stochastic dual dynamic programmingMath. Program.20161561–2343389345920510.1007/s10107-015-0884-31342.90116 BendersJFPartitioning procedures for solving mixed-variables programming problemsNumer. Math.19624123825214730310.1007/BF013863160109.38302 Ahmed, S., Cabral, F. G., Freitas Paulo da Costa, B.: Stochastic Lipschitz dynamic programming. Math. Programm. (2020) BelottiPKirchesCLeyfferSLinderothJLuedtkeJMahajanAMixed-integer nonlinear optimizationActa Numer.2013221131303869610.1017/S09624929130000321291.65172 GeoffrionAMGeneralized Benders decompositionJ. Optim. Theory Appl.197210423726032731010.1007/BF009348100229.90024 Hjelmeland, M.N., Zou, J., Helseth, A., Ahmed, S.: Nonconvex medium-term hydropower scheduling by stochastic dual dynamic integer programming. IEEE Trans. Sustain. Energy 10(1) (2019) SteegerGLohmannTRebennackSStrategic bidding for a price-maker hydroelectric producer: stochastic dual dynamic programming and Lagrangian relaxationIISE Trans.201847114 Bacci, T., Frangioni, A., Gentile, C., Tavlaridis-Gyparakis, K.: New MINLP formulations for the unit commitment problems with ramping constraints. Preprint, http://www.optimization-online.org/DB_FILE/2019/10/7426.pdf (2019) LLC Gurobi Optimization. Gurobi optimization reference manual (2021) GloverFImproved linear integer programming formulations of nonlinear integer problemsManage. Sci.197522445546040113810.1287/mnsc.22.4.4550318.90044 RebennackSComputing tight bounds via piecewise linear functions through the example of circle cutting problemsMath. Methods Oper. Res.2016841357353588210.1007/s00186-016-0546-01396.90051 BezansonJEdelmanAKarpinskiSShahVBJulia: a fresh approach to numerical computingSIAM Rev.20175916598360582610.1137/1410006711356.68030 RockafellarRTWetsRJBVariational Analysis2009BerlinSpringer0888.49001 VielmaJPAhmedSNemhauserGMixed-integer models for nonseparable piecewise-linear optimization: unifying framework and extensionsOper. Res.2010582303315267479810.1287/opre.1090.07211226.90046 BirgeJRDecomposition and partitioning methods for multistage stochastic linear programsOper. Res.1985335989100780691610.1287/opre.33.5.9890581.90065 SteegerGRebennackSDynamic convexification within nested Benders decomposition using Lagrangian relaxation: an application to the strategic bidding problemEur. J. Oper. Res.20172572669686356688610.1016/j.ejor.2016.08.0061394.90442 DunningIHuchetteJLubinMJuMP: a modeling language for mathematical optimizationSIAM Rev.2017592295320364649310.1137/15M10205751368.90002 Zhang, S., Sun, X. A.: Stochastic dual dynamic programming for multistage stochastic mixed-integer nonlinear optimization. (2021). Available at https://arxiv.org/abs/1912.13278. Accessed 29 Nov 2021 Van Dinter, J., Rebennack, S., Kallrath, J., Denholm, P., Newman, A.: The unit commitment model with concave emissions costs: a hybrid benders’ decomposition with nonconvex master problems. Ann. Oper. Res. 210(1), 361–386 (2013) KilinçMRSahinidisNVExploiting integrality in the global optimization of mixed-integer nonlinear programming problems with baronOptim. Methods Softw.2018333540562378367310.1080/10556788.2017.13501781398.90110 InfangerGMortonDPCut sharing for multistage stochastic linear programs with interstage dependencyMath. Program.1996752241256142664010.1007/BF025921540874.90147 van SlykeRMWetsRL-shaped linear programs with applications to optimal control and stochastic programmingJ. SIAM Appl. Math.196917463866325374110.1137/01170610197.45602 Kapelevich, L.: SDDiP.jl. Code released on GitHub https://github.com/lkapelevich/SDDiP.jl (2018) FeizollahiMJAhmedSSunAExact augmented Lagrangian duality for mixed integer linear programmingMath. Program.20171611–2365387359278210.1007/s10107-016-1012-81364.90226 Kallrath, J., Rebennack, S.: Computing area-tight piecewise linear overestimators, underestimators and tubes for univariate functions. In: Optimization in Science and Engineering, pp. 273–292. Springer (2014) LiCGrossmannIEA generalized Benders decomposition-based branch and cut algorithm for two-stage stochastic programs with nonconvex constraints and mixed-binary first and second stage variablesJ. Global Optim.2019752247272401243010.1007/s10898-019-00816-81428.90106 BurlacuRGeißlerBScheweLSolving mixed-integer nonlinear programmes using adaptively refined mixed-integer linear programmesOptim. Methods Softw.20203513764403294010.1080/10556788.2018.15566611432.90089 R Burlacu (1740_CR9) 2020; 35 1740_CR31 1740_CR34 1740_CR36 JE Beasley (1740_CR3) 1990; 41 J Bezanson (1740_CR7) 2017; 59 S Rebennack (1740_CR37) 2016; 156 J Zou (1740_CR55) 2019; 175 1740_CR1 1740_CR2 S Cerisola (1740_CR10) 2012; 218 MVF Pereira (1740_CR35) 1991; 52 1740_CR40 1740_CR42 1740_CR43 1740_CR44 RM van Slyke (1740_CR49) 1969; 17 P Belotti (1740_CR5) 2009; 24 AM Geoffrion (1740_CR19) 1972; 10 JF Benders (1740_CR6) 1962; 4 JP Vielma (1740_CR51) 2011; 128 JP Vielma (1740_CR50) 2010; 58 K Zhou (1740_CR53) 2018; 70 M Tawarmalani (1740_CR48) 2005; 103 S Rebennack (1740_CR38) 2016; 84 RT Rockafellar (1740_CR41) 2009 E Ogbe (1740_CR33) 2018; 75 X Li (1740_CR30) 2011; 151 1740_CR52 1740_CR54 1740_CR11 1740_CR12 JR Birge (1740_CR8) 1985; 33 1740_CR15 MR Kilinç (1740_CR27) 2018; 33 1740_CR16 1740_CR17 1740_CR18 R Misener (1740_CR32) 2014; 59 S Rebennack (1740_CR39) 2015; 167 F Glover (1740_CR21) 1975; 22 X Li (1740_CR29) 2012; 51 G Steeger (1740_CR45) 2018; 47 MJ Feizollahi (1740_CR14) 2017; 161 1740_CR20 P Belotti (1740_CR4) 2013; 22 1740_CR22 C Li (1740_CR28) 2019; 75 I Dunning (1740_CR13) 2017; 59 1740_CR24 1740_CR25 1740_CR26 G Steeger (1740_CR46) 2017; 257 M Tawarmalani (1740_CR47) 2002; 93 G Infanger (1740_CR23) 1996; 75
References_xml	– reference: Gamrath, G., Anderson, D., Bestuzheva, K., Chen, W.-K., Eifler, L., Gasse, M., Gemander, P., Gleixner, A., Gottwald, L., Halbig, K., Hendel, G., Hojny, C., Koch, T., Le Bodic, P., Maher, S. J., Matter, F., Miltenberger, M., Mühmer, E., Müller, B., Pfetsch, M. E., Schlösser, F., Serrano, F., Shinano, Y., Tawfik, C., Vigerske, S., Wegscheider, F., Weninger, D., Witzig, J.: The SCIP optimization suite 7.0. Technical report, Optimization Online (2020) – reference: InfangerGMortonDPCut sharing for multistage stochastic linear programs with interstage dependencyMath. Program.1996752241256142664010.1007/BF025921540874.90147 – reference: Schrage, L.: LindoSystems: LindoAPI (2004) – reference: ZhouKKilinçMRChenXSahinidisNVAn efficient strategy for the activation of mip relaxations in a multicore global minlp solverJ. Global Optim.2018703497516376461610.1007/s10898-017-0559-01393.90076 – reference: Schnetter, E.: Delaunay.jl. Code released on GitHub https://github.com/eschnett/Delaunay.jl (2020) – reference: Van Dinter, J., Rebennack, S., Kallrath, J., Denholm, P., Newman, A.: The unit commitment model with concave emissions costs: a hybrid benders’ decomposition with nonconvex master problems. Ann. Oper. Res. 210(1), 361–386 (2013) – reference: Philpott, A.B., Wahid, F., Bonnans, F.: MIDAS: a mixed integer dynamic approximation scheme. Math. Program. (2019) – reference: Zhang, S., Sun, X. A.: Stochastic dual dynamic programming for multistage stochastic mixed-integer nonlinear optimization. (2021). Available at https://arxiv.org/abs/1912.13278. Accessed 29 Nov 2021 – reference: TawarmalaniMSahinidisNConvex extensions and envelopes of lower semi-continuous functionsMath. Program.200293247263195265310.1007/s10107-002-0308-z1065.90062 – reference: BelottiPLeeJLibertiLMargotFWächterABranching and bounds tightening techniques for non-convex minlpOptim. Methods Softw.200924597634255490210.1080/105567809030871241179.90237 – reference: BelottiPKirchesCLeyfferSLinderothJLuedtkeJMahajanAMixed-integer nonlinear optimizationActa Numer.2013221131303869610.1017/S09624929130000321291.65172 – reference: Ahmed, S., Cabral, F. G., Freitas Paulo da Costa, B.: Stochastic Lipschitz dynamic programming. Math. Programm. (2020) – reference: Sahinidis, N.V.: BARON 17.8.9: global optimization of mixed-integer nonlinear programs, User’s Manual (2017) – reference: RockafellarRTWetsRJBVariational Analysis2009BerlinSpringer0888.49001 – reference: BeasleyJEOR-library: distributing test problems by electronic mailJ. Oper. Res. Soc.19904111106910.1057/jors.1990.166 – reference: BirgeJRDecomposition and partitioning methods for multistage stochastic linear programsOper. Res.1985335989100780691610.1287/opre.33.5.9890581.90065 – reference: BurlacuRGeißlerBScheweLSolving mixed-integer nonlinear programmes using adaptively refined mixed-integer linear programmesOptim. Methods Softw.20203513764403294010.1080/10556788.2018.15566611432.90089 – reference: ZouJAhmedSSunXAStochastic dual dynamic integer programmingMath. Program.20191751–2461502394289710.1007/s10107-018-1249-51412.90101 – reference: Dowson, O., Kapelevich, L.: SDDP.jl: a Julia package for stochastic dual dynamic programming. INFORMS J. Comput. (2020) – reference: LLC Gurobi Optimization. Gurobi optimization reference manual (2021) – reference: KilinçMRSahinidisNVExploiting integrality in the global optimization of mixed-integer nonlinear programming problems with baronOptim. Methods Softw.2018333540562378367310.1080/10556788.2017.13501781398.90110 – reference: Kapelevich, L.: SDDiP.jl. Code released on GitHub https://github.com/lkapelevich/SDDiP.jl (2018) – reference: Geißler, B.: Towards globally optimal solutions for MINLPs by discretization techniques with applications in gas network optimization. PhD thesis, Friedrich-Alexander-Universität Erlangen-Nürnberg (2011) – reference: PereiraMVFPintoLMVGMulti-stage stochastic optimization applied to energy planningMath. Program.1991521–3359375112617610.1007/BF015828950749.90057 – reference: GAMS Software GmbH. GAMS.jl. Code released on GitHub https://github.com/GAMS-dev/gams.jl (2020) – reference: DunningIHuchetteJLubinMJuMP: a modeling language for mathematical optimizationSIAM Rev.2017592295320364649310.1137/15M10205751368.90002 – reference: GloverFImproved linear integer programming formulations of nonlinear integer problemsManage. Sci.197522445546040113810.1287/mnsc.22.4.4550318.90044 – reference: Kannan, R.: Algorithms, analysis and software for the global optimization of two-stage stochastic programs. PhD thesis, Massachusetts Institute of Technology (2018) – reference: Bacci, T., Frangioni, A., Gentile, C., Tavlaridis-Gyparakis, K.: New MINLP formulations for the unit commitment problems with ramping constraints. Preprint, http://www.optimization-online.org/DB_FILE/2019/10/7426.pdf (2019) – reference: Pedroso, J. P., Kubo, M., Viana, A.: Unit commitment with valve-point loading effect. Technical report, Universidade do Porto (2014) – reference: BendersJFPartitioning procedures for solving mixed-variables programming problemsNumer. Math.19624123825214730310.1007/BF013863160109.38302 – reference: OgbeELiXA joint decomposition method for global optimization of multiscenario nonconvex mixed-integer nonlinear programsJ. Global Optim.201875595629401996110.1007/s10898-019-00786-x1432.90092 – reference: RebennackSCombining sampling-based and scenario-based nested Benders decomposition methods: application to stochastic dual dynamic programmingMath. Program.20161561–2343389345920510.1007/s10107-015-0884-31342.90116 – reference: MisenerRFloudasCAAntigone: algorithms for continuous/integer global optimization of nonlinear equationsJ. Global Optim.2014592–3503526321669010.1007/s10898-014-0166-21301.90063 – reference: TawarmalaniMSahinidisNVA polyhedral branch-and-cut approach to global optimizationMath. Program.2005103225249214637910.1007/s10107-005-0581-81099.90047 – reference: GeoffrionAMGeneralized Benders decompositionJ. Optim. Theory Appl.197210423726032731010.1007/BF009348100229.90024 – reference: Meyer, R.R.: Integer and mixed-integer programming models: general properties. J. Optim. Theory Appl. 3(4) (1975) – reference: SteegerGLohmannTRebennackSStrategic bidding for a price-maker hydroelectric producer: stochastic dual dynamic programming and Lagrangian relaxationIISE Trans.201847114 – reference: van SlykeRMWetsRL-shaped linear programs with applications to optimal control and stochastic programmingJ. SIAM Appl. Math.196917463866325374110.1137/01170610197.45602 – reference: BezansonJEdelmanAKarpinskiSShahVBJulia: a fresh approach to numerical computingSIAM Rev.20175916598360582610.1137/1410006711356.68030 – reference: Hjelmeland, M.N., Zou, J., Helseth, A., Ahmed, S.: Nonconvex medium-term hydropower scheduling by stochastic dual dynamic integer programming. IEEE Trans. Sustain. Energy 10(1) (2019) – reference: Kallrath, J., Rebennack, S.: Computing area-tight piecewise linear overestimators, underestimators and tubes for univariate functions. In: Optimization in Science and Engineering, pp. 273–292. Springer (2014) – reference: LiCGrossmannIEA generalized Benders decomposition-based branch and cut algorithm for two-stage stochastic programs with nonconvex constraints and mixed-binary first and second stage variablesJ. Global Optim.2019752247272401243010.1007/s10898-019-00816-81428.90106 – reference: FeizollahiMJAhmedSSunAExact augmented Lagrangian duality for mixed integer linear programmingMath. Program.20171611–2365387359278210.1007/s10107-016-1012-81364.90226 – reference: LiXTomasgardABartonPINonconvex generalized Benders decomposition for stochastic separable mixed-integer nonlinear programsJ. Optim. Theory Appl.2011151425454285122410.1007/s10957-011-9888-11245.90079 – reference: Zou, J., Ahmed, S., Sun, X.: Multistage stochastic unit commitment using stochastic dual dynamic integer programming. IEEE Trans. Power Syst. 34(3) (2019) – reference: CerisolaSLatorreJMRamosAStochastic dual dynamic programming applied to nonconvex hydrothermal modelsEur. J. Oper. Res.20122183687697288174110.1016/j.ejor.2011.11.0401244.90173 – reference: RebennackSKallrathJContinuous piecewise linear delta-approximations for univariate functions: computing minimal breakpoint systemsJ. Optim. Theory Appl.20151672617643341245310.1007/s10957-014-0687-31327.90245 – reference: LiXChenYBartonPINonconvex generalized Benders decomposition with piecewise convex relaxations for global optimization of integrated process design and operation problemsInd. Eng. Chem. Res.201251217287729910.1021/ie201262f – reference: RebennackSComputing tight bounds via piecewise linear functions through the example of circle cutting problemsMath. Methods Oper. Res.2016841357353588210.1007/s00186-016-0546-01396.90051 – reference: Rebennack, S., Krasko, V.: Piecewise linear function fitting via mixed-integer linear programming. INFORMS J. Comput. (2020) – reference: VielmaJPAhmedSNemhauserGMixed-integer models for nonseparable piecewise-linear optimization: unifying framework and extensionsOper. Res.2010582303315267479810.1287/opre.1090.07211226.90046 – reference: SteegerGRebennackSDynamic convexification within nested Benders decomposition using Lagrangian relaxation: an application to the strategic bidding problemEur. J. Oper. Res.20172572669686356688610.1016/j.ejor.2016.08.0061394.90442 – reference: VielmaJPNemhauserGLModeling disjunctive constraints with a logarithmic number of binary variables and constraintsMath. Program.20111281–24972281095210.1007/s10107-009-0295-41218.90137 – reference: Füllner, C.: NCNBD.jl. Code released on GitHub https://github.com/ChrisFuelOR/NCNBD.jl (2021) – ident: 1740_CR12 doi: 10.1287/ijoc.2020.0987 – volume: 75 start-page: 241 issue: 2 year: 1996 ident: 1740_CR23 publication-title: Math. Program. doi: 10.1007/BF02592154 – ident: 1740_CR36 doi: 10.1007/s10107-019-01368-1 – volume: 103 start-page: 225 year: 2005 ident: 1740_CR48 publication-title: Math. Program. doi: 10.1007/s10107-005-0581-8 – volume: 51 start-page: 7287 issue: 21 year: 2012 ident: 1740_CR29 publication-title: Ind. Eng. Chem. Res. doi: 10.1021/ie201262f – ident: 1740_CR26 – volume: 33 start-page: 989 issue: 5 year: 1985 ident: 1740_CR8 publication-title: Oper. Res. doi: 10.1287/opre.33.5.989 – volume: 84 start-page: 3 issue: 1 year: 2016 ident: 1740_CR38 publication-title: Math. Methods Oper. Res. doi: 10.1007/s00186-016-0546-0 – volume: 52 start-page: 359 issue: 1–3 year: 1991 ident: 1740_CR35 publication-title: Math. Program. doi: 10.1007/BF01582895 – ident: 1740_CR17 – volume: 33 start-page: 540 issue: 3 year: 2018 ident: 1740_CR27 publication-title: Optim. Methods Softw. doi: 10.1080/10556788.2017.1350178 – ident: 1740_CR42 – volume: 24 start-page: 597 year: 2009 ident: 1740_CR5 publication-title: Optim. Methods Softw. doi: 10.1080/10556780903087124 – ident: 1740_CR22 doi: 10.1109/TSTE.2018.2805164 – volume: 4 start-page: 238 issue: 1 year: 1962 ident: 1740_CR6 publication-title: Numer. Math. doi: 10.1007/BF01386316 – ident: 1740_CR40 doi: 10.1287/ijoc.2019.0890 – ident: 1740_CR52 doi: 10.1007/s10107-022-01875-8 – volume: 59 start-page: 503 issue: 2–3 year: 2014 ident: 1740_CR32 publication-title: J. Global Optim. doi: 10.1007/s10898-014-0166-2 – volume-title: Variational Analysis year: 2009 ident: 1740_CR41 – volume: 218 start-page: 687 issue: 3 year: 2012 ident: 1740_CR10 publication-title: Eur. J. Oper. Res. doi: 10.1016/j.ejor.2011.11.040 – volume: 10 start-page: 237 issue: 4 year: 1972 ident: 1740_CR19 publication-title: J. Optim. Theory Appl. doi: 10.1007/BF00934810 – ident: 1740_CR31 doi: 10.1007/BF01262932 – ident: 1740_CR18 – ident: 1740_CR2 – ident: 1740_CR54 doi: 10.1109/TPWRS.2018.2880996 – ident: 1740_CR20 – ident: 1740_CR24 – volume: 167 start-page: 617 issue: 2 year: 2015 ident: 1740_CR39 publication-title: J. Optim. Theory Appl. doi: 10.1007/s10957-014-0687-3 – volume: 156 start-page: 343 issue: 1–2 year: 2016 ident: 1740_CR37 publication-title: Math. Program. doi: 10.1007/s10107-015-0884-3 – volume: 257 start-page: 669 issue: 2 year: 2017 ident: 1740_CR46 publication-title: Eur. J. Oper. Res. doi: 10.1016/j.ejor.2016.08.006 – volume: 59 start-page: 295 issue: 2 year: 2017 ident: 1740_CR13 publication-title: SIAM Rev. doi: 10.1137/15M1020575 – ident: 1740_CR34 – volume: 17 start-page: 638 issue: 4 year: 1969 ident: 1740_CR49 publication-title: J. SIAM Appl. Math. doi: 10.1137/0117061 – volume: 70 start-page: 497 issue: 3 year: 2018 ident: 1740_CR53 publication-title: J. Global Optim. doi: 10.1007/s10898-017-0559-0 – ident: 1740_CR11 doi: 10.1007/s10479-012-1102-9 – volume: 128 start-page: 49 issue: 1–2 year: 2011 ident: 1740_CR51 publication-title: Math. Program. doi: 10.1007/s10107-009-0295-4 – ident: 1740_CR25 doi: 10.1007/978-1-4939-0808-0_14 – ident: 1740_CR44 – volume: 22 start-page: 455 issue: 4 year: 1975 ident: 1740_CR21 publication-title: Manage. Sci. doi: 10.1287/mnsc.22.4.455 – ident: 1740_CR15 – volume: 161 start-page: 365 issue: 1–2 year: 2017 ident: 1740_CR14 publication-title: Math. Program. doi: 10.1007/s10107-016-1012-8 – volume: 41 start-page: 1069 issue: 11 year: 1990 ident: 1740_CR3 publication-title: J. Oper. Res. Soc. doi: 10.1057/jors.1990.166 – volume: 175 start-page: 461 issue: 1–2 year: 2019 ident: 1740_CR55 publication-title: Math. Program. doi: 10.1007/s10107-018-1249-5 – ident: 1740_CR1 doi: 10.1007/s10107-020-01569-z – volume: 151 start-page: 425 year: 2011 ident: 1740_CR30 publication-title: J. Optim. Theory Appl. doi: 10.1007/s10957-011-9888-1 – volume: 59 start-page: 65 issue: 1 year: 2017 ident: 1740_CR7 publication-title: SIAM Rev. doi: 10.1137/141000671 – volume: 75 start-page: 595 year: 2018 ident: 1740_CR33 publication-title: J. Global Optim. doi: 10.1007/s10898-019-00786-x – volume: 75 start-page: 247 issue: 2 year: 2019 ident: 1740_CR28 publication-title: J. Global Optim. doi: 10.1007/s10898-019-00816-8 – volume: 47 start-page: 1 year: 2018 ident: 1740_CR45 publication-title: IISE Trans. – ident: 1740_CR43 – volume: 93 start-page: 247 year: 2002 ident: 1740_CR47 publication-title: Math. Program. doi: 10.1007/s10107-002-0308-z – ident: 1740_CR16 – volume: 22 start-page: 1 year: 2013 ident: 1740_CR4 publication-title: Acta Numer. doi: 10.1017/S0962492913000032 – volume: 58 start-page: 303 issue: 2 year: 2010 ident: 1740_CR50 publication-title: Oper. Res. doi: 10.1287/opre.1090.0721 – volume: 35 start-page: 37 issue: 1 year: 2020 ident: 1740_CR9 publication-title: Optim. Methods Softw. doi: 10.1080/10556788.2018.1556661
SSID	ssj0001388
Score	2.4488986
Snippet	We propose a new decomposition method to solve multistage non-convex mixed-integer (stochastic) nonlinear programming problems (MINLPs). We call this algorithm...
SourceID	proquest gale crossref springer
SourceType	Aggregation Database Enrichment Source Index Database Publisher
StartPage	987
SubjectTerms	Algorithms Approximation Benders decomposition Calculus of Variations and Optimal Control; Optimization Combinatorics Full Length Paper Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Mathematics of Computing Mixed integer Nonlinear programming Numerical Analysis Optimization Regularization Theoretical Unit commitment
Title	Non-convex nested Benders decomposition
URI	https://link.springer.com/article/10.1007/s10107-021-01740-0 https://www.proquest.com/docview/2733387237
Volume	196
WOSCitedRecordID	wos000740163700002&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/eLvHCXMwnV1LSwMxEA5aPejBt1itsgfBgwZ2N9smOVaxeLGIL7yF3TxAkK10q_jznUmzbX2CHsMm2TCTZL7JvAg51JzrDHAFzViuaWZtQaUwmgqWaMz4JKV2vtgE7_fFw4O8CkFhVe3tXpsk_U09E-yW4LNaiuovR6fEebIA4k5gwYbrm_vJ_ZswIepCrYgOQqjM93N8EEefL-Uv1lEvdHqr_1vuGlkJIDPqjnfFOpmz5QZZnkk9CK3LSb7WapMc9Qcl9Q7ob1HpX0CjU19jroqMRa_z4Nq1Re5657dnFzSUUKAaFMMRla5woDOwnIMSbKzAdlzEnVykrACs4kwheZwWVlpXpB0B-Mk4rrU1CctcGrNt0igHpd0hEUCfzEkQZU7mmTNtmFDkHK04Rra1iJskqSmpdMgvjmUuntQ0MzKSRAFJlCeJgjHHkzHP4-wav_Y-QgYpPHows85DBAGsD5NYqS4H_IZmW9EkrZqHKpzJSgFQA32cp4w3yUnNs-nnn_-7-7fue2QpxRgJH7DYIo3R8MXuk0X9Onqshgd-r74D7U3e8A
linkProvider	Springer Nature
linkToHtml	http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV1LSwMxEB60CurBt1itugehB13YV5vkWMWi2C6iVXoLu3mAIFvpVvHnO0l32_oEPYZNsmHy-iYz8w3AsSBERIgr3ChMhBsplbqMSuHS0BeG8YkxoW2yCRLHtN9nN0VQWF56u5cmSXtSzwS7-eZZLTDqLzFOifOwEOGNZRjzb-8eJuevH1JaJmo16KAIlfm-jw_X0edD-Yt11F467bX_DXcdVguQ6bTGq2ID5lS2CSsz1INY6k74WvMtqMeDzLUO6G9OZl9AnTObYy53pDJe54Vr1zbcty9655dukULBFagYjlymU406Q5gQVIKloqbspV4zoUGYIlbRMmXEC1LFlE6DJkX8JDURQkk_jHTghTtQyQaZ2gUHoU-kGV5lmiWRlg3skCbEWHEkawjqVcEvJclFwS9u0lw88SkzshEJR5FwKxKObU4mbZ7H7Bq_1q6bCeJm62HPIikiCHB8hsSKtwjiN2O2pVWolXPIiz2ZcwRqqI-TICRVOC3nbPr55__u_a36ESxd9rod3rmKr_dhOTDxEjZ4sQaV0fBFHcCieB095sNDu27fAWkz4dQ
linkToPdf	http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV1ZSwMxEB68EH3wFqtV90Hogy7dS5M8ehVFLQUPfAu7OUCQbWmr-POdSXfbeoL4GDbJhsn1TWbmG4A9xZhKEFf4SZwqPzEm8wXXyudxqIjxSQhlXbIJ1mzyx0fRGovid97upUlyENNALE15v97Rtj4W-BbSE1tEqjAjB8VJmE7IkZ709duH4VkcxpyXSVsJKRRhM9_38eFq-nxAf7GUuguosfj_oS_BQgE-vePBalmGCZOvwPwYJSGWboY8rr1VqDXbue8c09-83L2Meicu91zP04a80QuXrzW4b5zfnV74RWoFX6HC2PeFzSzqEnHKUDnWhlM5yIKjlEdxhhjG6kywIMqMMDaLjjjiKm2ZUkaHcWKjIF6Hqbydmw3wEBIlVuAVZ0WaWH2IHfKUkXVHi0PFgwqEpVSlKnjHKf3FsxwxJpNIJIpEOpFIbLM_bNMZsG78WrtGkyVpS2LPKi0iC3B8RG4ljxniOjLn8gpUy_mUxV7tSQRwqKezKGYVOCjnb_T55_9u_q36Lsy2zhry-rJ5tQVzEYVRuJjGKkz1uy9mG2bUa_-p191xS_gdYo3quA
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=Non-convex+nested+Benders+decomposition&rft.jtitle=Mathematical+programming&rft.au=F%C3%BCllner%2C+Christian&rft.au=Rebennack%2C+Steffen&rft.date=2022-11-01&rft.pub=Springer&rft.issn=0025-5610&rft.volume=196&rft.issue=1-2&rft.spage=987&rft_id=info:doi/10.1007%2Fs10107-021-01740-0&rft.externalDocID=A725670658
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